}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&&\leavevmode\hbox
to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip
5.83301pt\lower-3.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{0}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&1&&&&\framebox{1}&\\\
0&&0&1&&&\framebox{1}&&0\\\ 0&&\framebox{1}&0&0&&&&\\\
1&\framebox{1}&0&0&&0&&&\\\ \framebox{1}&0&0&0&&&&&\\\ \leavevmode\hbox
to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip
5.83301pt\lower-3.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{0}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&&0&\leavevmode\hbox
to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip
5.83301pt\lower-3.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\framebox{1}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&&&0&&\\\
\end{matrix}\enspace\right]\leavevmode\hbox to12.47pt{\vbox
to13.91pt{\pgfpicture\makeatletter\hbox{\hskip 6.233pt\lower-3.733pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } { {}{}{}}{}{{}}{} {
{}{}{}}{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{-6.033pt}{9.97743pt}\pgfsys@moveto{-6.033pt}{9.97743pt}\pgfsys@lineto{-6.033pt}{-3.533pt}\pgfsys@lineto{6.033pt}{-3.533pt}\pgfsys@lineto{6.033pt}{9.97743pt}\pgfsys@closepath\pgfsys@moveto{6.033pt}{-3.533pt}\pgfsys@stroke\pgfsys@invoke{
} { {}{}{}}{}{{}}{} {
{}{}{}}{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{-6.033pt}{9.97743pt}\pgfsys@moveto{-6.033pt}{9.97743pt}\pgfsys@lineto{-6.033pt}{-3.533pt}\pgfsys@lineto{6.033pt}{-3.533pt}\pgfsys@lineto{6.033pt}{9.97743pt}\pgfsys@closepath\pgfsys@moveto{6.033pt}{-3.533pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

(d) The reduced matrix $\bar{M}_{f}$, with $\sigma_{f}=[543621]$.

$\left[\enspace\begin{matrix}\leavevmode\hbox to11.67pt{\vbox
to13.11pt{\pgfpicture\makeatletter\hbox{\hskip 5.83301pt\lower-3.33301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{0}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&&\leavevmode\hbox
to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip
5.83301pt\lower-3.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{0}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&0&&&&\framebox{1}&\\\
0&&0&\framebox{1}&&&&&0\\\ 0&&\framebox{1}&0&0&&&&\\\
1&\framebox{1}&0&0&&0&&&\\\ \framebox{1}&0&0&0&&&&&\\\ \leavevmode\hbox
to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip
5.83301pt\lower-3.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{0}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&&0&\leavevmode\hbox
to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip
5.83301pt\lower-3.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{0}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&&&\framebox{1}&&\\\
\end{matrix}\enspace\right]\leavevmode\hbox to12.47pt{\vbox
to13.91pt{\pgfpicture\makeatletter\hbox{\hskip 6.233pt\lower-3.733pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } { {}{}{}}{}{{}}{} {
{}{}{}}{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{-6.033pt}{9.97743pt}\pgfsys@moveto{-6.033pt}{9.97743pt}\pgfsys@lineto{-6.033pt}{-3.533pt}\pgfsys@lineto{6.033pt}{-3.533pt}\pgfsys@lineto{6.033pt}{9.97743pt}\pgfsys@closepath\pgfsys@moveto{6.033pt}{-3.533pt}\pgfsys@stroke\pgfsys@invoke{
} { {}{}{}}{}{{}}{} {
{}{}{}}{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{-6.033pt}{9.97743pt}\pgfsys@moveto{-6.033pt}{9.97743pt}\pgfsys@lineto{-6.033pt}{-3.533pt}\pgfsys@lineto{6.033pt}{-3.533pt}\pgfsys@lineto{6.033pt}{9.97743pt}\pgfsys@closepath\pgfsys@moveto{6.033pt}{-3.533pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

(e) The reduced matrix $\bar{M}_{g}$, with $\sigma_{g}=[543261]$.

$\left[\enspace\begin{matrix}\leavevmode\hbox to11.67pt{\vbox
to13.11pt{\pgfpicture\makeatletter\hbox{\hskip 5.83301pt\lower-3.33301pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{0}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&&\leavevmode\hbox
to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip
5.83301pt\lower-3.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{0}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&0&&&&\framebox{1}&\\\
0&&0&\framebox{1}&&&&&0\\\ 0&&\framebox{1}&0&0&&&&\\\
\framebox{1}&&0&0&&0&&&\\\ 0&\framebox{1}&0&0&&&&&\\\ \leavevmode\hbox
to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip
5.83301pt\lower-3.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{0}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&&0&\leavevmode\hbox
to11.67pt{\vbox to13.11pt{\pgfpicture\makeatletter\hbox{\hskip
5.83301pt\lower-3.33301pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{0}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&&&\framebox{1}&&\\\
\end{matrix}\enspace\right]\leavevmode\hbox to12.47pt{\vbox
to13.91pt{\pgfpicture\makeatletter\hbox{\hskip 6.233pt\lower-3.733pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } { {}{}{}}{}{{}}{} {
{}{}{}}{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{-6.033pt}{9.97743pt}\pgfsys@moveto{-6.033pt}{9.97743pt}\pgfsys@lineto{-6.033pt}{-3.533pt}\pgfsys@lineto{6.033pt}{-3.533pt}\pgfsys@lineto{6.033pt}{9.97743pt}\pgfsys@closepath\pgfsys@moveto{6.033pt}{-3.533pt}\pgfsys@stroke\pgfsys@invoke{
} { {}{}{}}{}{{}}{} {
{}{}{}}{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{-6.033pt}{9.97743pt}\pgfsys@moveto{-6.033pt}{9.97743pt}\pgfsys@lineto{-6.033pt}{-3.533pt}\pgfsys@lineto{6.033pt}{-3.533pt}\pgfsys@lineto{6.033pt}{9.97743pt}\pgfsys@closepath\pgfsys@moveto{6.033pt}{-3.533pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

(f) The reduced matrix $\bar{M}_{b}$, with $\sigma_{b}=[453261]$.

Figure 4: Example matrices of the induction step of Proposition 3.9. We find
that $\sigma_{f}=\sigma_{g}(4\,5)$, $\sigma_{g}=\sigma_{f}^{\prime}(4\,5\,6)$,
and $\sigma_{b}=\sigma_{b}^{\prime}(6\,5\,4)$.

We state below the rearrangement inequality following [Vin90]. Since the
statement we need is slightly different from those we found in the literature,
we include a short proof, which is a slight modification of the argument in
[Vin90] and can be found also in [Ste04, p. 82].

######  Theorem 3.11 (Rearrangement inequality).

Let $g_{1},g_{2},\ldots,g_{n}$ be real valued functions defined on an interval
$I\subseteq\mathbb{R}$ such that $g_{k+1}-g_{k}$ is a non-decreasing function,
for all $k\in\\{1,\ldots,n-1\\}$, and let $b_{1}\leq b_{2}\leq\ldots\leq
b_{n}$ be a sequence of elements of $I$. If $\rho\leq\sigma$ in $S_{n}$, then

$\sum_{k=1}^{n}g_{k}(b_{\rho(k)})\geq\sum_{k=1}^{n}g_{k}(b_{\sigma(k)}).$

###### Proof.

Since the argument we present can be iterated, it is enough to prove the
statement for $\sigma=\rho\tau$ where $\tau=(i\,j)$ is an inversion: $i<j$ and
$\rho(i)<\rho(j)$. We have

$\displaystyle\sum_{k=1}^{n}g_{k}(b_{\rho(k)})-\sum_{k=1}^{n}g_{k}(b_{\sigma(k)})$
$\displaystyle=g_{i}(b_{\rho(i)})+g_{j}(b_{\rho(j)})-g_{i}(b_{\sigma(i)})-g_{j}(b_{\sigma(j)})$
$\displaystyle=g_{i}(b_{\rho(i)})+g_{j}(b_{\rho(j)})-g_{i}(b_{\rho(j)})-g_{j}(b_{\rho(i)})$
$\displaystyle=\left(g_{j}(b_{\rho(j)})-g_{i}(b_{\rho(j)})\right)-\left(g_{j}(b_{\rho(i)})-g_{i}(b_{\rho(i)})\right)\geq
0,$

where the last inequality follows from $b_{\rho(i)}\leq b_{\rho(j)}$ and from
the fact that $g_{j}-g_{i}$ is non-decreasing. ∎

######  Corollary 3.12.

Let $a_{1}\leq a_{2}\leq\ldots\leq a_{n}$ and $b_{1}\leq b_{2}\leq\ldots\leq
b_{n}$ be sequences of real numbers, and let $p\in[1,\infty)$. If
$\rho\leq\sigma$ in $S_{n}$, then

$\sum_{k=1}^{n}|a_{k}-b_{\rho(k)}|^{p}\leq\sum_{k=1}^{n}|a_{k}-b_{\sigma(k)}|^{p}.$

###### Proof.

Let $h_{k}(x)=|a_{k}-x|^{p}$. It is easy to check that the function
$h_{k+1}-h_{k}$ is non-increasing for all $k\in\\{1,\ldots,n-1\\}$, so we can
apply Theorem 3.11 to the sequence of functions $g_{k}:=-h_{k}$. ∎

######  Theorem 3.13.

For any monomorphism $f\colon Z\hookrightarrow X$ it is possible to determine
(via Algorithm 1) a bar-to-bar monomorphism $f_{b}\colon Z\hookrightarrow X$
such that
$\|\operatorname{coker}f_{b}\|_{p}\leq\|\operatorname{coker}f\|_{p}$, for all
$p\in[1,\infty]$.

###### Proof.

First, assume $p\in[1,\infty)$. The persistence modules
$\operatorname{coker}f$ and $\operatorname{coker}f_{b}$ have barcode
decompositions as in Corollary 3.10. Then, the claim follows from Corollary
3.12 applied to the permutations $\sigma_{b}\leq\sigma_{f}$ (Proposition 3.9).
The claim for $p=\infty$ follows from taking the limit for $p\to\infty$ of
both sides of the inequality
$\left\|\operatorname{coker}b\right\|_{p}\leq\left\|\operatorname{coker}f\right\|_{p}$,
recalling that $\displaystyle\lim_{p\to\infty}\|u\|_{p}=\|u\|_{\infty}$ for
any vector $u\in\mathbb{R}^{n}$ (Section 2.6). ∎

###  3.3 Bar-to-bar epimorphisms

The results on monomorphisms have analogous versions for epimorphisms, where
we need to work in a dual setting. For instance, we (co)present epimorphisms
by diagrams of cofree copresentations (the support of a cofree module is an
open interval $(-\infty,a)$)

${\ker}$${0}$${Z}$${G_{Z}}$${R_{Z}}$${0}$${X}$${G_{X}}$${R_{X}}$

where we obtain the equality $\ker=\ker(G_{Z}\to R_{Z}\oplus G_{X})$. Then,
given an epimorphism $f$, we can algorithmically define a bar-to-bar matrix
$M_{b}$, which is associated to a bar-to-bar epimorphism whose kernel is
smaller than that of $f$.

In particular, during matrix reduction, instead of doing left-to-right column
transformations, we reduce by top-to-bottom row transformations. The barcode
decomposition of the kernel is then given by reading the rightmost nonzero
coefficients of each row, as opposed to the lowest nonzero coefficients of
each column: bars are “cogenerated” at the end point associated to the row,
and they “covanish” at the start point associated to the column.

In conclusion, we obtain the following analogous result:

######  Theorem 3.14.

For any epimorphism $f\colon Z\twoheadrightarrow X$ it is possible to
determine a bar-to-bar epimorphism $f_{b}\colon Z\twoheadrightarrow X$ such
that $\left\|\ker f_{b}\right\|_{p}\leq\left\|\ker f\right\|_{p}$, for all
$p\in[1,\infty]$.

##  4 Noise systems and Wasserstein pseudometrics

In this section we study algebraic Wasserstein pseudometrics between
persistence modules. After introducing in Section 4.1 a generalization of the
pseudometrics associated with a noise system, we study in Section 4.2 noise
systems determined by $p$-norms of persistence modules and regular contours.
Section 4.3 is devoted to the associated algebraic Wasserstein pseudometrics.
For some choices of parameters, these pseudometrics have a combinatorial
interpretation, as we show in Section 4.4. Finally, in Section 4.5 we present
formulas to compute the algebraic Wasserstein pseudometric between persistence
modules in some specific cases.

###  4.1 Pseudometrics associated to noise systems

Given a noise system $\mathcal{S}$ and $p\in[1,\infty]$, in this section we
will introduce pseudometrics $d^{p}_{\mathcal{S}}$ between persistence
modules. These pseudometrics are a simple generalization to $p>1$ of the
pseudometric associated to a noise system in [SCL+17] (see Section 2.4), where
$p=1$. Although the statements in this section hold true for tame functors
indexed by $[0,\infty)^{r}$ for every positive natural number $r$, as in
[SCL+17], we will limit the presentation to $r=1$, since this is the setting
of the following sections.

######  Definition 4.1.

Let $X$ and $Y$ be persistence modules. A span of $X,Y$ is a triplet $(Z,f,g)$
with $Z$ a persistence module and $f:Z\to X$ and $g:Z\to Y$ morphisms between
persistence modules. A span of $X,Y$ is therefore a diagram in
$\operatorname{\mathsf{Tame}}$ of the form

$X\xleftarrow{f}Z\xrightarrow{g}Y$

######  Definition 4.2.

Let $X$ and $Y$ be persistence modules, and let $\mathcal{S}$ be a noise
system. A span $X\xleftarrow{f}Z\xrightarrow{g}Y$ is called a
$(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4})$-span if

$\ker
f\in\mathcal{S}_{\varepsilon_{1}},\quad\operatorname{coker}f\in\mathcal{S}_{\varepsilon_{2}},\quad\ker
g\in\mathcal{S}_{\varepsilon_{3}}\quad\text{and}\quad\operatorname{coker}g\in\mathcal{S}_{\varepsilon_{4}.}$

######  Definition 4.3.

Let $X$ and $Y$ be persistence modules, and let $\mathcal{S}$ be a noise
system. For $p\in[1,\infty]$ and $\varepsilon\in[0,\infty)$, we say that $X$
and $Y$ are $\varepsilon$-close in $p$-norm $\left\|\cdot\right\|_{p}$ if
there exists a
$(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4})$-span
$X\xleftarrow{f}Z\xrightarrow{g}Y$ for some
$\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\in[0,\infty)$
such that
$\left\|(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4})\right\|_{p}\leq\varepsilon$.
We define

$d^{p}_{\mathcal{S}}(X,Y)\coloneqq\inf\left\\{\varepsilon\in[0,\infty)\mid\text{$X$
and $Y$ are $\varepsilon$-close in $p$-norm}\right\\},$

adopting the convention $\inf\varnothing=\infty$.

Our next aim is to prove that $d^{p}_{\mathcal{S}}$ is a pseudometric on
$\operatorname{\mathsf{Tame}}$. We start by generalizing Proposition $8.5$ in
[SCL+17] to our current framework. Even if the generalization is not
difficult, we include the proof to highlight how the properties of $p$-norms
on $\mathbb{R}^{4}$ are used. We note that a similar result can be obtained
for a larger family of subadditive functions on $\mathbb{R}^{4}$ which include
$p$-norms (see [GNOW21], Proposition 4.11).

######  Proposition 4.4.

Let $F,G,H$ be persistence modules. Assume that $F$ and $G$ are
$\varepsilon$-close in $p$-norm, and that $G$ and $H$ are $\tau$-close in
$p$-norm. Then $F$ and $H$ are $(\varepsilon+\tau)$-close in $p$-norm.

###### Proof.

By assumption there exists a
$(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4})$-span
$F\xleftarrow{f^{\prime}}X\xrightarrow{f^{\prime\prime}}G$ with
$\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}\in[0,\infty)$
such that
$\left\|(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4})\right\|_{p}\leq\varepsilon$
and a $(\tau_{1},\tau_{2},\tau_{3},\tau_{4})$-span
$G\xleftarrow{g^{\prime}}Y\xrightarrow{g^{\prime\prime}}H$ with
$\tau_{1},\tau_{2},\tau_{3},\tau_{4}\in[0,\infty)$ such that
$\left\|(\tau_{1},\tau_{2},\tau_{3},\tau_{4})\right\|_{p}\leq\tau$. Consider
the following diagram, where the square is a pullback:

${Z}$${X}$${Y}$${F}$${G}$${H}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f^{\prime\prime}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{g^{\prime\prime}}$

By [SCL+17, Proposition 8.1], $\ker f\in\mathcal{S}_{\tau_{1}}$ and
$\operatorname{coker}f\in\mathcal{S}_{\tau_{2}}$, hence by [SCL+17,
Proposition 8.2] $\ker f^{\prime}f\in\mathcal{S}_{\varepsilon_{1}+\tau_{1}}$
and
$\operatorname{coker}f^{\prime}f\in\mathcal{S}_{\varepsilon_{2}+\tau_{2}}$. By
a similar argument, $\ker
g^{\prime\prime}g\in\mathcal{S}_{\varepsilon_{3}+\tau_{3}}$ and
$\operatorname{coker}g^{\prime\prime}g\in\mathcal{S}_{\varepsilon_{4}+\tau_{4}}$.
This proves that $F$ and $H$ are $\eta$-close in $p$-norm, where
$\eta\coloneqq\left\|(\varepsilon_{1}+\tau_{1},\varepsilon_{2}+\tau_{2},\varepsilon_{3}+\tau_{3},\varepsilon_{4}+\tau_{4})\right\|_{p}$.
Our claim follows from the inequality

$\left\|(\varepsilon_{1}+\tau_{1},\varepsilon_{2}+\tau_{2},\varepsilon_{3}+\tau_{3},\varepsilon_{4}+\tau_{4})\right\|_{p}\leq\left\|(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4})\right\|_{p}+\left\|(\tau_{1},\tau_{2},\tau_{3},\tau_{4})\right\|_{p}\leq\varepsilon+\tau$

which expresses the subadditivity of $\left\|\cdot\right\|_{p}$ and the
hypotheses. ∎

We are now ready to prove that $d^{p}_{\mathcal{S}}$ is a pseudometric on
$\operatorname{\mathsf{Tame}}$.

######  Proposition 4.5.

Given $p\in[1,\infty]$ and a noise system $\mathcal{S}$, the function
$d^{p}_{\mathcal{S}}$ in Definition 4.3 is a pseudometric on
$\operatorname{\mathsf{Tame}}$ (see Section 2.4).

###### Proof.

If $g:X\to Y$ is an isomorphism of persistence modules, the span
$X\xleftarrow{\operatorname{id}}X\xrightarrow{g}Y$ shows that
$d^{p}_{\mathcal{S}}(X,Y)=0$. For all persistence modules $X$ and $Y$, the
bijection between spans $X\xleftarrow{f}Z\xrightarrow{g}Y$ between $X$ and $Y$
and spans $Y\xleftarrow{g}Z\xrightarrow{f}X$ between $Y$ and $X$ implies that
$d^{p}_{\mathcal{S}}(X,Y)=d^{p}_{\mathcal{S}}(Y,X)$. Proposition 4.4 shows
that the triangle inequality holds true. ∎

######  Remark 4.6.

Given a noise system $\mathcal{S}$, the pseudometrics $d^{p}_{\mathcal{S}}$
for all $p\in[1,\infty]$ are strongly equivalent. Assuming $p\leq q$, for any
pair of persistence modules $X,Y$ we have

$d^{q}_{\mathcal{S}}(X,Y)\leq d^{p}_{\mathcal{S}}(X,Y)\leq
4^{\left(\frac{1}{p}-\frac{1}{q}\right)}d^{q}_{\mathcal{S}}(X,Y),$

as can be easily concluded from the properties on $p$-norms on
$\mathbb{R}^{4}$ stated in Section 2.1.

###  4.2 $p$-norms of persistence modules and contours

The aim or this section is to introduce and study a generalization of the
notion of $p$-norm of a persistence module (see Section 2.6) first introduced
in [ST20], that coincides with the original definition if $C$ is the standard
contour (see Section 2.2).

######  Definition 4.7.

Let $C$ be a regular contour. For $p\in[1,\infty]$, define the $(p,C)$-norm of
a persistence module $X\cong\bigoplus_{i=1}^{k}K(a_{i},b_{i})$ by

$\left\|X\right\|_{p,C}\coloneqq\begin{cases}\left(\sum_{i=1}^{k}\ell(a_{i},b_{i})^{p}\right)^{\frac{1}{p}}&\text{for
$p\in[1,\infty)$},\\\ \max\\{\ell(a_{i},b_{i})\\}_{i=1}^{k}&\text{for
$p=\infty$},\end{cases}$

where $\ell(a_{i},b_{i})$ denotes the lifetime of the bar $K(a_{i},b_{i})$
with respect to $C$ (see Section 2.2).

We see that $\left\|X\right\|_{p,C}$ does not depend on the choice of barcode
decomposition for $X$. For $p\in[1,\infty]$ and $\varepsilon\in[0,\infty)$,
consider the class of tame persistence modules

$\mathcal{S}^{p,C}_{\varepsilon}\coloneqq\\{X\in\operatorname{\mathsf{Tame}}\mid\left\|X\right\|_{p,C}\leq\varepsilon\\},$

and denote
$\mathcal{S}^{p,C}\coloneqq\\{\mathcal{S}^{p,C}_{\varepsilon}\\}_{\varepsilon\in[0,\infty)}$.
If $D$ is the standard contour (see Section 2.2), then
$\ell(a_{i},b_{i})=b_{i}-a_{i}$ and we have
$\left\|X\right\|_{p,D}=\left\|X\right\|_{p}$ and
$\mathcal{S}^{p,D}=\mathcal{S}^{p}$. The main result in this subsection is
showing that $\mathcal{S}^{p,C}$ is a noise system (see Section 2.3) whenever
$C$ is an action, for any $p\in[1,\infty]$. For the standard contour, this
result together with Proposition 4.5 provide an algebraic proof that the
algebraic $p$-Wasserstein distance introduced in [ST20] is a pseudometric, as
will be later highlighted in Remark 4.20.

Given a contour $C$, the function $C(0,-):[0,\infty)\to[0,\infty)$ is
nondecreasing. Hence it can be viewed as a functor from the poset category
$[0,\infty)$ to itself. For any persistence module $X$, the composition of
functors $T_{C}(X):=XC(0,-):[0,\infty)\to\operatorname{vect}_{K}$ is a
persistence module. As we will show, $T_{C}(X)$ is in
$\operatorname{\mathsf{Tame}}$ whenever $X$ is in
$\operatorname{\mathsf{Tame}}$ and $C$ is a regular contour (Corollary 4.11).
The assignment $X\mapsto T_{C}(X)$ can be extended to a functor
$T_{C}:\operatorname{\mathsf{Tame}}\to\operatorname{\mathsf{Tame}}$ sending a
morphism $f:X\to Y$ of persistence modules to the morphism
$T_{C}(f):T_{C}(X)\to T_{C}(Y)$ defined as the natural transformation between
$T_{C}(X)$ and $T_{C}(Y)$ whose component at $a\in[0,\infty)$ is
$T_{C}(f)_{a}=f_{C(0,a)}:X_{C(0,a)}\to Y_{C(0,a)}$.

Since direct sums in $\operatorname{\mathsf{Tame}}$ are defined pointwise
(Section 2.1), if $\\{X_{i}\\}_{i\in I}$ is a finite collection of persistence
modules and $C$ is a contour, then $T_{C}(\bigoplus_{i\in
I}X_{i})\cong\bigoplus_{i\in I}T_{C}(X_{i})$. Similarly, we show that the
functor $T_{C}$ preserves kernels and cokernels.

######  Proposition 4.8.

Let $0\to X\to Y\to Z\to 0$ be an exact sequence in
$\operatorname{\mathsf{Tame}}$, and let $C$ be a regular contour. Then the
sequence $0\to T_{C}(X)\to T_{C}(Y)\to T_{C}(Z)\to 0$ is also exact.

###### Proof.

Exactness in $\operatorname{\mathsf{Tame}}$ is defined pointwise: $0\to X\to
Y\to Z\to 0$ is exact if and only if $0\to X_{a}\to Y_{a}\to Z_{a}\to 0$ is
exact in $\operatorname{vect}_{K}$, for every $a\in[0,\infty)$. As a
consequence, $0\to X_{C(0,b)}\to Y_{C(0,b)}\to Z_{C(0,b)}\to 0$ is exact in
$\operatorname{vect}_{K}$, for every $b\in[0,\infty)$, hence by definition the
sequence $0\to T_{C}(X)\to T_{C}(Y)\to T_{C}(Z)\to 0$ is exact. ∎

######  Remark 4.9.

As is clear from its proof, Proposition 4.8 holds for the precomposition of
persistence modules by any increasing bijection of $[0,\infty)$ other than
$C(0,-)$ for a regular contour $C$.

We now explain the relationship between the barcode decompositions of $X$ and
$T_{C}(X)$ when $C$ is a regular contour.

######  Proposition 4.10.

Let $C$ be a regular contour, and let $\ell$ be the associated lifetime
function. Consider a bar $K(a,b)$. Then

$T_{C}(K(a,b))\cong K(\ell(0,a),\ell(0,b)).$

###### Proof.

The functor $T_{C}(K(a,b)):[0,\infty)\to\operatorname{vect}_{K}$ sends $c\leq
d$ in $[0,\infty)$ to the linear function

$K(a,b)_{C(0,c)\leq C(0,d)}:K(a,b)_{C(0,c)}\to K(a,b)_{C(0,d)},$

which is the identity on $K$ if $a\leq C(0,c)\leq C(0,d)<b$ and the zero
function otherwise. Since $C$ is regular, $\ell(0,-)$ is a strictly increasing
function, hence the condition $a\leq C(0,c)\leq C(0,d)<b$ is equivalent to
$\ell(0,a)\leq c\leq d<\ell(0,b)$. ∎

######  Corollary 4.11.

Let $X$ be a persistence module with barcode decomposition
$\bigoplus_{i=1}^{k}K(a_{i},b_{i})$, and let $C$ be a regular contour. Then
$T_{C}(X)\cong\bigoplus_{i=1}^{k}K(\ell(0,a_{i}),\ell(0,b_{i}))$.

###### Proof.

We have

$\displaystyle T_{C}(X)$ $\displaystyle\cong
T_{C}(\bigoplus_{i=1}^{k}K(a_{i},b_{i}))$
$\displaystyle\cong\bigoplus_{i=1}^{k}T_{C}(K(a_{i},b_{i}))$
$\displaystyle=\bigoplus_{i=1}^{k}K(\ell(0,a_{i}),\ell(0,b_{i})).$ (by
Proposition 4.10)

∎

In the rest of the article, we will focus on contours that are regular and
actions (see Section 2.2). We prove here a simple but important property of
such contours, and the associated lifetime function $\ell$, which is used to
prove the subsequent results.

######  Lemma 4.12.

If $C$ is a regular contour and an action, then
$\ell(a,c)=\ell(a,b)+\ell(b,c)$ for any $a\leq b\leq c$ in $[0,\infty)$.

###### Proof.

Let $a\leq b\leq c$. Using the definitions and the assumption that $C$ is an
action, we have $C(C(a,\ell(a,b)),\ell(b,c))=C(a,\ell(a,b)+\ell(b,c))$. Again
by definition, we observe that the left-hand side equals $c$, and that
$c=C(a,\ell(a,b)+\ell(b,c))$ implies $\ell(a,c)=\ell(a,b)+\ell(b,c)$. ∎

######  Proposition 4.13.

Let $X$ be a persistence module, let $p\in[1,\infty]$, and let $C$ be a
regular contour that is an action. Then
$\left\|X\right\|_{p,C}=\left\|T_{C}(X)\right\|_{p}$.

###### Proof.

Let $X\cong\bigoplus_{i=1}^{k}K(a_{i},b_{i})$. For any fixed $p\in[1,\infty)$,
we have

$\displaystyle\left\|T_{C}(X)\right\|_{p}$
$\displaystyle=\left(\sum_{i=1}^{k}(\ell(0,b_{i})-\ell(0,a_{i}))^{p}\right)^{\frac{1}{p}}$
$\displaystyle=\left(\sum_{i=1}^{k}\ell(a_{i},b_{i})^{p}\right)^{\frac{1}{p}}$
$\displaystyle=\left\|X\right\|_{p,C},$

where the first equality is by Corollary 4.11, the second one is by Lemma
4.12, and the third one is by definition of $\left\|\cdot\right\|_{p,C}$. The
case $p=\infty$ is similar. ∎

We are now ready to prove that $\mathcal{S}^{p,C}$, with $C$ a regular contour
that is an action, satisfies the axioms in the definition of noise system (see
Section 2.3).

######  Lemma 4.14.

Let $0\to X\to Y\to Z\to 0$ be an exact sequence in
$\operatorname{\mathsf{Tame}}$, and let $C$ be a regular contour. Then
$\left\|X\right\|_{p,C}\leq\left\|Y\right\|_{p,C}$ and
$\left\|Z\right\|_{p,C}\leq\left\|Y\right\|_{p,C}$.

For the standard contour, our statement coincides with Lemma 8.8 in [ST20],
which is proven using the induced matchings [BL15] for monomorphisms and
epimorphisms of persistence modules. Although structurally similar, our proof
leverages bar-to-bar morphisms and our results of Section 3.

###### Proof.

By Theorem 3.13, the existence of a monomorphism from $X$ to $Y$ implies the
existence of a bar-to-bar monomorphism $f:X\hookrightarrow Y$. By Remark 3.2,
there are barcode decompositions $X\cong\bigoplus_{i=1}^{m}X_{i}$ and
$Y\cong\bigoplus_{j=1}^{n}Y_{j}$ with $m\leq n$ such that $f=\bigoplus_{i\in
I}f_{i}\oplus\bigoplus_{j=m+1}^{n}h_{j}$, where each $f_{i}$ is a
monomorphisms $X_{i}\hookrightarrow Y_{i}$ between bars and each $h_{j}$ is
the zero morphism $0\to Y_{j}$. Now recall (see Section 2.1) that a
monomorphism between bars $K(a,b)$ and $K(a^{\prime},b^{\prime})$ exists if
and only if $a^{\prime}\leq a$ and $b^{\prime}=b$, and observe that this
implies $\ell(a,b)\leq\ell(a^{\prime},b^{\prime})$ by monotonicity of
contours. Considering the definition of $\left\|\cdot\right\|_{p,C}$, we see
that every term in the expression for $\left\|X\right\|_{p,C}$ is upper
bounded by a term in the expression for $\left\|Y\right\|_{p,C}$, implying
that $\left\|X\right\|_{p,C}\leq\left\|Y\right\|_{p,C}$.

The proof of the inequality $\left\|Z\right\|_{p,C}\leq\left\|Y\right\|_{p,C}$
is obtained similarly, using Theorem 3.14 and dual arguments on epimorphisms.
∎

######  Lemma 4.15.

Let $0\to X\to Y\to Z\to 0$ be an exact sequence in
$\operatorname{\mathsf{Tame}}$, and let $C$ be a regular contour that is an
action. Then
$\left\|Y\right\|_{p,C}\leq\left\|X\right\|_{p,C}+\left\|Z\right\|_{p,C}$.

###### Proof.

First, we prove the statement assuming that $C$ is the standard contour. Let
$0\to X\xrightarrow{f}Y\xrightarrow{g}Z\to 0$ be a short exact sequence of
persistence modules, and let us show that
$\left\|Y\right\|_{p}\leq\left\|X\right\|_{p}+\left\|Z\right\|_{p}$. We
consider the monomorphism $f$ and observe that $Z\cong\operatorname{coker}f$
implies that $Z$ and $\operatorname{coker}f$ have the same barcode
decomposition, hence
$\left\|Z\right\|_{p}=\left\|\operatorname{coker}f\right\|_{p}$. Theorem 3.13
tells us that, among all monomorphisms between two fixed persistence modules,
the norm $\left\|\cdot\right\|_{p}$ of the cokernel is minimized by a bar-to-
bar monomorphism. We therefore just need to prove that
$\left\|Y\right\|_{p}\leq\left\|X\right\|_{p}+\left\|\operatorname{coker}f\right\|_{p}$,
for any bar-to-bar monomorphism $f$ between $X$ and $Y$.

Let $\bigoplus_{i=1}^{m}X_{i}$ and $\bigoplus_{j=1}^{n}Y_{j}$ be barcode
decompositions of $X$ and $Y$, respectively. By Remark 3.2, if $f:X\to Y$ is a
bar-to-bar monomorphisms, then $m\leq n$ and, up to permutation of the
$Y_{j}$, there are monomorphisms $f_{i}:X_{i}\to Y_{i}$ between bars such that
$\operatorname{coker}f=\bigoplus_{i=1}^{m}\operatorname{coker}f_{i}\oplus\bigoplus_{j=m+1}^{n}Y_{j}$.
We observe that, for each bar $Y_{i}=K(a_{i},b_{i})$ of $Y$ with
$i\in\\{1,\dots,m\\}$, there is a bar $X_{i}=K(a^{\prime}_{i},b_{i})$ of $X$
and a corresponding summand $\operatorname{coker}f_{i}$ of
$\operatorname{coker}f$, which is a bar $K(a_{i},a^{\prime}_{i})$ if
$a_{i}<a^{\prime}_{i}$, and it is the zero module if $a_{i}=a^{\prime}_{i}$.
Similarly, we observe that each bar $Y_{j}=K(a_{j},b_{j})$ of $Y$ with
$j\in\\{m+1,\ldots,n\\}$ is also a bar of $\operatorname{coker}f$. By
definition, $\left\|Y\right\|_{p}$ is the $p$-norm of the following element of
$\mathbb{R}^{n}$:

$(b_{j}-a_{j})_{j\in\\{1,\ldots,n\\}}=(((b_{i}-a^{\prime}_{i})+(a^{\prime}_{i}-a_{i}))_{i\in\\{1,\ldots,m\\}},(b_{j}-a_{j})_{j\in\\{m+1,\ldots,n\\}}).$

Then, by the triangular inequality of $p$-norms in $\mathbb{R}^{n}$, we have
$\left\|Y\right\|_{p}\leq\left\|X\right\|_{p}+\left\|\operatorname{coker}f\right\|_{p}$,
which completes the proof when $C$ is the standard contour.

Let now $C$ be any contour that is regular and an action. By Proposition 4.8,
exactness of $0\to X\to Y\to Z\to 0$ implies exactness of $0\to T_{C}(X)\to
T_{C}(Y)\to T_{C}(Z)\to 0$. Applying the previous part of the proof yields
$\left\|T_{C}(Y)\right\|_{p}\leq\left\|T_{C}(X)\right\|_{p}+\left\|T_{C}(Z)\right\|_{p}$,
which by Proposition 4.13 coincides with our claim. ∎

For the standard contour, the statement of Lemma 4.15 is given in Remark 7.32
of [ST20]. However, to our knowledge, we provide the first algebraic proof of
this inequality, which is the most difficult to show among the axioms of a
noise system.

We can now prove the main result of this subsection.

######  Theorem 4.16.

For any $p\in[1,\infty]$ and any regular contour $C$ that is an action,
$\mathcal{S}^{p,C}$ is a noise system.

###### Proof.

We show that $\mathcal{S}^{p,C}$ satisfies all axioms of the definition of
noise system (see Section 2.3). Since the norm $\left\|\cdot\right\|_{p,C}$ of
the zero module $0$ is zero, we have $0\in\mathcal{S}_{\varepsilon}^{p,C}$,
for all $\varepsilon\in[0,\infty)$. By definition of $\mathcal{S}^{p,C}$, it
is clear that
$\mathcal{S}_{\tau}^{p,C}\subseteq\mathcal{S}_{\varepsilon}^{p,C}$ whenever
$\tau\leq\varepsilon$. Lemma 4.14 and Lemma 4.15 complete the proof, showing
that $\mathcal{S}^{p,C}$ satisfies both conditions on short exact sequences of
persistence modules. ∎

######  Remark 4.17.

For $p<\infty$, the noise system $\mathcal{S}^{p,C}$ in not closed under
direct sums (Section 2.3), since $\left\|X\oplus
Y\right\|_{p,C}=\left\|\left(\|X\|_{p,C},\|Y\|_{p,C}\right)\right\|_{p}$ by
equation (2.2).

######  Remark 4.18.

Let us briefly highlight the role of our hypotheses on contours, which are
required to be regular and actions in Theorem 4.16. The regularity assumption
ensures for instance that the associated lifetime function $\ell$ is well-
defined, and that the functor $T_{C}$ is an endofunctor on
$\operatorname{\mathsf{Tame}}$. The weaker assumption that
$C(0,-):[0,\infty)\to[0,\infty)$ is an increasing bijection is sufficient to
prove many results of this subsection, but we choose to assume the stronger
condition of regularity to facilitate a comparison with the results of [CR20],
observing in addition that many examples of regular contours can be found, for
example the contours of distance type (Section 2.2) that are used in our
experiments (see Section 5). The hypothesis that the considered contours are
actions is necessary to obtain the main results of this subsection. In
particular, it is important in Lemma 4.12, since otherwise we can only
conclude $\ell(a,c)\leq\ell(a,b)+\ell(b,c)$ for any $a\leq b\leq c$. As is
easy to show, Lemma 4.15 (and consequently Theorem 4.16) are not true if we
remove the action hypothesis on contours. Consider for example the contour
$C(a,\varepsilon):=a+\varepsilon^{2}$, which is regular but not an action
[CR20]. Let $r$ be a positive real number, and consider an exact sequence of
bars $0\to K(0,r^{2})\to K(0,2r^{2})\to K(r^{2},2r^{2})\to 0$. Assuming the
claim of Lemma 4.15 holds for the contour $C$ would imply
$2^{\frac{1}{2}}r\leq 2^{\frac{1}{p}}r$ for any $p$, but the inequality holds
if, and only if, $p\leq 2$.

###  4.3 Contours and algebraic Wasserstein distances

We now turn to considering the pseudometrics $d^{q}_{\mathcal{S}^{p,C}}$
associated (as in Section 4.1) with the noise systems $\mathcal{S}^{p,C}$
introduced in Section 4.2, for fixed $p,q\in[1,\infty]$ and a regular contour
$C$ that is an action. We also refer to these pseudometrics as algebraic
Wasserstein distances. First, we show that the functor $T_{C}$ introduced in
Section 4.2 allows us to switch between a pseudometric
$d^{q}_{\mathcal{S}^{p,C}}$ and the pseudometric $d^{q}_{\mathcal{S}^{p}}$
associated with the standard contour. More precisely, we show that $T_{C}$ can
be viewed as an isometry

$T_{C}:(\operatorname{\mathsf{Tame}},d^{q}_{\mathcal{S}^{p,C}})\to(\operatorname{\mathsf{Tame}},d^{q}_{\mathcal{S}^{p}}).$

Let us recall that, if $C$ is a regular contour, the function
$C(0,-):[0,\infty)\to[0,\infty)$ is an increasing bijection. Its inverse
$\ell(0,-):=C^{-1}(0,-)$ is therefore an increasing bijection as well.
Mimicking the definition of $T_{C}$ given in Section 4.2, we can define a
functor $T_{\ell}:\operatorname{\mathsf{Tame}}\to\operatorname{\mathsf{Tame}}$
given by precomposition by the increasing function $\ell(0,-)$. By Proposition
4.8, the functor
$T_{C}:\operatorname{\mathsf{Tame}}\to\operatorname{\mathsf{Tame}}$ preserves
kernels and cokernels, and $T_{\ell}$ has the same property by Remark 4.9.
Furthermore, since $C(0,-)$ and $\ell(0,-)$ are inverse to each other, the
compositions $T_{C}T_{\ell}$ and $T_{\ell}T_{C}$ are the identity functor
$1_{\operatorname{\mathsf{Tame}}}$ on $\operatorname{\mathsf{Tame}}$.

To prove the following result, it is convenient to define the $(p,q,C)$-cost
of a span $X\xleftarrow{f}Z\xrightarrow{g}Y$ of persistence modules as the
element $c\in[0,\infty]$ defined by

$c:=\left\|\left(\left\|\ker
f\right\|_{p,C},\left\|\operatorname{coker}f\right\|_{p,C},\left\|\ker
g\right\|_{p,C},\left\|\operatorname{coker}g\right\|_{p,C}\right)\right\|_{q}.$

######  Proposition 4.19.

Let $C$ be a regular contour that is an action, and let $X,Y$ be persistence
modules. Then

$d^{q}_{\mathcal{S}^{p,C}}(X,Y)=d^{q}_{\mathcal{S}^{p}}(T_{C}(X),T_{C}(Y)).$

###### Proof.

Let $D$ denote the standard contour, and let us recall that the $(p,D)$-norm
of a persistence module coincides with its $p$-norm (Section 4.2). We describe
a correspondence between spans having the same cost, calculated with respect
to $(p,q,C)$ and $(p,q,D)$ respectively.

Let $X\xleftarrow{f}Z\xrightarrow{g}Y$ be a span and let $c$ be its
$(p,q,C)$-cost. Applying the functor $T_{C}$, we obtain the span
$T_{C}(X)\xleftarrow{T_{C}(f)}T_{C}(Z)\xrightarrow{T_{C}(g)}T_{C}(Y)$, whose
$(p,q,D)$-cost is

$\displaystyle c^{\prime}$ $\displaystyle=\left\|\left(\left\|\ker
T_{C}(f)\right\|_{p},\left\|\operatorname{coker}T_{C}(f)\right\|_{p},\left\|\ker
T_{C}(g)\right\|_{p},\left\|\operatorname{coker}T_{C}(g)\right\|_{p}\right)\right\|_{q}$
$\displaystyle=\left\|\left(\left\|T_{C}(\ker
f)\right\|_{p},\left\|T_{C}(\operatorname{coker}f)\right\|_{p},\left\|T_{C}(\ker
g)\right\|_{p},\left\|T_{C}(\operatorname{coker}g)\right\|_{p}\right)\right\|_{q}$
$\displaystyle=c,$

where the second equality holds because the functor $T_{C}$ preserves kernels
and cokernels, and the last equality holds by Proposition 4.13.

To prove the other direction of the correspondence, we start from a span
$T_{C}(X)\xleftarrow{\varphi}T_{C}(Z)\xrightarrow{\psi}T_{C}(Y)$ whose
$(p,q,D)$-cost is

$k:=\left\|\left(\left\|\ker\varphi\right\|_{p},\left\|\operatorname{coker}\varphi\right\|_{p},\left\|\ker\psi\right\|_{p},\left\|\operatorname{coker}\psi\right\|_{p}\right)\right\|_{q},$

and we exhibit a span between $X$ and $Y$ whose $(p,q,C)$-cost equals $k$.
Applying the functor $T_{\ell}$, we obtain the span
$X\xleftarrow{T_{\ell}(\varphi)}Z\xrightarrow{T_{\ell}(\psi)}Y$. To determine
the $(p,q,C)$-cost of this span we observe that

$\left\|\ker
T_{\ell}(\varphi)\right\|_{p,C}=\left\|T_{\ell}(\ker\varphi)\right\|_{p,C}=\left\|T_{C}T_{\ell}(\ker\varphi)\right\|_{p}=\left\|\ker\varphi\right\|_{p},$

where the first equality holds because $T_{\ell}$ preserves kernels, the
second equality is by Proposition 4.13, and the third equality holds because
$T_{C}T_{\ell}=1_{\operatorname{\mathsf{Tame}}}$. Since similar equalities
hold for $\operatorname{coker}T_{\ell}(\varphi)$, $\ker T_{\ell}(\psi)$ and
$\operatorname{coker}T_{\ell}(\psi)$, the $(p,q,C)$-cost of the span
$X\xleftarrow{T_{\ell}(\varphi)}Z\xrightarrow{T_{\ell}(\psi)}Y$ equals $k$. ∎

######  Remark 4.20.

Some of the pseudometrics between persistence modules that have been studied
by other authors fall within the framework we have presented in this
subsection and in Section 4.1. If $C$ is a regular contour, the pseudometric
denoted by $d_{C}$ in [CR20, Sect. 6] coincide with our pseudometrics of the
type $d^{1}_{\mathcal{S}^{\infty,C}}$. In particular, for the standard contour
(Section 2.2) the pseudometric $d^{1}_{\mathcal{S}^{\infty}}$ coincides with
the standard pseudometric already introduced in [SCL+17]. As we already
mentioned, the algebraic pseudometrics introduced in [ST20, Sect. 7] are of
the form $d^{p}_{\mathcal{S}^{p}}$, thus coinciding with our pseudometrics
with the choice $p=q$ and for the standard contour. In [GNOW21], the authors
propose a framework to study distances on abelian categories which is
equivalent to noise systems on abelian categories. The authors of [BSS22] also
study distances on abelian categories, introducing the notion of exact weight,
which is more general than noise systems as the first axiom on short exact
sequences is relaxed. The so-called path metric associated with an exact
weight is defined for zigzags of morphisms of arbitrary finite length, but for
the particular case of path metrics on noise systems considering spans is
sufficient. In this case, the path metric coincides with a pseudometric of the
form $d^{1}_{\mathcal{S}}$. In particular, the path metric $d_{\mu\circ\dim}$
between persistence modules studied in [BSS22, Sect. 5] coincides with
$d^{1}_{\mathcal{S}^{1}}$ in our notations, while the $p$-Wasserstein
distances introduced by the authors are different from our pseudometrics
$d^{q}_{\mathcal{S}^{p,C}}$.

###  4.4 Algebraic and combinatorial $(p,C)$-Wasserstein distances

In this subsection we consider Wasserstein distances between persistence
diagrams. Here, we call these pseudometrics _combinatorial_ Wasserstein
distances, to distinguish them from the _algebraic_ pseudometrics
$d^{q}_{\mathcal{S}^{p,C}}$ defined on the class of persistence modules. We
introduce a new family of combinatorial Wasserstein distances, parametrized by
$p,q\in[1,\infty]$ and a regular contour $C$ that is an action, which
generalize the Wasserstein distances commonly used in persistence theory.
Finally, we prove isometry results involving the combinatorial Wasserstein
distances and the algebraic Wasserstein distances $d^{q}_{\mathcal{S}^{p,C}}$
introduced in Section 4.2.

Let $U:=\\{(a,b)\in[0,\infty)\times[0,\infty]\mid a<b\\}$ be a subset of the
extended plane. A persistence diagram is a finite multiset
$D=\\{x_{i}\\}_{i\in S}$ of elements of $U$. Since $D$ is a multiset, it may
happen that $x_{i}=x_{k}$ for some $i\neq k$. The diagonal $\Delta$ of
$[0,\infty)$ is the set $\Delta:=\\{(a,a)\mid a\in[0,\infty)\\}\subset U$. For
all $p\in[1,\infty]$, we denote by $d_{p}$ the metric on $U$ induced by the
$p$-norm, defined by $d_{p}(x,y):=\left\|x-y\right\|_{p}$ for all $x,y\in U$,
and we denote $d_{p}(x,\Delta):=\inf_{z\in\Delta}d_{p}(x,z)$. As is easy to
show, if $x=(a,b)$, then $d_{p}(x,\Delta)=d_{p}(x,\overline{z})$ with
$\overline{z}:=(\frac{a+b}{2},\frac{a+b}{2})$.

Let $D=\\{x_{i}\\}_{i\in\\{1,\ldots,m\\}}$ and
$D^{\prime}=\\{x^{\prime}_{j}\\}_{j\in\\{1,\ldots,n\\}}$ be persistence
diagrams. For any $p,q\in[1,\infty]$, the $(p,q)$-Wasserstein distance between
$D$ and $D^{\prime}$ is defined by

$W^{q}_{p}(D,D^{\prime}):=\\\
\inf_{\alpha}\left\|\left(\left\|(d_{p}(x_{i},x^{\prime}_{\alpha(i)}))_{i\in
I}\right\|_{q},\left\|(d_{p}(x_{i},\Delta))_{i\in\\{1,\ldots,m\\}\setminus
I}\right\|_{q},\left\|(d_{p}(\Delta,x^{\prime}_{j}))_{j\in\\{1,\ldots,n\\}\setminus\alpha(I)}\right\|_{q}\right)\right\|_{q},$

where the infimum is over all injective functions
$\alpha:I\to\\{1,\ldots,n\\}$, with $I\subseteq\\{1,\ldots,m\\}$.

######  Remark 4.21.

We note that in the literature, the letters $p$ and $q$ are sometimes
interchanged with respect to our notation of the parameters of Wasserstein
distances between persistence diagrams. This is the case for instance in
[ST20, Def. 2.5]. Our choice of notation is motivated by symmetry with the
definition of algebraic Wasserstein distances, where a norm $\|\cdot\|_{q}$ is
used to “aggregate” costs expressed with respect to a norm $\|\cdot\|_{p}$.

Let $\mathcal{D}$ denote the set of all persistence diagrams. We define the
function $\operatorname{Dgm}:\operatorname{\mathsf{Tame}}\to\mathcal{D}$
sending any persistence module $X$ to the persistence diagram
$\operatorname{Dgm}(X)$ such that
$X\cong\bigoplus_{(a,b)\in\operatorname{Dgm}(X)}K(a,b)$, where we note that in
the right-hand term each bar $K(a,b)$ appears the same number of times as the
multiplicity of $(a,b)$ in the multiset $\operatorname{Dgm}(X)$. By virtue of
the barcode decomposition theorem (Theorem 2.2), the function
$\operatorname{Dgm}:\operatorname{\mathsf{Tame}}\to\mathcal{D}$ induces a
bijection between the set $\operatorname{\mathsf{Tame}}/_{\sim}$ of
isomorphism classes of persistence modules and $\mathcal{D}$.

As proven in [ST20], if $p=q$ then the algebraic distance
$d^{q}_{\mathcal{S}^{p}}$ between persistence modules coincides with the
combinatorial distance $W^{q}_{p}$ between the associated persistence
diagrams.

######  Theorem 4.22 ([ST20]).

For any $p\in[1,\infty]$ and for any persistence modules $X$ and $Y$ we have

$d^{p}_{\mathcal{S}^{p}}(X,Y)=W^{p}_{p}(\operatorname{Dgm}(X),\operatorname{Dgm}(Y)).$

It is worth observing that the equality of Theorem 4.22 does not hold when
$p\neq q$. For example, we can consider the persistence modules

$X=K(a_{1},a_{1}+\ell_{1})\oplus K(a_{2},a_{2}+\ell_{2})\oplus
K(a_{3},a_{3}+\ell_{3})$

with $\ell_{1},\ell_{2},\ell_{3}$ positive real numbers, and $0$, the zero
module. Then, assuming $q<\infty$,

$d^{q}_{\mathcal{S}^{p}}(X,0)=\left(\left\|\left(\frac{\ell_{1}}{2},\frac{\ell_{2}}{2},\frac{\ell_{3}}{2}\right)\right\|_{p}^{q}+\left\|\left(\frac{\ell_{1}}{2},\frac{\ell_{2}}{2},\frac{\ell_{3}}{2}\right)\right\|_{p}^{q}\right)^{\frac{1}{q}}$

(as we will prove in Lemma 4.23), while

$W^{q}_{p}(\operatorname{Dgm}(X),\operatorname{Dgm}(0))=\left(\left\|\left(\frac{\ell_{1}}{2},\frac{\ell_{1}}{2}\right)\right\|_{p}^{q}+\left\|\left(\frac{\ell_{2}}{2},\frac{\ell_{2}}{2}\right)\right\|_{p}^{q}+\left\|\left(\frac{\ell_{3}}{2},\frac{\ell_{3}}{2}\right)\right\|_{p}^{q}\right)^{\frac{1}{q}}.$

Given a contour $C$, we now define a function $\tau_{C}:U\to U$ as follows:
for $x=(a,b)\in U$, we set $\tau_{C}(x)=(\ell(0,a),\ell(0,b))$, where
$\ell(0,-)$ is the lifetime function associated with $C$ (Section 2.2). If $D$
is a persistence diagram, then by applying $\tau_{C}$ to each element of $D$
we obtain a persistence diagram that we denote by $\tau_{C}(D)$. Hence, we
have a function $\mathcal{D}\to\mathcal{D}$ which we denote again by
$\tau_{C}$, with a slight abuse of notation. If $C$ is the standard contour,
then $\tau_{C}$ is the identity function and in particular $\tau_{C}(D)=D$.
Figure 5 illustrates a persistence diagram transformed by applying $\tau_{C}$
for a contour $C$ of distance type.

Given a regular contour $C$, we define the combinatorial $(p,C)$-Wasserstein
distance $W^{p}_{p,C}$ pulling back the pseudometric $W^{p}_{p}$ via
$\tau_{C}:\mathcal{D}\to\mathcal{D}$. Explicitly, for all persistence diagrams
$D$ and $D^{\prime}$, we define
$W^{p}_{p,C}(D,D^{\prime}):=W^{p}_{p}(\tau_{C}(D),\tau_{C}(D^{\prime}))$. If
$C$ is regular and an action, then as a consequence of Corollary 4.11 we have
$\operatorname{Dgm}(T_{C}(X))=\tau_{C}(\operatorname{Dgm}(X))$, for every
persistence module $X$. This implies, by virtue of Proposition 4.19 and
Theorem 4.22, that

$d^{p}_{\mathcal{S}^{p,C}}(X,Y)=W^{p}_{p,C}(\operatorname{Dgm}(X),\operatorname{Dgm}(Y)),$

for all persistence modules $X$ and $Y$.

To summarize, for any $p\in[1,\infty]$ and any contour $C$ that is regular and
an action, we have a commutative diagram of isometries

${{(\operatorname{\mathsf{Tame}},d^{p}_{\mathcal{S}^{p,C}})}}$${({\mathcal{D},W^{p}_{p,C})}}$${{(\operatorname{\mathsf{Tame}},d^{p}_{\mathcal{S}^{p}})}}$${({\mathcal{D},W^{p}_{p})}}$$\scriptstyle{T_{C}}$$\scriptstyle{\tau_{C}}$$\scriptstyle{\operatorname{Dgm}}$$\scriptstyle{\operatorname{Dgm}}$

Figure 5: Left: A persistence diagram $D=\\{(0.2,0.4),(0.4,0.6),(0.6,0.8)\\}$.
A contour $C$ of distance type parametrized by a Gaussian density
($\mu=0.5,\sigma=0.15$) is chosen and the corresponding function
$f(x)=\ell(0,x)$ (i.e. the Gaussian cumulative distribution function) is shown
above and to the right of the persistence diagram. Right: The transformed
persistence diagram
$\tau_{C}(D)=\\{(\ell(0,0.2),\ell(0,0.4)),(\ell(0,0.4),\ell(0,0.6)),(\ell(0,0.6),\ell(0,0.8))\\}$.
The regular grid from the left diagram has also been transformed to illustrate
how $\tau_{C}$ stretches the plane.

###  4.5 Algebraic parametrized Wasserstein distances

The equivalence between algebraic and combinatorial Wasserstein distances for
the case $p=q$, described in Section 4.4 or in [ST20] for the standard
contour, implies that in general Wasserstein distances have no closed form.
However, for specific classes of persistence modules the distance can be
computed by a formula depending on the barcode decompositions of the
persistence modules we are comparing. The focus of this section is to present
such formulas for the exact computation of algebraic Wasserstein distances. To
avoid distinguishing the cases $q<\infty$ and $q=\infty$ in stating the
results of this subsection, for $q=\infty$ we set by convention
$\frac{1}{q}=0$ and $2^{\frac{1-q}{q}}=2^{-1}$.

######  Lemma 4.23.

For all persistence modules $X$ and all $p,q\in[1,\infty]$ we have

$d^{q}_{\mathcal{S}^{p}}(X,0)=2^{\frac{1-q}{q}}\left\|X\right\|_{p}.$

###### Proof.

Let $X=\bigoplus_{i=1}^{k}K(a_{i},b_{i})$ be a barcode decomposition of $X$,
consider a persistence module of the form
$Z=\bigoplus_{i=1}^{k}K(\frac{a_{i}+b_{i}}{2},b_{i})$ and a bar-to-bar
morphism $f=\bigoplus_{i=1}^{k}f_{i}:Z\to X$, with each
$f_{i}:K(\frac{a_{i}+b_{i}}{2},b_{i})\rightarrow K(a_{i},b_{i})$ a
monomorphism between bars. The existence of the span
$X\xhookleftarrow{f}Z\twoheadrightarrow 0$ implies that $X$ and $0$ are
$2^{\frac{1-q}{q}}\left\|X\right\|_{p}$ close in $q$-norm (Definition 4.3),
proving that $d^{q}_{\mathcal{S}^{p}}(X,0)\leq
2^{\frac{1-q}{q}}\left\|X\right\|_{p}$.

To prove the converse inequality, let us show that if
$d^{q}_{\mathcal{S}^{p}}(X,0)<\varepsilon$ then
$2^{\frac{1-q}{q}}\left\|X\right\|_{p}<\varepsilon$. If
$d^{q}_{\mathcal{S}^{p}}(X,0)<\varepsilon$, then there exists a
$(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},0)$-span
$X\xleftarrow{\varphi}Z\twoheadrightarrow 0$ for some
$\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}$ in $[0,\infty)$ such that
$\left\|(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3})\right\|_{q}<\varepsilon$.
Note that $X\hookleftarrow\operatorname{im}\varphi\twoheadrightarrow 0$ is
then a $(0,\varepsilon_{2},\varepsilon_{3},0)$-span. Consider the short exact
sequence $\operatorname{im}\varphi\hookrightarrow
X\twoheadrightarrow\operatorname{coker}\varphi$. Since
$\operatorname{coker}\varphi\in\mathcal{S}^{p}_{\varepsilon_{2}}$ and
$\operatorname{im}\varphi\in\mathcal{S}^{p}_{\varepsilon_{3}}$, by the third
axiom of noise systems we get
$X\in\mathcal{S}^{p}_{\varepsilon_{2}+\varepsilon_{3}}$, and so we get
$\left\|X\right\|_{p}\leq\varepsilon_{1}+\varepsilon_{2}$ by definition of
$\mathcal{S}^{p}$. Furthermore, by inequalities (2.1) between $p$-norms on
$\mathbb{R}^{2}$,
$\varepsilon_{2}+\varepsilon_{3}=\left\|(\varepsilon_{2},\varepsilon_{3})\right\|_{1}\leq
2^{1-\frac{1}{q}}\left\|(\varepsilon_{2},\varepsilon_{3})\right\|_{q}<2^{1-\frac{1}{q}}\varepsilon$.
Therefore we have $\left\|X\right\|_{p}<2^{1-\frac{1}{q}}\varepsilon$ or
equivalently $2^{\frac{1-q}{q}}\left\|X\right\|_{p}\ <\varepsilon$. We
conclude that $d^{q}_{\mathcal{S}^{p}}(X,0)\geq
2^{\frac{1-q}{q}}\left\|X\right\|_{p}$, and therefore
$d^{q}_{\mathcal{S}^{p}}(X,0)=2^{\frac{1-q}{q}}\left\|X\right\|_{p}$. ∎

######  Remark 4.24.

The formula
$d^{q}_{\mathcal{S}^{p}}(X,0)=2^{\frac{1-q}{q}}\left\|X\right\|_{p}$ of Lemma
4.23 was already shown for the case $p=q$ in [ST20] by using the
correspondence between combinatorial and algebraic Wasserstein distances.

The proof of Lemma 4.23 can be easily extended to the case of a regular
contour $C$ that is an action. In this case, we have

$d^{q}_{\mathcal{S}^{p,C}}(X,0)=d^{q}_{\mathcal{S}^{p}}(T_{C}(X),0)=2^{\frac{1-q}{q}}\left\|T_{C}(X)\right\|_{p}=2^{\frac{1-q}{q}}\left\|X\right\|_{p,C},$
(4.1)

where the first equality holds by Proposition 4.19, the second by Lemma 4.23
and the third by Proposition 4.13. Similar arguments can be applied to all the
results of this subsection. For exposition purposes we consider the case of
the standard contour throughout the section and collect generalizations of the
main results at the end of the subsection in Proposition 4.32.

######  Proposition 4.25.

Let $X,Y,V$ be persistence modules. Then, for every $p,q\in[1,\infty]$,

$d^{q}_{\mathcal{S}^{p}}(X\oplus V,Y\oplus V)\leq
d^{q}_{\mathcal{S}^{p}}(X,Y).$

###### Proof.

It suffices to observe that for any span $X\xleftarrow{f}Z\xrightarrow{g}Y$,
the span $X\oplus V\xleftarrow{f\oplus 1}Z\oplus V\xrightarrow{g\oplus
1}Y\oplus V$ has the same cost. ∎

######  Remark 4.26.

Note that by considering $Y=0$, Proposition 4.25 gives

$d^{q}_{\mathcal{S}^{p}}(X\oplus V,V)\leq
d^{q}_{\mathcal{S}^{p}}(X,0)=2^{\frac{1-q}{q}}\left\|X\right\|_{p}$

The converse inequality $d^{q}_{\mathcal{S}^{p}}(X\oplus V,V)\geq
d^{q}_{\mathcal{S}^{p}}(X,0)=2^{\frac{1-q}{q}}\left\|X\right\|_{p}$ does not
hold in general, as illustrated in the following example. Consider $p=q=2$,
$X=K(0,6)$ and $V=K(1,5)\oplus K(2,4)$. By Lemma 4.23 we have that
$d^{q}_{\mathcal{S}^{p}}(X,0)=\frac{1}{\sqrt{2}}\cdot 6=\sqrt{18}$. However,
$X\oplus V$ and $Y\oplus V$ are $\sqrt{6}$-close via the following
$(0,\sqrt{3},\sqrt{3},0)$-span

$K(0,6)\oplus K(1,5)\oplus K(2,4)\xleftarrow{f_{1}\oplus f_{2}\oplus
f_{3}}K(1,6)\oplus K(2,5)\oplus K(3,4)\xrightarrow{g_{1}\oplus g_{2}\oplus
g_{3}}K(1,5)\oplus K(2,4)\oplus 0$

implying that $d^{q}_{\mathcal{S}^{p}}(X\oplus V,Y\oplus
V)\leq\sqrt{6}<\sqrt{18}=d^{q}_{\mathcal{S}^{p}}(X,Y).$ This example is based
on the fact that given a span $X\xleftarrow{f}Z\xrightarrow{g}Y$ realizing the
distance between $X$ and $Y$, the span $X\oplus V\xleftarrow{f\oplus 1}Z\oplus
V\xrightarrow{g\oplus 1}Y\oplus V$ not always is the one achieving the
distance between $X\oplus V$ and $Y\oplus V$.

Let $\\{K(a_{i},b_{i})\\}_{i\in\\{1,\ldots,k\\}}$ be a sequence of bars
ordered non-decreasingly by length, that is, $b_{1}-a_{1}\leq
b_{2}-a_{2}\leq\cdots\leq b_{k}-a_{k}$. For $j\in\\{1,\ldots,k\\}$, consider
$Z:=\bigoplus_{i=1}^{j}K(a_{i},b_{i})$ and
$Y:=\bigoplus_{i=j+1}^{k}K(a_{i},b_{i})$. The remainder of this section is
devoted to proving that, in this case $d^{q}_{\mathcal{S}^{p}}(Y\oplus
Z,Y)=d^{q}_{\mathcal{S}^{p}}(Z,0)=2^{\frac{1-q}{q}}\left\|Z\right\|_{p}.$ In
Section 5, this result will be used for the computation of the stable rank of
a persistence module with respect to $d^{q}_{\mathcal{S}^{p}}$.

######  Proposition 4.27.

Let $\mathcal{S}$ be a noise system. For any
$(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4})$-span
$X\xleftarrow{}Z\xrightarrow{}Y$ of persistence modules there is a mono-epi
$(0,\varepsilon_{2},\varepsilon_{3},0)$-span
$X\xhookleftarrow{}\operatorname{im}f\twoheadrightarrow P$ such that
$\operatorname{rank}(P)\leq\operatorname{rank}(Y)$.

###### Proof.

By Theorem 3.13 and Remark 3.2, if $U\hookrightarrow V$ is a monomorphism
between persistence modules, than
$\operatorname{rank}(U)\leq\operatorname{rank}(V)$, and similarly if
$V\twoheadrightarrow U$ is an epimorphism, then
$\operatorname{rank}(U)\leq\operatorname{rank}(V)$. Let
$X\xleftarrow{f}Z\xrightarrow{g}Y$ be a
$(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4})$-span of
persistence modules, and consider the following diagram in
$\operatorname{\mathsf{Tame}}$, where the square is a push-out:

${Z}$${\operatorname{im}f}$${\operatorname{im}g}$${X}$${P}$${Y}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{j}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{i}$

Since $f^{\prime}$ is an epimorphisms and $i$ is a monomorphism,
$\operatorname{rank}(P)\leq\operatorname{rank}(\operatorname{im}g)\leq\operatorname{rank}(Y)$.
We consider the span
$X\xhookleftarrow{j}\operatorname{im}f\overset{g^{\prime}}{\twoheadrightarrow}P$.
Clearly, the kernel of the corestriction
$g:Z\twoheadrightarrow\operatorname{im}g$ still belongs to
$\mathcal{S}_{\varepsilon_{3}}$, and its cokernel is zero. Then, by
Proposition $8.1$ in [SCL+17], $\ker
g^{\prime}\in\mathcal{S}_{\varepsilon_{3}}$ and
$\operatorname{coker}g^{\prime}=0$. The kernel of $j$ is $0$, while its
cokernel belongs to $\mathcal{S}_{\varepsilon_{2}}$, as it coincides with the
cokernel of $f:Z\to X$. ∎

######  Lemma 4.28.

Let $p,q\in[1,\infty]$, and let $[a_{i},b_{i}]$ be nonempty intervals in
$[0,\infty)$, for $i\in\\{1,\ldots,j\\}$. The function
$\gamma:\prod_{i=1}^{j}[a_{i},b_{i}]\to[0,\infty)$ defined by

$\gamma(x_{1},\ldots,x_{j}):=\left\|\left(\left\|(x_{1}-a_{1},\ldots,x_{j}-a_{j})\right\|_{p},\left\|(b_{1}-x_{1},\ldots,b_{j}-x_{j})\right\|_{p}\right)\right\|_{q}$

has a global minimum at
$(\frac{a_{1}+b_{1}}{2},\ldots,\frac{a_{j}+b_{j}}{2})$.

###### Proof.

The function $\gamma$ is continuous with a compact domain, so it admits a
global minimum by the extreme value theorem. Moreover, it is convex because
norms are convex functions.

Write $a=(a_{1},\ldots,a_{j})$, $b=(b_{1},\ldots,b_{j})$ and
$x=(x_{1},\ldots,x_{j})$ in $\mathbb{R}^{j}$. Since $\gamma(x)=\gamma(a+b-x)$
for every $x$, the function $\gamma$ is invariant under point reflection
through $\frac{a+b}{2}$. By convexity, we conclude that $\frac{a+b}{2}$ is a
global minimum of $\gamma$. ∎

######  Proposition 4.29.

Let $X=\bigoplus_{i=1}^{k}K(a_{i},b_{i})$, with the bars ordered non-
decreasingly by length. Let $j\in\\{1,\ldots,k\\}$, and let
$p,q\in[1,\infty]$. Then, any persistence module $Y$ with
$\operatorname{rank}(Y)\leq\operatorname{rank}(X)-j$ is such that

$d^{q}_{\mathcal{S}^{p}}(X,Y)\geq
2^{\frac{1-q}{q}}\textstyle{\left\|\bigoplus_{i=1}^{j}K(a_{i},b_{i})\right\|_{p}}.$

###### Proof.

We prove the claim by contradiction. Suppose that there exists a persistence
module $Y$ such that $\operatorname{rank}(Y)\leq\operatorname{rank}(X)-j$ and

$d^{q}_{\mathcal{S}^{p}}(X,Y)<2^{\frac{1-q}{q}}\textstyle{\left\|\bigoplus_{i=1}^{j}K(a_{i},b_{i})\right\|_{p}}.$

By definition, there exists a span $X\xleftarrow{f}Z\xrightarrow{g}Y$ such
that

$\left\|\left(\left\|\ker
f\right\|_{p},\left\|\operatorname{coker}f\right\|_{p},\left\|\ker
g\right\|_{p},\left\|\operatorname{coker}g\right\|_{p}\right)\right\|_{q}<2^{\frac{1-q}{q}}\textstyle{\left\|\bigoplus_{i=1}^{j}K(a_{i},b_{i})\right\|_{p}}.$
(4.2)

By Proposition 4.27 we can assume (possibly after replacing $Y$ with a
persistence module of smaller or equal rank) that the span above is mono-epi,
that is, of the form $X\xhookleftarrow{f}Z\overset{g}{\twoheadrightarrow}Y$.
By Theorems 3.13 and 3.14, we can moreover assume that $f$ and $g$ are bar-to-
bar morphisms.

Thus, we can consider a barcode decomposition $Z=\bigoplus_{i=1}^{k}Z_{i}$,
with some of the $Z_{i}$ possibly zero, and a barcode decomposition
$Y=\bigoplus_{i=1}^{k}Y_{i}$, with at least $j$ of the $Y_{i}$ equal to zero
by assumption, together with morphisms between bars
$K(a_{i},b_{i})\xhookleftarrow{f_{i}}Z_{i}\overset{g_{i}}{\twoheadrightarrow}Y_{i}$
such that $f=\bigoplus_{i=1}^{k}f_{i}$ and $g=\bigoplus_{i=1}^{k}g_{i}$. Let
$I\subseteq\\{1,\ldots,k\\}$, with $|I|\geq j$, be the subset of the indices
$i$ such that $Y_{i}=0$. For every $i\in I$, we have
$K(a_{i},b_{i})\xhookleftarrow{f_{i}}Z_{i}\overset{g_{i}}{\twoheadrightarrow}0$,
with $Z_{i}=K(x_{i},b_{i})$ for some $a_{i}\leq x_{i}\leq b_{i}$, where
$K(b_{i},b_{i})$ denotes the zero module. Since $\ker
f=\bigoplus_{i=1}^{k}\ker f_{i}$ and
$\operatorname{coker}f=\bigoplus_{i=1}^{k}\operatorname{coker}f_{i}$, by
Remark 3.2 we observe that $\bigoplus_{i\in I}K(a_{i},x_{i})$ is a direct
summand of $\operatorname{coker}f$, and similarly that $\bigoplus_{i\in
I}K(x_{i},b_{i})$ is a direct summand of $\ker g$, which gives

$\displaystyle\left\|\operatorname{coker}f\right\|_{p}$
$\displaystyle\geq\left\|\oplus_{i\in
I}K(a_{i},x_{i})\right\|_{p}=\left\|(x_{i}-a_{i})_{i\in I}\right\|_{p},$
$\displaystyle\left\|\ker g\right\|_{p}$ $\displaystyle\geq\left\|\oplus_{i\in
I}K(x_{i},b_{i})\right\|_{p}=\left\|(b_{i}-x_{i})_{i\in I}\right\|_{p}.$

If $b_{i}<\infty$ for all $i\in I$, it is easy to show using Lemma 4.28 that
the cost of the span is

$\left\|\left(\left\|\operatorname{coker}f\right\|_{p},\left\|\ker
g\right\|_{p}\right)\right\|_{q}\geq
2^{\frac{1-q}{q}}\textstyle{\left\|(b_{i}-a_{i})_{i\in
I}\right\|_{p}}=2^{\frac{1-q}{q}}\textstyle{\left\|\bigoplus_{i\in
I}K(a_{i},b_{i})\right\|_{p}},$

and the same inequality clearly holds if $b_{i}=\infty$ for some $i\in I$.
However, since $|I|\geq j$, the right-hand side of the inequality cannot be
smaller than

$2^{\frac{1-q}{q}}\textstyle{\left\|(b_{i}-a_{i})_{i\in\\{1,\ldots,j\\}}\right\|_{p}}=2^{\frac{1-q}{q}}\textstyle{\left\|\bigoplus_{i=1}^{j}K(a_{i},b_{i})\right\|_{p}},$

and this contradicts (4.2). ∎

######  Proposition 4.30.

Let $X=\bigoplus_{i=1}^{k}K(a_{i},b_{i})$, with the bars ordered non-
decreasingly by length. Let $j\in\\{1,\ldots,k\\}$, and let
$Y=\bigoplus_{i=j+1}^{k}K(a_{i},b_{i})$ (with $Y=0$ when $j=k$). Then, for all
$p,q\in[1,\infty]$,

$d^{q}_{\mathcal{S}^{p}}(X,Y)=2^{\frac{1-q}{q}}\textstyle{\left\|\bigoplus_{i=1}^{j}K(a_{i},b_{i})\right\|_{p}}.$
(4.3)

###### Proof.

Since $\operatorname{rank}(Y)=\operatorname{rank}(X)-j$, Proposition 4.29
gives us the inequality

$d^{q}_{\mathcal{S}^{p}}(X,Y)\geq
2^{\frac{1-q}{q}}\textstyle{\left\|\bigoplus_{i=1}^{j}K(a_{i},b_{i})\right\|_{p}}.$

To prove the other direction, it is enough to exhibit a span between $X$ and
$Y$ with cost equal to the right-hand side of (4.3). We construct a mono-epi
and bar-to-bar span $X\xhookleftarrow{f}Z\overset{g}{\twoheadrightarrow}Y$ as
follows. Let

$\textstyle{Z:=\bigoplus_{i=1}^{j}K(\frac{a_{i}+b_{i}}{2},b_{i})\oplus\bigoplus_{i=j+1}^{k}K(a_{i},b_{i}),}$

and let $f=\bigoplus_{i=1}^{k}f_{i}$ and $g=\bigoplus_{i=1}^{k}g_{i}$ with

$\displaystyle f_{i}$
$\displaystyle=\begin{cases}K(\frac{a_{i}+b_{i}}{2},b_{i})\xhookrightarrow{}K(a_{i},b_{i})&\text{
if $1\leq i\leq j$}\\\
K(a_{i},b_{i})\xrightarrow{\operatorname{id}}K(a_{i},b_{i})&\text{ if $j+1\leq
i\leq k$},\end{cases}$ $\displaystyle g_{i}$
$\displaystyle=\begin{cases}K(\frac{a_{i}+b_{i}}{2},b_{i})\xrightarrow{}0&\text{
if $1\leq i\leq j$}\\\
K(a_{i},b_{i})\xrightarrow{\operatorname{id}}K(a_{i},b_{i})&\text{ if $j+1\leq
i\leq k$}.\end{cases}$

Recalling Remark 3.2, we observe that $\ker f=\operatorname{coker}g=0$ and
$\left\|\operatorname{coker}f\right\|_{p}=\left\|\ker
g\right\|_{p}=\left\|(\frac{b_{i}-a_{i}}{2})_{i\in\\{1,\ldots,j\\}}\right\|_{p}$.
The cost $\left\|\left(\left\|\operatorname{coker}f\right\|_{p},\left\|\ker
g\right\|_{p}\right)\right\|_{q}$ of this span is therefore as claimed. ∎

In the final part of this subsection we generalize some results from the case
of the standard contour to the case of any regular contour $C$ that is an
action.

######  Definition 4.31.

Let $C$ be a regular contour, and let $X=\bigoplus_{i=1}^{k}K(a_{i},b_{i})$.
We say that (the barcode decomposition of) $X$ has bars ordered non-
decreasingly by lifetime if
$\ell(a_{1},b_{1})\leq\ell(a_{2},b_{2})\leq\cdots\leq\ell(a_{k},b_{k})$, where
$\ell$ denotes the lifetime function associated with $C$ (see Section 2.2).

######  Proposition 4.32.

Let $C$ be a regular contour and an action, and let $p,q\in[1,\infty]$. Let
$X=\bigoplus_{i=1}^{k}K(a_{i},b_{i})$, with bars ordered non-decreasingly by
lifetime, and let $j\in\\{1,\ldots,k\\}$. Then, for all persistence modules
$Y$,

  1. 1.

if $\operatorname{rank}(Y)\leq\operatorname{rank}(X)-j$, then

$d^{q}_{\mathcal{S}^{p,C}}(X,Y)\geq
2^{\frac{1-q}{q}}\textstyle{\left\|\bigoplus_{i=1}^{j}K(a_{i},b_{i})\right\|_{p,C}};$

  2. 2.

if $Y=\bigoplus_{i=j+1}^{k}K(a_{i},b_{i})$ (with the convention $Y=0$ when
$j=k$), then

$d^{q}_{\mathcal{S}^{p,C}}(X,Y)=2^{\frac{1-q}{q}}\textstyle{\left\|\bigoplus_{i=1}^{j}K(a_{i},b_{i})\right\|_{p,C}}.$

###### Proof.

The first statement follows from

$\displaystyle d^{q}_{\mathcal{S}^{p,C}}(X,Y)$
$\displaystyle=d^{q}_{\mathcal{S}^{p}}(T_{C}(X),T_{C}(Y))$ $\displaystyle\geq
2^{\frac{1-q}{q}}\textstyle{\left\|\bigoplus_{i=1}^{j}T_{C}(K(a_{i},b_{i}))\right\|_{p}}$
$\displaystyle=2^{\frac{1-q}{q}}\textstyle{\left\|\bigoplus_{i=1}^{j}K(a_{i},b_{i})\right\|_{p,C}}$

where we are using in sequence Proposition 4.19, Proposition 4.29 (observing
that the length of a bar $T_{C}(K(a,b))$ coincides with the lifetime
$\ell(a,b)$ of $K(a,b)$, see Proposition 4.10), and Proposition 4.13. The
second statement is proven similarly, using Proposition 4.30. ∎

##  5 Wasserstein stable ranks: computations and stability

In Section 4 it was shown that the Wasserstein distances
$d^{q}_{\mathcal{S}^{p,C}}$ are pseudometrics on
$\operatorname{\mathsf{Tame}}$. They can therefore be used in the framework of
hierarchical stabilization (see Section 2.5) to produce stable invariants of
persistence modules. The focus of this section is on one type of such
invariants, the Wasserstein stable ranks, which are the hierarchical
stabilization of the rank function with respect to Wasserstein distances
$d^{q}_{\mathcal{S}^{p,C}}.$ Denoting $d^{q}_{\mathcal{S}^{p,C}}$ by $d$, the
stability result for stable ranks (Proposition 2.5) states that for every pair
of persistence modules $X$ and $Y$

$d(X,Y)\geq
d_{\bowtie}(\widehat{\operatorname{rank}}_{d}(X),\widehat{\operatorname{rank}}_{d}(Y)).$

In the case where $p=q$ and $C$ is the standard contour, combining the above
inequality with the stability results of [ST20] gives several stability
results of Wasserstein stable ranks with respect to perturbation of the
original data. In particular, [ST20, Theorem $4.8$] expresses stability with
respect to sublevel set filtrations of monotone functions on cellular
complexes, [ST20, Theorem $5.1$] expresses stability with respect to the
construction of cubical complexes from grey scale images, and [ST20, Theorem
$5.9$], expresses stability with respect to Wasserstein distance between point
clouds when using the Vietoris-Rips construction.

In order to use the Wasserstein stable ranks in applications, it is important
to be able to efficiently compute them as well as distances between them. In
this section we use computations of Wasserstein distances from Section 4 to
derive a formula for the Wasserstein stable rank and propose a convenient
formulation of the interleaving distance between stable ranks.

Having defined a rich family of Wasserstein distances
$d^{q}_{\mathcal{S}^{p,C}}$, it is natural to ask whether we can in a
supervised learning context search for an optimal distance for a problem at
hand. Choosing a suitable parametrization of a contour and leveraging the
simple expression of the interleaving distance between Wasserstein stable
ranks, in Section 5.3 we set up a simple metric learning problem with the aim
of observing the interaction between the parameter $p$ and the parameters
related to the contour $C$ within the learning. Preliminary results on the
optimization of only a contour in a metric learning framework are presented in
[Gäv18].

###  5.1 Computation of the stable rank with Wasserstein distances

The results of this subsection provide explicit formulas to compute the stable
rank with respect to the Wasserstein distances $d^{q}_{\mathcal{S}^{p,C}}$
introduced in Section 4. We begin by showing that the computation of the
stable rank of a persistence module $X$ can be reduced to the computation of
the stable rank of the submodule $X^{\prime}$ formed by all finite bars of
$X$. As in the previous section, if $q=\infty$ we set by convention
$\frac{1}{q}=0$ and $2^{\frac{1-q}{q}}=2^{-1}$.

######  Proposition 5.1.

Let $X=\bigoplus_{i=1}^{k}K(a_{i},b_{i})$, and let $d$ denote the pseudometric
$d^{q}_{\mathcal{S}^{p,C}}$, for some $p,q\in[1,\infty]$ and a regular contour
$C$ that is an action. Let $I:=\\{i\in\\{1,\ldots,k\\}\mid b_{i}<\infty\\}$,
$J:=\\{i\in\\{1,\ldots,k\\}\mid b_{i}=\infty\\}$, and consider the submodules
$X^{\prime}:=\bigoplus_{i\in I}K(a_{i},b_{i})$ and
$X^{\prime\prime}:=\bigoplus_{i\in J}K(a_{i},b_{i})$ such that
$X=X^{\prime}\oplus X^{\prime\prime}$. Then
$\widehat{\operatorname{rank}}_{d}(X)=\widehat{\operatorname{rank}}_{d}(X^{\prime})+\operatorname{rank}(X^{\prime\prime})$.

###### Proof.

We prove that, for any $t\in[0,\infty)$, we have
$\widehat{\operatorname{rank}}_{d}(X)(t)=\widehat{\operatorname{rank}}_{d}(X^{\prime})(t)+\operatorname{rank}(X^{\prime\prime})$.
Let $r:=\widehat{\operatorname{rank}}_{d}(X^{\prime})(t)$, meaning that there
exists a persistence module $Y^{\prime}$ such that
$\operatorname{rank}(Y^{\prime})=r$ and $d(X^{\prime},Y^{\prime})\leq t$. By
Proposition 4.32(2), since $X^{\prime\prime}$ contains only infinite bars,
$d(X^{\prime}\oplus X^{\prime\prime},Y^{\prime}\oplus
X^{\prime\prime})=d(X^{\prime},Y^{\prime})$. Furthermore,
$\operatorname{rank}(Y^{\prime}\oplus
X^{\prime\prime})=r+\operatorname{rank}(X^{\prime\prime})$. This proves
$\widehat{\operatorname{rank}}_{d}(X)\leq\widehat{\operatorname{rank}}_{d}(X^{\prime})+\operatorname{rank}(X^{\prime\prime})$.

To prove the converse inequality, let $t\in[0,\infty)$ and let
$n:=\widehat{\operatorname{rank}}_{d}(X)(t)$, meaning that there is a
persistence module $Y$ such that $\operatorname{rank}(Y)=n$ and $d(X,Y)\leq
t$. We can assume that $X$ has bars ordered non-decreasingly by lifetime
(Definition 4.31). By Proposition 4.32, we can take
$Y=\bigoplus_{i=k-n+1}^{k}K(a_{i},b_{i})$, and we know that
$d(X,Y)=2^{\frac{1-q}{q}}\left\|\bigoplus_{i=1}^{k-n}K(a_{i},b_{i})\right\|_{p,C}\leq
t$, which implies that $n\geq\operatorname{rank}(X^{\prime\prime})$. We can
consider $X^{\prime}$ and its $m:=n-\operatorname{rank}(X^{\prime\prime})$
bars with greatest lifetime, and define
$Y^{\prime}:=\bigoplus_{i=\operatorname{rank}(X^{\prime})-m+1}^{\operatorname{rank}(X^{\prime})}K(a_{i},b_{i})$.
Using Proposition 4.32 again and observing that
$\operatorname{rank}(X^{\prime})-m=k-n$, we have

$d(X^{\prime},Y^{\prime})=2^{\frac{1-q}{q}}\left\|\textstyle\bigoplus_{i=1}^{\operatorname{rank}(X^{\prime})-m}K(a_{i},b_{i})\right\|_{p,C}=d(X,Y),$

so
$\widehat{\operatorname{rank}}_{d}(X^{\prime})(t)\leq\operatorname{rank}(Y^{\prime})=n-\operatorname{rank}(X^{\prime\prime})$.
Rearranging the terms of this inequality, we obtain
$\widehat{\operatorname{rank}}_{d}(X)\geq\widehat{\operatorname{rank}}_{d}(X^{\prime})+\operatorname{rank}(X^{\prime\prime})$.
∎

######  Corollary 5.2.

Let $X=\bigoplus_{i=1}^{k}K(a_{i},b_{i})$, and let $d$ denote the metric
$d^{q}_{\mathcal{S}^{p,C}}$, for some $p,q\in[1,\infty]$ and a regular contour
$C$ that is an action. Then,

$\lim_{t\to\infty}\widehat{\operatorname{rank}}_{d}(X)(t)=|\\{i\in\\{1,\ldots,k\\}\mid
b_{i}=\infty\\}|.$

###### Proof.

We apply Proposition 5.1, observing that
$\widehat{\operatorname{rank}}_{d}(X^{\prime})(t)=0$ for a sufficiently large
$t\in[0,\infty)$ since $d(X^{\prime},0)$ is finite by equation (4.1). ∎

We now prove a formula to compute the stable rank of a persistence module $X$.
By Proposition 5.1, we can restrict to the case in which $X$ has only finite
bars. First, we consider the case $p<\infty$.

######  Proposition 5.3.

Let $p\in[1,\infty)$ and $q\in[1,\infty]$, let $C$ be a regular contour that
is an action, and let $d$ denote the metric $d^{q}_{\mathcal{S}^{p,C}}$. Let
$X=\bigoplus_{i=1}^{k}K(a_{i},b_{i})$, with $b_{i}<\infty$ for every
$i\in\\{1,\ldots,k\\}$ and assume that bars in the barcode decomposition of
$X$ are ordered non-decreasingly by lifetime (Definition 4.31). Then, there
exist real numbers $0=t_{0}<t_{1}<t_{2}<\cdots<t_{k}$ such that the stable
rank function $\widehat{\operatorname{rank}}_{d}(X):[0,\infty)\to[0,\infty)$
is constant on the intervals $[t_{0},t_{1})$, $[t_{1},t_{2})$,…,
$[t_{k-1},t_{k})$, $[t_{k},\infty)$, and

$\widehat{\operatorname{rank}}_{d}(X)(t_{j})=\operatorname{rank}(X)-j,$

for every $j\in\\{0,1,\ldots,k\\}$. Furthermore,

$t_{j}=2^{\frac{1-q}{q}}\textstyle{\left\|\bigoplus_{i=1}^{j}K(a_{i},b_{i})\right\|_{p,C}}=2^{\frac{1-q}{q}}\left\|(\ell(a_{1},b_{1}),\ldots,\ell(a_{j},b_{j}))\right\|_{p}$

for every $j\in\\{1,\ldots,k\\}$, where $\ell$ is the lifetime function
associated with $C$.

###### Proof.

For every $j\in\\{1,\ldots,k\\}$, by Proposition 4.32
$Y_{j}:=\bigoplus_{i=j+1}^{k}K(a_{i},b_{i})$ is the closest persistence module
to $X$ (in the metric $d^{q}_{\mathcal{S}^{p,C}}$) such that
$\operatorname{rank}(Y_{j})=\operatorname{rank}(X)-j$. We have

$d^{q}_{\mathcal{S}^{p,C}}(X,Y_{j})=2^{\frac{1-q}{q}}\textstyle{\left\|\bigoplus_{i=1}^{j}K(a_{i},b_{i})\right\|_{p,C}}=:t_{j},$

and we observe that $0=t_{0}<t_{1}<t_{2}<\cdots<t_{k}$ as a consequence of the
assumption $p<\infty$. ∎

In particular, when $p<\infty$, the value of the piecewise constant function
$\widehat{\operatorname{rank}}_{d}(X)$ can only decrease by 1 at every
discontinuity point $t_{j}$. For $p=\infty$, the stable rank has a slightly
different behavior. Even though we can define the sequence of real numbers
$(t_{j})_{j}$ as in Proposition 5.3, we only have $0=t_{0}\leq t_{1}\leq
t_{2}\leq\cdots\leq t_{k}$ instead of strict inequalities. Letting $s_{m}$
denote the $m^{\text{th}}$ smallest value in $\\{t_{j}\\}_{j}$ we obtain a
sequence $0=s_{0}<s_{1}<s_{2}<\cdots<s_{k^{\prime}}$ such that the stable rank
with respect to the metric $d:=d^{q}_{\mathcal{S}^{\infty,C}}$ is constant on
the intervals $[s_{0},s_{1})$,…, $[s_{k^{\prime}},\infty)$, taking the values

$\widehat{\operatorname{rank}}_{d}(X)(s_{m})=\operatorname{rank}(X)-\max\\{j\mid
t_{j}=s_{m}\\}.$

An explicit formula for the stable rank in the case $p=\infty$ and $q=1$ was
first given in [CR20].

######  Remark 5.4.

We observe that for a persistence module $X$ of rank $k$, once the $k$ bars in
the barcode decomposition of $X$ have been ordered non-decreasingly by
lifetime, the complexity of computing the discontinuity points of the the
Wasserstein stable rank using Proposition 5.3 is linear in $k$. Therefore the
computational complexity of the Wasserstein stable rank is $O(k\log k)$,
determined by the complexity of the sorting algorithm to order the bars non-
decreasingly by lifetime.

###  5.2 Interleaving distance between stable ranks

The aim of this subsection is to propose a convenient expression for the
interleaving distance (Section 2.5) between two non-increasing piecewise
constant functions. We assume functions to take only finitely many values,
that is the case of stable ranks which will be the object of our study. Let
$f,g:[0,\infty)\to[0,\infty)$ be non-increasing piecewise constant functions.
If $\lim_{t\to\infty}f(t)\neq\lim_{t\to\infty}g(t)$, then
$d_{\bowtie}(f,g)=\infty$. For the computation of the interleaving distance we
can therefore assume that the functions $f$ and $g$ have the same limit value
and denote it by $L$. Given a non-increasing piecewise constant function
$f:[0,\infty)\to[0,\infty)$ with limit value $L$, we define the non-increasing
piecewise constant function $f^{-1}:[L,\infty)\to[0,\infty)$ with values
$f^{-1}(y):=\inf\\{t\mid f(t)\leq y\\}$. If in addition the function $f$ is
right-continuous, then $f^{-1}(y)=\min\\{t\mid f(t)\leq y\\}$. We observe that
for every right-continuous non-increasing piecewise constant function $f$ we
have $f^{-1}(f(t))\leq t$ for all $t$, and equality holds if $t$ is a
discontinuity point of $f$. Moreover, $f(f^{-1}(y))\leq y$ for all $y\geq L$,
and equality holds if $y\in\operatorname{im}f$. Our focus in this subsection
will be on the discontinuity points $\\{t_{i}\\}$ of $f$ and on the values in
$\operatorname{im}f$, rather than on the full domain and codomain of $f$, thus
justifying our use of the notation $f^{-1}$.

######  Proposition 5.5.

Consider two right-continuous non-increasing piecewise constant functions
$f,g:[0,\infty)\to[0,\infty)$ having the same limit value $L$. Using the
notation introduced above, we have:

$d_{\bowtie}(f,g)=\|f^{-1}-g^{-1}\|_{\infty}.$

###### Proof.

Let us define the following subset of $[0,\infty)$,

$A(f,g):=\\{\varepsilon\in[0,\infty)\mid f(t)\geq g(t+\varepsilon)\text{ and
}g(t)\geq f(t+\varepsilon),\text{ for all }t\in[0,\infty)\\}.$

Remember that, by definition, $d_{\bowtie}(f,g)=\inf A(f,g)$.

We first prove that $d_{\bowtie}(f,g)\geq\|f^{-1}-g^{-1}\|_{\infty}$. Let
$\varepsilon\in A(f,g)$. Then, for all $y\geq L$, we have $y\geq
f(f^{-1}(y))\geq g(f^{-1}(y)+\varepsilon)$. Composing by the non-increasing
function $g^{-1}$ and recalling that $g^{-1}(g(t))\leq t$ for all $t$, we
obtain $f^{-1}(y)+\varepsilon\geq g^{-1}(y)$. We have thus shown that
$g^{-1}(y)-f^{-1}(y)\leq\varepsilon$, for all $y\geq L$ and $\varepsilon\in
A(f,g)$, which implies $g^{-1}(y)-f^{-1}(y)\leq d_{\bowtie}(f,g)$, for all
$y\geq L$. By symmetry in the roles of $f$ and $g$, we conclude that
$|g^{-1}(y)-f^{-1}(y)|\leq d_{\bowtie}(f,g)$, for all $y\geq L$.

We now prove that $d_{\bowtie}(f,g)\leq\|f^{-1}-g^{-1}\|_{\infty}$ by showing
that $\varepsilon:=\|f^{-1}-g^{-1}\|_{\infty}$ is in $A(f,g)$. For any
$t\in[0,\infty)$, there exists $y\geq L$ such that $f(t)=f(f^{-1}(y))$ (which
implies $f^{-1}(y)\leq t$), as it is enough to take $y=f(t)$. Since
$g^{-1}(y)\leq f^{-1}(y)+\varepsilon$ by definition of $\varepsilon$,
$f^{-1}(y)\leq t$, and $g$ is non-increasing, we get the following
inequalities:

$f(t)=y\geq g(g^{-1}(y))\geq g(f^{-1}(y)+\varepsilon)\geq g(t+\varepsilon).$

By symmetry, we also get $g(t)\geq f(t+\varepsilon)$, and we conclude that
$\varepsilon\in A(f,g)$. ∎

If $f$ is the Wasserstein stable rank of a persistence module, by Proposition
5.3 it is right-continuous, which implies that the values of $f^{-1}$ can be
computed as minima and correspond the discontinuity points of $f$. If we
denote by $\\{t_{1},\ldots,t_{k}\\}$ the set of discontinuity points of $f$
and set $t_{0}:=0$, then $f^{-1}(f(t_{i}))=t_{i}$ for $i\in\\{0,\ldots,k\\}$,
and these values are sufficient to encode $f^{-1}$. Indeed, by definition
$f^{-1}(y)=t_{i}$ for $f(t_{i})\leq y<f(t_{i-1})$ and $i\in\\{1,\ldots,k\\}$,
and $f^{-1}(y)=t_{0}$ for $y\geq f(t_{0})$. This implies that the interleaving
distance between two stable ranks $f$ and $g$ can be realized as the
$L^{\infty}$ norm between two finite vectors. Explicitly, evaluating $f^{-1}$
and $g^{-1}$ on the union of the discontinuity points of these two functions,
one obtains finite vectors $\hat{f}^{-1}$ and $\hat{g}^{-1}$ such that
$\|\hat{f}^{-1}-\hat{g}^{-1}\|_{\infty}=\|f^{-1}-g^{-1}\|_{\infty}$. By using
the characterization in Proposition 5.3 of discontinuity points of Wasserstein
stable ranks, the interleaving distance between the Wasserstein stable ranks
of persistence modules $X=\bigoplus_{i=1}^{n}K(a_{i},b_{i})$ and
$Y=\bigoplus_{i=1}^{m}K(c_{i},d_{i})$, both with bars ordered non-decreasingly
by lifetime, can be written as the $L^{\infty}$ norm of the vector
$(\hat{f}_{i}^{-1}-\hat{g}_{i}^{-1})_{i=0,\ldots,\text{min}(n,m)}$ with
components:

$\displaystyle\hat{f}_{i}^{-1}-\hat{g}_{i}^{-1}=2^{\frac{1-q}{q}}(\|(\ell(a_{1},b_{1}),\ldots,\ell(a_{n-i},b_{n-i}))\|_{p}-\|(\ell(c_{1},d_{1}),\ldots,\ell(c_{m-i},d_{m-i}))\|_{p}).$
(5.1)

######  Remark 5.6.

For two persistence modules $X$ and $Y$ both of rank $k$, the complexity of
computing the interleaving distance is dominated by the sorting of the bars in
the respective barcode decompositions of $X$ and $Y$, since forming the vector
as in (5.1) and computing its $L^{\infty}$ norm can be done linearly in $k$.
The computational complexity of the interleaving distance between Wasserstein
stable ranks is thus $O(k\log k)$.

######  Example 5.7.

Consider a persistence module $Y=\bigoplus_{i=1}^{3}K(a_{i},b_{i})$ with bars
ordered non-decreasingly by lifetime and $X=K(a_{0},b_{0})\oplus Y$ such that
$\varepsilon:=\ell(a_{0},b_{0})\leq\ell(a_{1},b_{1})$. By using the formula
(5.1) and observing that

$\|(\ell(a_{0},b_{0}),\ldots,\ell(a_{i},b_{i}))\|_{p}-\|(\ell(a_{1},b_{1}),\ldots,\ell(a_{i},b_{i}))\|_{p}\leq\ell(a_{0},b_{0})$

for $i\in\\{1,2,3\\}$ by properties (2.1) and (2.2) of $p$-norms, we see that
the interleaving distance between $\widehat{\operatorname{rank}}_{d}(X)$ and
$\widehat{\operatorname{rank}}_{d}(Y)$ with $d=d^{q}_{\mathcal{S}^{p,C}}$ is
given by $2^{\frac{1-q}{q}}\varepsilon$. Note that by Proposition 4.32 we know
$d^{q}_{\mathcal{S}^{p,C}}(X,Y)=2^{\frac{1-q}{q}}\textstyle{\left\|K(a_{0},b_{0})\right\|_{p,C}}=2^{\frac{1-q}{q}}\varepsilon$.
Therefore in this case the interleaving distance between stable ranks with
respect to Wasserstein distance coincides with the Wasserstein distance
between $X$ and $Y$. Note however that this is not always the case. The
Wasserstein stable ranks of $X$ and $Y$ with respect to
$d^{q}_{\mathcal{S}^{p,C}}$, with parameters $q=1$, $p=2$ and $C$ the standard
contour, are shown in Figure 6, together with their “inverse” functions which
are used for the computation of the interleaving distance.

Figure 6: Schematic representation of the computation of the interleaving
distance in Example 5.7. Left: Barcode decomposition of $Y$ in orange and bar
$K(a_{0},b_{0})$ in blue. Middle: Stable ranks computed with standard contour,
$q=1$ and $p=2$. The functions $\widehat{\operatorname{rank}}_{d}(X)$ and
$\widehat{\operatorname{rank}}_{d}(Y)$ are represented in blue and orange,
respectively. Right: Inverse stable ranks for the computation of interleaving
distance, with $\widehat{\operatorname{rank}}_{d}^{-1}(X)$ in blue and
$\widehat{\operatorname{rank}}_{d}^{-1}(Y)$ in orange. The interleaving
distance between stable ranks can be computed as
$\|\widehat{\operatorname{rank}}_{d}^{-1}(X)-\widehat{\operatorname{rank}}_{d}^{-1}(Y)\|_{\infty}=2^{\frac{1-q}{q}}\varepsilon$,
illustrated with the pink arrow.

Let us keep denoting $d^{q}_{\mathcal{S}^{p,C}}$ by $d$. It follows from
triangle inequality and Lemma 4.23 that:

$d(X,Y)\geq 2^{\frac{1-q}{q}}|\left\|X\right\|_{p}-\left\|Y\right\|_{p}|.$

However this inequality can be refined by

$d(X,Y)\geq
d_{\bowtie}(\widehat{\text{rank}}_{d}(X),\widehat{\text{rank}}_{d}(Y))\geq
2^{\frac{1-q}{q}}|\left\|X\right\|_{p}-\left\|Y\right\|_{p}|,$

where the first inequality is given by the stability theorem of hierarchical
stabilization (Proposition 2.5) and the second inequality is provided by the
characterization of interleaving distances between stable ranks in Proposition
5.5. An example where the second inequality is strict is provided by Example
5.7 for $p>1$, while an example where this is an equality is provided in the
case $Y=0$ by Lemma 4.23. A simple example in which the first inequality is
strict is provided instead by $X=K(0,1)$, $Y=K(0,2)$ and $q=2$.

######  Remark 5.8.

Since stable ranks are measurable functions $[0,\infty)\to[0,\infty)$, there
are many pseudometrics to compare them other than the interleaving distance
$d_{\bowtie}$. In particular, one can consider the standard
$L^{p}$-pseudometrics, here denoted by
$d_{p}(f,g):=\left(\int_{0}^{\infty}|f(t)-g(t)|^{p}\,dt\right)^{\frac{1}{p}}$.
As shown in [CR20, Prop. 2.1], the stability theorem of hierarchical
stabilization implies the following bounds for $d_{p}$:

$c\,d(X,Y)^{\frac{1}{p}}\geq
d_{p}(\widehat{\text{rank}}_{d}(X),\widehat{\text{rank}}_{d}(Y)),$

for any persistence modules $X$ and $Y$, where
$c:=\max\\{\operatorname{rank}(X),\operatorname{rank}(Y)\\}$ and $d$ denotes
any pseudometric between persistence modules. In this article we have chosen
to work with the interleaving distance between Wasserstein stable ranks
because of the strong stability result, expressed as a 1-Lipschitz condition.
Lipschitz stability for Wasserstein distances other than $W_{1}$ can not be
obtained for example by considering linear representations of persistence
diagrams [HKNU17, AEK+17, CWRW15, KFH17, RHBK15] as proved in Theorem 6.3 in
[ST20]. The trade-off between stability and the possibility of exploiting a
Banach or Hilbert space structure is still to be explored.

###  5.3 Metric learning

We have defined distances $d^{q}_{\mathcal{S}^{p,C}}$ between persistence
modules, parametrized by $q$, $p$ and by a contour $C$, and computable stable
rank invariants with corresponding stable interleaving distances. These
distances can be pulled back in $\operatorname{\mathsf{Tame}}$ via the
function $\widehat{\operatorname{rank}}_{d}$, with
$d=d^{q}_{\mathcal{S}^{p,C}}$, so that one can speak of interleaving distances
between persistence modules.

Recalling that the stable ranks depend on the pseudometric
$d^{q}_{\mathcal{S}^{p,C}}$, we now turn to the question of how to choose $p$
and $C$. The optimization of the parameter $q$ is not relevant, since it
determines a constant multiplicative factor to the distance of each pair of
persistence modules. We thus fix $q=1$ for a direct comparison with the
original framework of noise systems.

For brevity, we write $d:=d^{1}_{\mathcal{S}^{p,C}}$ and
$d_{\bowtie,p,C}(X,Y):=d_{\bowtie}(\widehat{\operatorname{rank}}_{d}(X),\widehat{\operatorname{rank}}_{d}(Y))$.
The field of metric learning provides a variety of loss functions suited for
different machine learning problems. For example, if we consider a simple
binary classification problem we have a dataset of persistence modules
$\\{X_{i}\\}_{i\in I}$ and the index set $I$ is partitioned into two sets $A$
and $B$, to represent the labeling. For this problem, a loss function (from
[ZW19]), designed to yield small intra-class distances and large inter-class
distances can be formulated as:

$\mathcal{L}=\frac{\sum_{i,j\in
A}(d_{\bowtie,p,C}(X_{i},X_{j}))^{2}}{\sum_{i\in A,j\in
I}(d_{\bowtie,p,C}(X_{i},X_{j}))^{2}}+\frac{\sum_{i,j\in
B}(d_{\bowtie,p,C}(X_{i},X_{j}))^{2}}{\sum_{i\in B,j\in
I}(d_{\bowtie,p,C}(X_{i},X_{j}))^{2}}$ (5.2)

In order to proceed we need to choose a family of contours that is practically
searchable when minimizing the loss function above. We work with contours of
distance type which are parametrized by densities (see Section 2.2). In turn,
in order to use gradient optimization methods, we want the densities to be
parametrized by a finite real-valued parameter vector. To this aim we choose
as densities unnormalized Gaussian mixtures
$f(x)=\sum_{i=1}^{k}\lambda_{i}\mathcal{N}(x|\mu_{i},\sigma_{i})$ for some
chosen $k$, where $\mathcal{N}$ is Gaussian with mean $\mu_{i}$ and standard
deviation $\sigma_{i}$, and $\lambda_{1}=1$.

In summary, the metric learning problem amounts to minimizing the loss
function with respect to a parameter vector $\theta\in\mathbb{R}^{3k}$, i.e.
$\theta=(\mu_{1},\ldots,\mu_{k},\sigma_{1},\ldots,\sigma_{k},\lambda_{2},\ldots,\lambda_{k},p)$,
designed to learn conjointly the parameter $p$ and the parameters of the
contour of the algebraic Wasserstein distance. The loss function is a simple
function of the pairwise interleaving distances between Wasserstein stable
ranks of persistence modules in the dataset. As can be seen in the expression
(5.1), the interleaving distance is the $L^{\infty}$ norm of differentiable
functions with respect to $\theta$ and is therefore differentiable almost
everywhere with respect to $\theta$, implying the same behavior for the loss
function. Hence the metric learning problem is amenable to gradient-based
optimization methods such as gradient descent.

##  6 Examples of analyses with Wasserstein stable ranks

In a first experiment, we show how varying the parameter $p$ affects the
distance space of the Wasserstein stable ranks and can serve as a way to
weight the importance of long versus short bars, for a set of synthetic
persistence modules. In a second experiment, we illustrate on a real-world
dataset how learning the parameter $p$ together with the parameters of a
contour can lead to more discriminative Wasserstein stable ranks in a
classification problem.

###  6.1 Synthetic data

A straightforward way to apply persistent homology in the context of computer
vision is to construct a complex (e.g. cubical complex) from the grid of
pixels constituting an image. The complex is then filtered based on the
grayscale intensity of the pixels (or based on the color channels for color
images).

It is easy to see that what should be considered as signal versus noise in
this context is highly dependent on the application. For example, for
classification of handwritten digits from the MNIST dataset [GT19, TNVL21] the
dominant topological features are often the most discriminative (for instance
the existence of a 1-dimensional cycle may be enough to distinguish between
digits $0$ and $1$). On the other hand, in biomedical imaging [CHLS18, QTT+19]
pathological states can translate into images with irregularities or lack of
homogeneity, associated with high numbers of short-lived components as
observed in [GHMM19].

Inspired by these applications, we construct two much simpler synthetic
datasets of images and associated persistence modules, with the goal of
illustrating the effect of choosing the parameter $p$ when using Wasserstein
stable ranks. The parameter $q$ is set to $1$ and the contour is fixed to be
the standard contour. In other words, we study the effect of the parameter $p$
on how the function $\widehat{\operatorname{rank}}_{d}$, with
$d=d^{1}_{\mathcal{S}^{p}}$, maps persistence modules onto the space of stable
ranks, endowed with the interleaving distance. Each dataset is composed of
$100$ images together with their class label, A or B. Each image is composed
of one block of high-intensity pixels and a number of blocks of low-intensity
pixels (while the size of the pixel blocks does not have a direct impact on
the following persistent homology analysis, the high-intensity block is made
larger for visual clarity, see Figures 7, 8). The images are represented as
cubical complexes on which super-level set filtration is performed and we
analyze the $H_{0}$ barcodes obtained from this process. Since we use pixel
intensity $[0,255]$ and super-level sets are used, the resulting filtration
scale is $[255,-\infty)$. This is capped to the minimum pixel value, $0$, and
transformed as $255-x$ to obtain a filtration scale $[0,255]$ as can be seen
in the barcodes in Figures 7, 8.

  * •

In Dataset 1 the pixels in the high-intensity block have slightly higher
intensity in images from class A (uniformly distributed between $245$ and
$255$) compared to images of class B (between $200$ and $210$). The low-
intensity blocks however follow the same distribution for images of both
classes (the number of blocks is uniformly distributed between $50$ and $100$
and the intensity is between $1$ and $10$). Sample images and barcodes are
shown in Figure 7.

  * •

In Dataset 2 on the other hand, the intensity of the high-intensity blocks
follows the same distribution for both classes (uniformly distributed between
$100$ and $255$). The number of low-intensity blocks however follows a
different distribution for Class A (between $20$ and $30$) and Class B
(between $120$ and $130$). Their intensity is the same for both classes
(between $1$ and $10$). Sample images and barcodes are shown in Figure 8.

Figure 7: Dataset 1. Left: Sample images from classes A and B. Right: $H_{0}$
barcodes corresponding to the sample images. Figure 8: Dataset 2. Left: Sample
images from classes A and B. Right: $H_{0}$ barcodes corresponding to the
sample images.

In terms of the barcodes, for Dataset 1 the signal is the single dominant
topological feature (the long bar, which varies slightly between the classes)
while the noise is composed of the numerous short bars. Choosing a value of
$p=\infty$ when generating the stable ranks thus effectively “denoises” the
barcodes and organizes the space of Wasserstein stable ranks in a way where
stable ranks of the same class are close to each other in interleaving
distance but far from elements of the other class. Stable ranks corresponding
to $p=1$ however fail to organize the corresponding distance space in this
clear-cut way, being too sensitive to the noisy short bars in the barcodes. To
illustrate this effect, in Figure 9 we show the hierarchical clustering (with
average linkage, similar results were observed for complete and single
linkage) corresponding to the distance spaces of Wasserstein stable ranks for
$p=1$ and $p=\infty$.

On the contrary, for Dataset 2 the signal is the number of short bars while
the noise is the single long bar. In this case a choice of $p=1$ organizes the
space of stable ranks such that elements of the same class cluster together,
while $p=\infty$, being too sensitive to the (for this dataset) noisy long
bar, fails to do so. This is illustrated in Figure 10. While with our
parameters the effect on the distance space is clear, some class-based
structure remains. By increasing the amount of noise it is however possible to
induce e.g. a nearest neighbor classifier to perform arbitrary poorly for the
$p=\infty$ while still distinguishing the classes for $p=1$ (and vice versa
for Dataset 1).

Figure 9: Dataset 1. Hierarchical clustering on the Wasserstein stable ranks
for $p=1$ (left) and $p=\infty$ (right) with respect to the interleaving
distance. The leaves (stable ranks in the dataset) are labeled and colored
according to their class.

Figure 10: Dataset 2. Hierarchical clustering on the Wasserstein stable ranks
for $p=1$ (left) and $p=\infty$ (right) with respect to the interleaving
distance. The leaves (stable ranks in the dataset) are labeled and colored
according to their class.

The choice of the parameter value $p$, which we have demonstrated can have a
large impact, is essentially related to the underlying distance between
persistence modules. Using Wasserstein-stable invariants however has
computational advantages, facilitates learning the right parameters for a
particular problems and allows for a richer use of machine learning methods as
we illustrate in the next section on a real-world dataset.

###  6.2 Brain artery data

In [BZM+10] a dataset of brain artery trees corresponding to 97 subjects aged
18 to 72 is introduced. Each data point is modeled as a tree embedded in
$\mathbb{R}^{3}$. In [BMM+16] the dataset is further analyzed with Topological
Data Analysis. To be able to apply sublevel set filtration on the tree, a
real-valued function is defined on the vertices as the height of the vertex in
the 3D-embedding. This is extended to a function on the edges by taking the
maximum value of the vertices connected by the edge. After applying persistent
homology, each data point is represented by a vector containing the sorted
lengths of the $100$ longest bars in a barcode decomposition of the
corresponding persistence module. This feature is further used to demonstrate,
among other things, an age effect of brain artery structure, by showing that
the projection of the vectors on the first principal component of the dataset
is correlated with age.

The authors note that using vectors of sorted length was computationally more
feasible than computing Wasserstein distances between the persistence diagrams
and they are more amenable to statistical analysis. In addition, the authors
observed that it was not necessary to use the whole vector of lengths to
establish the correlation and in fact the topological features of medium
length, rather than the longest ones, were the most discriminatory. Finally,
since only length is used, potentially discriminative aspects of persistent
homology related to when in the filtration scale a feature appears are not
considered.

Analyzing the dataset with stable ranks offers computational and statistical
advantages. Moreover, for this problem where the discriminative information is
not contained in the most persistent feature, considering other distances than
the bottleneck ($p=\infty$) and more generally tuning the parameter $p$ might
be beneficial. Finally, combining the tuning of the parameter $p$ with a
contour might increase the power of the method. Indeed the parameter $p$ and
the contour, intuitively are related to different features of a persistence
barcode: while the parameter $p$ globally weights the importance of long
versus short bars as illustrated in Section 6.1, the contour highlights parts
of the parameter space. While we also study age effects of brain artery
structure, we choose to binarize the problem by creating two classes: young
($age<45$, 50 subjects) and old ($age\geq 45$, 47 subjects) and treat the
problem as a classification, allowing us to apply the metric learning method
described in Section 5.3. Using folded cross-validation (with $97$ folds, i.e.
LOOCV), for each training fold we learn the metric that optimally separates
training samples from the two classes by minimizing the loss defined in (5.2).
For simplicity we then classify using the $k$-nearest neighbors algorithm
[PVG+11]: for each sample in the test fold we compute its distance to all the
samples in the training fold, using the learned metric. We can then classify
the test sample to the majority class of the $k$ nearest elements in the
learned distance space ($k$ is treated as a hyperparameter and tuned on the
training fold).

For the metric learning, the contours are parametrized by densities which are
unnormalized Gaussian mixtures with two components. The loss function is
implemented in PyTorch [PGM+19]. After a random initialization of the
parameters, projected gradient descent (to respect the constraints $p\geq
1,\lambda_{i},\sigma_{i}>0$) with momentum is used to achieve a lower loss. An
example of an optimization on a training fold over $25000$ iterations is shown
in Figure 11.

Figure 11: Results for one example run of the metric learning optimization for
Wasserstein stable ranks (see Section 5.3) over $25000$ iterations. Top Left:
Progression of the loss function over the iterations. Top Middle, Top Right,
Bottom Left, Bottom Middle: Progression of the parameters in
$\theta=(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\lambda_{2},p)$ parametrizing
Wasserstein stable ranks: $p$, mean $\mu_{i}$, standard deviation $\sigma_{i}$
and $\lambda_{2}$ respectively over the iterations. Bottom Right: Density at
different iterations.

The metric learning is effective in finding distances that improve the
classification performance: running the optimization problem not only
decreases the loss (as is seen in Figure 11 in the top left plot) but also
decreases the corresponding classification error, from errors between $28.9\%$
and $55.7\%$ (for 10 random initialization points used) to an error of $24\%$
corresponding to the parameters that the optimization converged to. This is an
improvement also compared to the standard stable rank (standard contour and
$p=\infty$), for which the classification error is $38.1\%$. While varying the
parameter $p$ only (when holding the standard contour fixed) has an impact
(error varies between $27.8\%$ and $45.4\%$), it is when we learn $p$ and the
contour conjointly that the lowest loss and corresponding classification error
is achieved.

Figure 12: Sample barcodes from the two classes with superposed learned
density. Bars are colored according to the density.

The optimal parameters found with the metric learning method are of interest
because they allow to construct a distance space in which machine learning
methods can be carried out, but they are also interpretable: they contain
information about which features of the dataset are important to distinguish
the two classes. This is illustrated in Figure 12 where two sample barcodes –
one from each class – are displayed with the optimal density superposed and
the bars colored according to the density. From the insight that some parts of
the filtration scale are more important in distinguishing younger from older
subjects, one may pursue the analysis by looking for characteristics of bars
in that region of the barcode. One can also take the analysis a step further
by looking at the object from which the filtered simplicial complex was
created. In our case, since the filtration scale corresponds to the height
(z-coordinate) in the 3D-embedding of the brain artery tree, one may for
example investigate whether differences in brain artery between subjects of
different ages in this particular region carries a biological meaning.

## Acknowledgements

This work was partially supported by the Swedish Research Council
(Vetenskapsrådet) and the Wallenberg AI, Autonomous Systems and Software
Program (WASP) funded by the Knut and Alice Wallenberg Foundation.

## References

  * [AEK+17] Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier.  Persistence images: A stable vector representation of persistent homology.  Journal of Machine Learning Research, 18, 2017.
  * [AM21] Henry Adams and Michael Moy.  Topology applied to machine learning: from global to local.  Frontiers in Artificial Intelligence, page 54, 2021.
  * [ARSC21] Jens Agerberg, Ryan Ramanujam, Martina Scolamiero, and Wojciech Chachólski.  Supervised learning using homology stable rank kernels.  Frontiers in Applied Mathematics and Statistics, 7, 2021.
  * [BDSS15] Peter Bubenik, Vin De Silva, and Jonathan Scott.  Metrics for generalized persistence modules.  Foundations of Computational Mathematics, 15(6):1501–1531, 2015\.
  * [BL15] Ulrich Bauer and Michael Lesnick.  Induced matchings and the algebraic stability of persistence barcodes.  Journal of Computational Geometry, 6(2):162–191, 2015.
  * [BM21] Peter Bubenik and Nikola Milićević.  Homological algebra for persistence modules.  Foundations of Computational Mathematics, 21(5):1233–1278, 2021\.
  * [BMM+16] Paul Bendich, James S Marron, Ezra Miller, Alex Pieloch, and Sean Skwerer.  Persistent homology analysis of brain artery trees.  The annals of applied statistics, 10(1):198, 2016.
  * [BPP+22] Lev Buhovsky, Jordan Payette, Iosif Polterovich, Leonid Polterovich, Egor Shelukhin, and Vukašin Stojisavljević.  Coarse nodal count and topological persistence.  arXiv preprint arXiv:2206.06347, 2022.
  * [BSS22] Peter Bubenik, Jonathan Scott, and Donald Stanley.  Exact weights, path metrics, and algebraic Wasserstein distances.  Journal of Applied and Computational Topology, 2022.
  * [BZM+10] Elizabeth Bullitt, Donglin Zeng, Benedicte Mortamet, Arpita Ghosh, Stephen R Aylward, Weili Lin, Bonita L Marks, and Keith Smith.  The effects of healthy aging on intracerebral blood vessels visualized by magnetic resonance angiography.  Neurobiology of aging, 31(2):290–300, 2010.
  * [CCI+20] Mathieu Carrière, Frédéric Chazal, Yuichi Ike, Théo Lacombe, Martin Royer, and Yuhei Umeda.  Perslay: A neural network layer for persistence diagrams and new graph topological signatures.  In International Conference on Artificial Intelligence and Statistics, pages 2786–2796. PMLR, 2020.
  * [CDSGO16] Frédéric Chazal, Vin De Silva, Marc Glisse, and Steve Oudot.  The structure and stability of persistence modules, volume 10.  Springer, 2016.
  * [CHLS18] Yu-Min Chung, Chuan-Shen Hu, Austin Lawson, and Clifford Smyth.  Topological approaches to skin disease image analysis.  In 2018 IEEE International Conference on Big Data (Big Data), pages 100–105. IEEE, 2018.
  * [CR20] Wojciech Chachólski and Henri Riihimaki.  Metrics and stabilization in one parameter persistence.  SIAM Journal on Applied Algebra and Geometry, 4(1):69–98, 2020\.
  * [CSEHM10] David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Yuriy Mileyko.  Lipschitz functions have l p-stable persistence.  Foundations of computational mathematics, 10(2):127–139, 2010.
  * [CSEM06] David Cohen-Steiner, Herbert Edelsbrunner, and Dmitriy Morozov.  Vines and vineyards by updating persistence in linear time.  In Proceedings of the twenty-second annual symposium on Computational geometry, pages 119–126, 2006.
  * [CWRW15] Yen-Chi Chen, Daren Wang, Alessandro Rinaldo, and Larry Wasserman.  Statistical analysis of persistence intensity functions.  arXiv preprint arXiv:1510.02502, 2015.
  * [dSMS18] Vin de Silva, Elizabeth Munch, and Anastasios Stefanou.  Theory of interleavings on categories with a flow.  Theory and Applications of Categories, 33(21):583–607, 2018.
  * [ELZ00] Herbert Edelsbrunner, David Letscher, and Afra Zomorodian.  Topological persistence and simplification.  In Proceedings 41st annual symposium on foundations of computer science, pages 454–463. IEEE, 2000.
  * [Gäv18] Oliver Gävfert.  Topology-based metric learning.  Conference poster. Available: https://people.kth.se/~oliverg/TAGS_poster.pdf, 2018\.
  * [GC17] Oliver Gäfvert and Wojciech Chachólski.  Stable invariants for multidimensional persistence.  arXiv preprint arXiv:1703.03632, 2017.
  * [GHMM19] Kathryn Garside, Robin Henderson, Irina Makarenko, and Cristina Masoller.  Topological data analysis of high resolution diabetic retinopathy images.  PloS one, 14(5):e0217413, 2019.
  * [GNOW21] Barbara Giunti, John S Nolan, Nina Otter, and Lukas Waas.  Amplitudes in abelian categories.  arXiv preprint arXiv:2107.09036, 2021.
  * [GT19] Adélie Garin and Guillaume Tauzin.  A topological" reading" lesson: Classification of mnist using tda.  In 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA), pages 1551–1556. IEEE, 2019.
  * [HKNU17] Christoph Hofer, Roland Kwitt, Marc Niethammer, and Andreas Uhl.  Deep learning with topological signatures.  Advances in neural information processing systems, 30, 2017.
  * [HNH+16] Yasuaki Hiraoka, Takenobu Nakamura, Akihiko Hirata, Emerson G Escolar, Kaname Matsue, and Yasumasa Nishiura.  Hierarchical structures of amorphous solids characterized by persistent homology.  Proceedings of the National Academy of Sciences, 113(26):7035–7040, 2016.
  * [KFH17] Genki Kusano, Kenji Fukumizu, and Yasuaki Hiraoka.  Kernel method for persistence diagrams via kernel embedding and weight factor.  The Journal of Machine Learning Research, 18(1):6947–6987, 2017\.
  * [KMN17] Michael Kerber, Dmitriy Morozov, and Arnur Nigmetov.  Geometry helps to compare persistence diagrams, 2017.
  * [Les15] Michael Lesnick.  The theory of the interleaving distance on multidimensional persistence modules.  Foundations of Computational Mathematics, 15(3):613–650, 2015.
  * [OPT+17] Nina Otter, Mason A Porter, Ulrike Tillmann, Peter Grindrod, and Heather A Harrington.  A roadmap for the computation of persistent homology.  EPJ Data Science, 6:1–38, 2017.
  * [PGM+19] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al.  Pytorch: An imperative style, high-performance deep learning library.  Advances in neural information processing systems, 32, 2019.
  * [PVG+11] Fabian Pedregosa, Gaël Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, et al.  Scikit-learn: Machine learning in python.  the Journal of machine Learning research, 12:2825–2830, 2011.
  * [QTT+19] Talha Qaiser, Yee-Wah Tsang, Daiki Taniyama, Naoya Sakamoto, Kazuaki Nakane, David Epstein, and Nasir Rajpoot.  Fast and accurate tumor segmentation of histology images using persistent homology and deep convolutional features.  Medical image analysis, 55:1–14, 2019.
  * [RCB21] Raphael Reinauer, Matteo Caorsi, and Nicolas Berkouk.  Persformer: A transformer architecture for topological machine learning.  arXiv preprint arXiv:2112.15210, 2021.
  * [RHBK15] Jan Reininghaus, Stefan Huber, Ulrich Bauer, and Roland Kwitt.  A stable multi-scale kernel for topological machine learning.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4741–4748, 2015.
  * [Rot09] Joseph Rotman.  An Introduction to Homological Algebra.  Springer-Verlag New York, 2nd edition, 2009.
  * [SCL+17] Martina Scolamiero, Wojciech Chachólski, Anders Lundman, Ryan Ramanujam, and Sebastian Öberg.  Multidimensional persistence and noise.  Foundations of Computational Mathematics, 17(6):1367–1406, 2017\.
  * [SHP17] Bernadette J Stolz, Heather A Harrington, and Mason A Porter.  Persistent homology of time-dependent functional networks constructed from coupled time series.  Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(4):047410, 2017.
  * [ST20] Primoz Skraba and Katharine Turner.  Wasserstein stability for persistence diagrams.  arXiv preprint arXiv:2006.16824, 2020.
  * [Ste04] J Michael Steele.  The Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities.  Cambridge University Press, 2004.
  * [TNVL21] Renata Turkeš, Jannes Nys, Tim Verdonck, and Steven Latré.  Noise robustness of persistent homology on greyscale images, across filtrations and signatures.  PloS one, 16(9):e0257215, 2021.
  * [Vin90] A Vince.  A rearrangement inequality and the permutahedron.  The American mathematical monthly, 97(4):319–323, 1990.
  * [ZC05] Afra Zomorodian and Gunnar Carlsson.  Computing persistent homology.  Discrete Comput Geom, 33:249–274, 2005.
  * [ZW19] Qi Zhao and Yusu Wang.  Learning metrics for persistence-based summaries and applications for graph classification.  arXiv preprint arXiv:1904.12189, 2019.

Department of Mathematics, KTH, S-10044 Stockholm, Sweden
<EMAIL_ADDRESS>
Corresponding author: Martina Scolamiero<EMAIL_ADDRESS>