# Hamiltonian Vector Fields on Multiphase Spaces
of Classical Field Theory ††thanks: Work partially supported by CNPq (Conselho
Nacional de Desenvolvimento Científico e Tecnológico), Brazil

Michael Forger 1 and Mário Otávio Salles 2 Work done in partial fulfillment of
the requirements for the degree of Doctor in Science

(${}^{1}\,$ Departamento de Matemática Aplicada,
Instituto de Matemática e Estatística,
Universidade de São Paulo,
Caixa Postal 66281,
BR–05315-970 São Paulo, S.P., Brazil
${}^{2}\,$ Faculdade de Tecnologia de Carapicuiba
Av. Francisco Pignatari, 650,
BR–06390-310 Carapicuiba, S.P., Brazil )

###### Abstract

We present a classification of hamiltonian vector fields on multisymplectic
and polysymplectic fiber bundles closely analogous to the one known for the
corresponding dual jet bundles that appear in the multisymplectic and
polysymplectic approach to first order classical field theories.

Universidade de São Paulo

RT-MAP-0802

August 2024

##  1 Introduction

The quest for a fully covariant hamiltonian formulation of classical field
theory has a long history. In particular, the search for a first order
formalism analogous to that for classical mechanics in terms of concepts from
symplectic geometry has stimulated the development of new geometric tools
usually referred to as “multisymplectic” or “polysymplectic” structures whose
real significance is emerging only gradually. In fact, for many years there
has not even been a convincing general definition, although a standard class
of examples in terms of duals of jet bundles has long been known and widely
used.111This situation has been strikingly similar to that in classical
mechanics before it was realized that symplectic manifolds, rather than just
cotangent bundles, provide an adequate framework if one wants to accomodate
phenomena such as half-integral spin within classical mechanics. This defect
has recently been overcome [1], and it has been realized that both
multisymplectic and polysymplectic structures (which are not the same thing)
play an important role in the formalism; in particular, a multisymplectic
structure always induces a special kind of polysymplectic structure by means
of a construction called the “symbol”. But the fact that both types come
together to form a pair has been anticipated almost 20 years ago [2], when it
became apparent that the covariant hamiltonian formulation of first order
classical field theories requires the simultaneous use of two types of
“multiphase space” that we shall refer to as “ordinary multiphase space” and
“extended multiphase space”, respectively.

To be more precise, let us briefly recall the cornerstones of the construction
of the two types of multiphase space for first order lagrangian field
theories; for more details, the reader is referred to [2, 3, 4]. The starting
point is the choice of a fiber bundle $E$ over the space-time manifold $M$
called the configuration bundle because its sections represent the basic
fields of the theory at hand. Next, one takes the first order jet bundle $JE$
of $E$ to accomodate first order derivatives of these fields: this is an
affine bundle over $E$ and is also the domain of definition of the lagrangian.
Besides, one also considers the linearized first order jet bundle $\vec{J}E$
of $E$: this is a vector bundle over $E$ defined as the difference vector
bundle of $JE$. Finally, as in mechanics, one uses appropriate versions of the
Legendre transformation induced by the given lagrangian to pass to the
(twisted) affine dual
$J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$ of $JE$ and to
the (twisted) linear dual
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$ of
$\vec{J}E$: the former is the extended multiphase space and the latter is the
ordinary multiphase space of the theory. Note that the former is an affine
line bundle over the latter and that the hamiltonian obtained from the given
lagrangian through Legendre transformation is not a function but rather a
section of this affine line bundle, so both of these multiphase spaces are
essential ingredients for defining the concept of a hamiltonian system in
field theory! Moreover, it is well known that
$J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$ carries a
naturally defined multisymplectic form $\,\omega$. However, what does not seem
to have been so widely noticed is the fact that
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$ carries a
naturally defined polysymplectic form $\,\hat{\omega}$ – even though this form
already appears explicitly in Ref. [5]. As has been shown more recently [1],
it can be derived from the multisymplectic form $\,\omega$ on
$J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$ by taking its
symbol, which turns out to be degenerate precisely along the fibers of the
aforementioned affine line bundle, and then passing to the corresponding
quotient of $J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$ by
the kernel of $\,\omega$, which is precisely
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$. Note
that this polysymplectic form $\,\hat{\omega}$ on
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$ is
canonical, whereas the form $\,\omega_{\mathscr{H}}$ on
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$ obtained
as the pull-back of $\,\omega$ by means of a hamiltonian section
$\;\mathscr{H}:\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E\rightarrow
J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E\;$ is not, since
it depends on the choice of hamiltonian.222It should be noted that the form
$\,\omega_{\mathscr{H}}$ is closed and non-degenerate but not multisymplectic
in the sense of the definition given in Ref. [1].

In terms of adapted local coordinates $(x^{\mu},q^{i},p\>\\!_{i}^{\mu},p)$ for
$J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$ and
$(x^{\mu},q^{i},p\>\\!_{i}^{\mu})$ for
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$, induced
by local coordinates $x^{\mu}$ for $M$, local coordinates $q^{i}$ for the
typical fiber $Q$ of $E$ and a local trivialization of $E$ [4], we have

$\begin{array}[]{c}\mbox{extended multiphase space
$J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$}\\\\[2.84526pt]
\mbox{adapted local coordinates
$(x^{\mu},q^{i},p\>\\!_{i}^{\mu},p)$}\\\\[2.84526pt] \mbox{multisymplectic
form}\quad\omega~{}=~{}dq^{i}\,{\scriptstyle\wedge\,}\,dp\>\\!_{i}^{\mu}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu}\,-\;dp\,\,{\scriptstyle\wedge\,}\,d^{\,n}x\end{array}$
(1)

and

$\begin{array}[]{c}\mbox{ordinary multiphase space
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$}\\\\[2.84526pt]
\mbox{adapted local coordinates
$(x^{\mu},q^{i},p\>\\!_{i}^{\mu})$}\\\\[2.84526pt] \mbox{polysymplectic
form}\quad\hat{\omega}~{}=~{}dq^{i}\,{\scriptstyle\wedge\,}\,dp\>\\!_{i}^{\mu}\,\,{\scriptstyle\otimes}\,\,d^{\,n}x_{\mu}\end{array}$
(2)

where $p\,$ is (except for a sign) a scalar energy variable and $d^{\,n}x$ is
the (local) volume form induced by the $x^{\mu}$ while $d^{\,n}x_{\mu}$ is the
(local) $(n\\!-\\!1)$-form obtained by contracting $d^{\,n}x$ with
$\,\partial_{\mu}\equiv\partial/\partial x^{\mu}$:

$d^{\,n}x_{\mu}~{}=~{}i_{\partial_{\mu}}\,d^{\,n}x~{}.$

The same picture prevails in the general case if we replace adapted local
coordinates by Darboux coordinates; see [1]. _Extended multiphase space is
multisymplectic, ordinary multiphase space is polysymplectic._

A crucial role in the development of the hamiltonian formalism is played by
the notion of a hamiltonian vector field. According to the picture outlined
above, this comes in two variants: a multisymplectic one and a polysymplectic
one. We shall deal with the two versions separately, beginning with the
pertinent definitions.

##  2 The multisymplectic case

According to Ref. [1], a multisymplectic fiber bundle of rank $N$ can be
defined as a fiber bundle $P$ over an $n$-dimensional base manifold $M$
equipped with a closed, non-degenerate $(n+1)$-form $\,\omega$ on its total
space $P$ which (a) is $(n-1)$-horizontal, i.e., such that its contraction
with any three vertical vector fields vanishes, and (b) admits a
multilagrangian distribution, i.e., an isotropic vector subbundle $L$ of the
vertical bundle $VP$ of $P$ of codimension $N$ and dimension $Nn+1$. (It then
turns out that $P$ has dimension $(N+1)(n+1)$.) Assuming this distribution to
be involutive, which is automatic as soon as $\,n\geqslant 3\,$ but has to be
imposed as a separate condition when $\,n=2\,$, Darboux’s theorem assures that
there exist local coordinates, called canonical local coordinates or Darboux
coordinates, in which $\,\omega$ assumes the form

$\omega~{}=~{}dq^{i}\,{\scriptstyle\wedge\,}\,dp\>\\!_{i}^{\mu}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu}\,-\;dp\,\,{\scriptstyle\wedge\,}\,d^{\,n}x~{}.$
(3)

Locally, $\,\omega$ is exact, i.e.,

$\omega\;=\;-\,d\theta~{},$ (4)

where $d$ denotes the exterior derivative, with

$\theta~{}=~{}p\>\\!_{i}^{\mu}\;dq^{i}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu}\,+\;p\;d^{\,n}x~{}.$
(5)

The standard example is that of the extended multiphase space
$J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$ mentioned above,
for which $\,\omega$ is also globally exact, i.e., the so-called
multicanonical form $\theta$ in equations (4) and (5) is globally defined, and
$L$ is the vector subbundle of $VP$ generated by the vector fields
$\partial/\partial p\>\\!_{i}^{\mu}$ and $\partial/\partial p$, that is, the
vertical bundle for the projection of
$J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$ onto $E$ (with
respect to which $J^{\raisebox{0.1507pt}{$\scriptscriptstyle\bigcirc$}\star}E$
is a vector bundle).

Given this situation, we say that a vector field $X$ on $P$ is _locally
hamiltonian_ if $i_{X}\omega$ is closed, or equivalently, if

$L_{X}\omega~{}=~{}0~{}.$ (6)

It is called _globally hamiltonian_ if $i_{X}\omega$ is exact, that is, if
there exists an $(n-1)$-form $f$ on $P$ such that

$i_{X}\omega~{}=~{}df~{}.$ (7)

In this case, $f$ is said to be a _hamiltonian form associated with_ $X$.
Finally, when $\,\omega$ is exact and given by equation (4), $X$ is called
_exact hamiltonian_ if

$L_{X}\theta~{}=~{}0~{}.$ (8)

The main theorem states that these vector fields can be classified in terms of
their components with respect to canonical local coordinates, which are given
by the expansion

$X~{}=~{}X^{\mu}\,\frac{\partial}{\partial
x_{\phantom{i}}^{\mu}}\,+\,X^{i}\,\frac{\partial}{\partial
q^{i}}\,+\,X_{i}^{\mu}\,\frac{\partial}{\partial
p\>\\!_{i}^{\mu}}\,+\,X_{0}\,\frac{\partial}{\partial p}~{},$ (9)

whereas, locally, the hamiltonian form corresponding to such a vector field,
which is determined up to an arbitrary closed form, can be assumed to have an
expansion of the form

$f~{}=~{}f^{\mu}\;d^{\,n}x_{\mu}\,+\,{\textstyle\frac{1}{2}}\,f_{i}^{\mu\nu}\;dq^{i}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu\nu}~{},$
(10)

where

$d^{\,n}x_{\mu\nu}~{}=~{}i_{\partial_{\nu}}i_{\partial_{\mu}}\,d^{\,n}x~{}.$

An easy calculation gives

$\begin{array}[]{rcl}i_{X}\omega\\!\\!&=&\\!\\!X^{\nu}\;dq^{i}\,{\scriptstyle\wedge\,}\,dp\>\\!_{i}^{\mu}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu\nu}\,-\,X_{i}^{\mu}\;dq^{i}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu}\,+\,X^{i}\;dp\>\\!_{i}^{\mu}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu}\\\\[5.69054pt]
&&\\!\\!\mbox{}+\,X^{\mu}\;dp\,\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu}\,-\,X_{0}\;d^{\,n}x~{},\end{array}$
(11)

and in the exact case

$i_{X}\theta~{}=~{}(p\>\\!_{i}^{\mu}X^{i}\,+\,p\,X^{\mu})\;d^{\,n}x_{\mu}\,-\,p\>\\!_{i}^{\mu}X^{\nu}\;dq^{i}\,{\scriptstyle\wedge\,}\,d^{\,n}x_{\mu\nu}~{}.$
(12)

These formulas constitute the starting point for the proof of the following

###### Theorem 1

A vector field $X$ on $P$ is locally hamiltonian if and only if its components
$X^{\mu}$, $X^{i}$, $X_{i}^{\mu}$ and $X_{0}$ with respect to canonical local
coordinates, as defined by equation (9), satisfy the following conditions:

  1. 1.

the coefficients $X^{\mu}$ and $X^{i}$ are independent of the multimomentum
variables $p\>\\!_{k}^{\kappa}$ and of the energy variable $p$, with the
coefficients $X^{\mu}$ depending only on the local coordinates $x^{\kappa}$ of
the base manifold $M$ as soon as $\,N>1$,

  2. 2.

the remaining coefficients $X_{i}^{\mu}$ and $X_{0}$ can be expressed in terms
of the previous ones and of new coefficients $f_{0}^{\mu}$ which are also
independent of the multimomentum variables $p\>\\!_{k}^{\kappa}$ and of the
energy variable $p$, according to

$X_{i}^{\mu}~{}=\;-\,p\;\frac{\partial X^{\mu}}{\partial
q^{i}}\,-\,p\>\\!_{j}^{\mu}\;\frac{\partial X^{j}}{\partial
q^{i}}\,+\,p\>\\!_{i}^{\nu}\;\frac{\partial X^{\mu}}{\partial
x^{\nu}}\,-\,p\>\\!_{i}^{\mu}\;\frac{\partial X^{\nu}}{\partial
x^{\nu}}\,+\,\frac{\partial f_{0}^{\mu}}{\partial q^{i}}~{},$ (13)

(the first term being absent as soon as $\,N>1$) and

$X_{0}~{}=\;-\,p\;\frac{\partial X^{\mu}}{\partial
x^{\mu}}\,-\,p\>\\!_{i}^{\mu}\;\frac{\partial X^{i}}{\partial
x^{\mu}}\,+\,\frac{\partial f_{0}^{\mu}}{\partial x^{\mu}}~{}.$ (14)

The components of the corresponding hamiltonian form $f$ are given by

$f^{\mu}~{}=~{}p\>\\!_{i}^{\mu}X^{i}\,+\,p\,X^{\mu}\,+\,f_{0}^{\mu}~{},$ (15)

and

$f_{i}^{\mu\nu}~{}=~{}p\>\\!_{i}^{\nu}X^{\mu}\,-\,p\>\\!_{i}^{\mu}X^{\nu}~{}.$
(16)

In addition, when $\,\omega$ is exact and given by equation (4), $X$ is exact
hamiltonian if and only if the coefficients $f_{0}^{\mu}$ vanish.

This theorem has been explicitly stated in Ref. [6] and an explicit proof of a
more general theorem (where vector fields are replaced by multivector fields)
can be found in Ref. [7].

##  3 The polysymplectic case

According to Ref. [1], a polysymplectic fiber bundle of rank $N$ can be
defined as a fiber bundle $P$ over an $n$-dimensional base manifold $M$
equipped with a vertically closed, non-degenerate vertical $2$-form
$\,\hat{\omega}$ on its total space $P$ which (a) takes values in (the pull-
back $\pi^{*}\hat{T}$ to $P$ of) some given $\hat{n}$-dimensional coefficient
vector bundle $\hat{T}$ over $M$ and (b) admits a polylagrangian distribution,
i.e., an isotropic vector subbundle $L$ of the vertical bundle $VP$ of $P$ of
codimension $N$ and dimension $N\hat{n}$. (It then turns out that $P$ has
dimension $N\hat{n}+N+\hat{n}$.) Assuming this distribution to be involutive,
which is automatic as soon as $\,\hat{n}\geqslant 3\,$ but has to be imposed
as a separate condition when $\,\hat{n}=2\,$, Darboux’s theorem assures that
given any basis $\,\\{\,\hat{e}_{a}\,/\,1\leqslant a\leqslant\hat{n}\,\\}\,$
of local sections of $\hat{T}$, there exist local coordinates, called
canonical local coordinates or Darboux coordinates, in which $\,\hat{\omega}$
assumes the form

$\hat{\omega}~{}=~{}dq^{i}\,{\scriptstyle\wedge\,}\,dp\>\\!_{i}^{a}\,\,{\scriptstyle\otimes}\,\,\hat{e}_{a}~{}.$
(17)

Locally, $\,\hat{\omega}$ is vertically exact, i.e.,

$\hat{\omega}\;=\;-\,d_{V}\hat{\theta}~{},$ (18)

where $d_{V}$ denotes the vertical exterior derivative, with

$\hat{\theta}~{}=~{}p\>\\!_{i}^{a}\;dq^{i}\,\,{\scriptstyle\otimes}\,\,\hat{e}_{a}~{}.$
(19)

The standard example is that of the ordinary multiphase space
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$ mentioned
above, with
$\,\hat{T}=\raisebox{0.86108pt}{${\textstyle\bigwedge}$}^{n-1}T^{*}M\,$ and
$\,e_{a}=d^{\,n}x_{\mu}$, for which $\,\hat{\omega}$ is also globally
vertically exact, i.e., the so-called polycanonical form $\hat{\theta}$ in
equations (18) and (19) is globally defined, and $L$ is the vector subbundle
of $VP$ generated by the vector fields $\partial/\partial p\>\\!_{i}^{\mu}$,
that is, the vertical bundle for the projection of
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$ onto $E$
(with respect to which
$\vec{J}^{\,\raisebox{0.3014pt}{$\scriptscriptstyle\bigcirc$}\ast}E$ is a
vector bundle).

Given this situation, we say that a vertical vector field $X$ on $P$ is
_locally hamiltonian_ if $i_{X}\hat{\omega}$ is vertically closed, or
equivalently, if

$L_{X}\hat{\omega}~{}=~{}0~{}.$ (20)

It is called _globally hamiltonian_ if $i_{X}\hat{\omega}$ is vertically
exact, that is, if there exists a section $f$ of the vector bundle
$\pi^{*}\hat{T}$ over $P$ such that

$i_{X}\hat{\omega}~{}=~{}d_{V}f~{}.$ (21)

In this case, $f$ is said to be a _hamiltonian section associated with_ $X$.
Finally, when $\,\hat{\omega}$ is vertically exact and given by equation (18),
$X$ is called _exact hamiltonian_ if

$L_{X}\hat{\theta}~{}=~{}0~{}.$ (22)

The main theorem states that these vector fields can be classified in terms of
their components with respect to canonical local coordinates, which are given
by the expansion

$X~{}=~{}X^{i}\,\frac{\partial}{\partial
q^{i}}\,+\,X_{i}^{a}\,\frac{\partial}{\partial p\>\\!_{i}^{a}}$ (23)

whereas, locally, the hamiltonian section corresponding to such a vector
field, which is determined up to (the pull-back to $P$ of) an arbitrary
section of $\hat{T}$, can be assumed to have an expansion of the form

$f~{}=~{}f^{a}\;\hat{e}_{a}~{}.$ (24)

An easy calculation gives

$i_{X}\hat{\omega}~{}=\;-\,X_{i}^{a}\;dq^{i}\,\,{\scriptstyle\otimes}\,\,\hat{e}_{a}\,+\,X^{i}\;dp\>\\!_{i}^{a}\,\,{\scriptstyle\otimes}\,\,\hat{e}_{a}~{},$
(25)

and in the exact case

$i_{X}\hat{\theta}~{}=~{}X^{i}p\>\\!_{i}^{a}\;\hat{e}_{a}~{}.$ (26)

These formulas constitute the starting point for the proof of the following

###### Theorem 2

A vector field $X$ on $P$ is locally hamiltonian if and only if its components
$X^{i}$ and $X_{i}^{\mu}$ with respect to canonical local coordinates, as
defined by equation (23), satisfy the following conditions:

  1. 1.

the coefficients $X^{i}$ are independent of the multimomentum variables
$p\>\\!_{k}^{\kappa}$,

  2. 2.

the remaining coefficients $X_{i}^{a}$ can be expressed in term of the
previous ones and of new coefficients $f_{0}^{a}$ which are also independent
of the multimomentum variables $p\>\\!_{k}^{\kappa}$, according to

$X_{i}^{a}~{}=\;-\,p\>\\!_{j}^{a}\;\frac{\partial X^{j}}{\partial
q^{i}}\,+\,\frac{\partial f_{0}^{a}}{\partial q^{i}}~{}.$ (27)

The components of the corresponding hamiltonian section $f$ are given by

$f^{a}~{}=~{}p\>\\!_{i}^{a}X^{i}\,+\,f_{0}^{a}~{},$ (28)

In addition, when $\,\hat{\omega}$ is vertically exact and given by equation
(18), $X$ is exact hamiltonian if and only if the coefficients $f_{0}^{a}$
vanish.

The proof of this theorem is entirely analogous to that of the previous one,
except that it is much simpler. The essence of the argument can already be
found in Ref. [8, 9], but the proper global context of the result is not
adequately treated there.

##  4 Outlook

The analogous problem of determining hamiltonian vector fields with respect to
the form $\,\omega_{\mathscr{H}}$ on ordinary multiphase space mentioned in
the introduction has been addressed and solved in Ref. [10], but the results
are somewhat complicated and not very enlightening. We now believe this to be
related to the fact that, according to the structurally natural definition
given in Ref. [1], $\omega_{\mathscr{H}}$ is _not_ multisymplectic.

One problem that, for the time being, remains open is to give a global,
coordinate independent formulation of the results of Theorems 1 and 2. This
question is presently under investigation.

## References

  * [1] M. Forger & L. Gomes: _Multisymplectic and Polysymplectic Structures on Fiber Bundles_ , arXiv:0708.1586.
  * [2] J.F. Cariñena, M. Crampin & L.A. Ibort: _On the Multisymplectic Formalism for First Order Field Theories_ , Diff. Geom. Appl. 1 (1991) 345-374.
  * [3] M.J. Gotay, J. Isenberg, J.E. Marsden & R. Montgomery: _Momentum Maps and Classical Relativistic Fields. Part I: Covariant Field Theory_, arXiv:physics/9801019v2.
  * [4] M. Forger & S.V. Romero: _Covariant Poisson Brackets in Geometric Field Theory_ , Commun. Math. Phys. 256 (2005) 375-410, arXiv:math-ph/0408008.
  * [5] C. Günther: _The Polysymplectic Hamiltonian Formalism in Field Theory and Calculus of Variations I: the Local Case_ , J. Diff. Geom. 25 (1987) 23-53.
  * [6] M. Forger & H. Römer: _A Poisson Bracket on Multisymplectic Phase Space_ , Rep. Math. Phys. 48 (2001) 211-218, arXiv:math-ph/0009037.
  * [7] M. Forger, C. Paufler & H. Römer: _Hamiltonian Multivector Fields and Poisson Forms in Multisymplectic Field Theory_ , J. Math. Phys. 46 (2005) 112903, 29 pp., arXiv:math-ph/0407057.
  * [8] I. Kanatchikov: _On Field Theoretic Generalizations of a Poisson Algebra_ , Rep. Math. Phys. 40 (1997) 225-234, arXiv:hep-th/9710069.
  * [9] I. Kanatchikov: _Canonical Structure of Classical Field Theory in the Polymomentum Phase Space_ , Rep. Math. Phys. 41 (1998) 49-90, arXiv:hep-th/9709229.
  * [10] M.O. Salles: _Campos Hamiltonianos e Colchete de Poisson na Teoria Geométrica dos Campos_ , PhD thesis, IME-USP, June 2004.


11institutetext: SAILAB, University of Siena http://sailab.diism.unisi.it
22institutetext: University of Pavia 22email:
<EMAIL_ADDRESS>MAASAI, Universitè
Côte d’Azur 33email<EMAIL_ADDRESS>
# Clustering-Based Interpretation of Deep ReLU Network

Nicola Picchiotti 11 2 2 0000-0003-3454-7250 Marco Gori 11 3 3

###### Abstract

Amongst others, the adoption of Rectified Linear Units (ReLUs) is regarded as
one of the ingredients of the success of deep learning. ReLU activation has
been shown to mitigate the vanishing gradient issue, to encourage sparsity in
the learned parameters, and to allow for efficient backpropagation. In this
paper, we recognize that the non-linear behavior of the ReLU function gives
rise to a natural clustering when the pattern of active neurons is considered.
This observation helps to deepen the learning mechanism of the network; in
fact, we demonstrate that, within each cluster, the network can be fully
represented as an affine map. The consequence is that we are able to recover
an explanation, in the form of feature importance, for the predictions done by
the network to the instances belonging to the cluster. Therefore, the
methodology we propose is able to increase the level of interpretability of a
fully connected feedforward ReLU neural network, downstream from the fitting
phase of the model, without altering the structure of the network. A
simulation study and the empirical application to the Titanic dataset, show
the capability of the method to bridge the gap between the algorithm
optimization and the human understandability of the black box deep ReLU
networks.

###### Keywords:

ReLU clustering explainability linearity feature importance

##  1 Introduction

The ”black box” nature of deep neural network models is often a limit for
high-stakes applications like diagnostic techniques, autonomous guide etc.
since the reliability of the model predictions can be affected by the
incompleteness in the optimization problem formalization [1]. Recent works
have shown that an adequate level of interpretability could enforce the neural
network trustworthiness (see [2]). However, this is generally difficult to
achieve without alter the mechanism of deep learning [3]. The topic has became
particularly relevant in the last decades, since technology development and
data availability led to the adoption of more and more complex models, and
widened the gap between performances on train/test data and interpretability.

Much work has been done in order to explain and interpret the models developed
by AI in a human comprehensible manner. The main reason behind these effort is
that the human experience and capacity for abstraction allow to monitor the
process of the model decisions in a sound way, trying to mitigate the risks of
data-driven models. On the other hand, the recent developments of deep neural
networks offer enormous progress in artificial intelligence in various
sectors. In particular, the ReLU functions have been shown to mitigate the
vanishing gradient issue, encourage sparsity in the learned parameters and
allow for efficient backpropagation.

In this paper, we tackle the explainability problem of deep ReLU networks, by
characterizing the natural predisposition of the network to partition the
input dataset in different clusters. The direct consequence of the clustering
process is that, within each cluster, the network can be simplified and
represented as an affine map. Surprisingly, the deep network provides a notion
of cluster-specific importance for each feature, thus easier to interpret.

##  2 Bibliographic review

Despite the benefits and the expressiveness of Rectifier Networks have been
widely investigated, e.g. in [4], the cluster analysis and the consequent
interpretability of the networks via the modelling of the pattern of active
neurons has not been discussed in the literature, to our knowledge.

On the other hand, many model-specific methodologies can be applied for
gaining interpretability in neural network models (see [5] and [6] for a
review). The importance of a feature is one of the most used strategies to
gain local explainability from an opaque machine learning model. Permutation
feature importance methods evaluate the feature importance through the
variation of a loss metric by permuting the feature’s values on a set of
instances in the training or validation set. The approach is introduced for
random forest in [7] and for neural network in [8]. Other methods, such as
class model visualization [9], compute the partial derivative of the score
function with respect to the input, and [10] introduces expert distribution
for the input giving activation maximization. In [11] the author introduces
deep lift and computes the discrete gradients with respect to a baseline
point, by backpropagating the scoring difference through each unit. Integrated
gradients [12] cumulates the gradients with respect to inputs along the path
from a given baseline to the instance. Finally, a set of well known methods
called additive feature attribution methods defined in [13], rely on the
redistribution of the predicted value $\hat{f}(\bm{x})$ over the input
features. They are designed to mimic the behaviour of a predictive function
$\hat{f}$ with a surrogate Boolean linear function $g$.

##  3 Deep ReLU networks for the partition of the input space

In this section, we demonstrate that a deep ReLU neural network gives rise to
a partition of the input dataset into a set of clusters, each one
characterized by an affine map.

Let us denote by $W_{i},\ \ i\in[1,\ldots,p]$ the weight matrices associated
with the $p$ layers of a given multilayer network with predictor $\tilde{f}$
(of a $q$-dim target variable) and collect111The $\hat{}$ in the notation
means that the bias term is incorporated in the variable. $\hat{W}_{i}$ in
$\hat{W}=[\hat{W}_{1},\ldots,\hat{W}_{p}]$. For any input
$\bm{u}\in\mathbb{R}^{d}$, the initial Directed Acyclic Graph (DAG) ${\cal G}$
of the deep network is reduced to ${\cal G}_{\bm{u}}$ which only keeps the
units corresponding to active neurons222A neuron is considered active for a
particular pattern if the input falls in the right linear part of the domain’s
function. and the corresponding arcs. This DAG is clearly associated with a
given set of weights $\hat{W}$. We can formally state this pruning for the
given neural network, characterized by ${\cal G}$, paired with input $\bm{u}$
with weights $\hat{W}$ by

${\cal G}_{\bm{u}}=\gamma({\cal G},\hat{W},\bm{u}),$

where, since all neurons operate in “linear regime” (affine functions), as
stated in the following, the output, the composition of affine functions, is
in fact an affine function itself.

###### Theorem 3.1

Let $\mathscr{X}_{i}\subset\mathbb{R}^{d_{i}},\ \ i\in[1,\ldots,p]$ be, where
$d_{1}=d$. Let $\\{h_{1},\ldots,h_{p}\\}$ be a collection of affine functions,
where

$h_{i}:\mathscr{X}_{i}\mapsto\mathscr{Y}_{i}:x\mapsto
W_{i}\,x+\bm{b}_{i}=\hat{W}_{i}\,\hat{x},$

and assume that $\mathscr{Y}_{i}$ is chosen in such a way that $\forall
i=1,\ldots,p-1:\ \ \mathscr{X}_{i+1}\subset\mathscr{Y}_{i}$, whereas
$\mathscr{Y}_{p}\subset\mathbb{R}^{d_{q}}$. Then we have that

${\tilde{f}}(\hat{W},\bm{u})=h_{p}\circ h_{p-1}\circ\dots\circ h_{2}\circ
h_{1}(\bm{u})$

is affine and we have
${\tilde{f}}(W,\bm{u})=f(\hat{\Omega},\hat{\bm{u}})=\Omega\,\bm{u}+\bm{b},$
where333We stress the dependence of $\Omega$ and $\bm{b}$ from the number of
layer (p) since it will be useful for the proof.

$\displaystyle\hat{\Omega}:=[\Omega,\bm{b}]$ (1)
$\displaystyle\Omega(p)=\prod_{i=p}^{1}W_{i}$ (2)
$\displaystyle\bm{b}(p)=\sum_{i=1}^{p}\left(\prod_{t=p+1}^{i+1}W_{t}\right)\cdot\bm{b}_{i}$
(3)

being $W_{p+1}:=\mathbb{1}$.

###### Proof

The proof is given by induction on $p$.

  * •

Basis: For $p=1$ we have ${\tilde{f}}=W_{1}\,\bm{u}+\bm{b}_{1}$ and
$\Omega(1)=W_{1}$ which confirms (2), and when considering
$W_{2}:=\mathbb{1}$, we have

$\bm{b}(1)=W_{2}\cdot\bm{b}_{1}=\bm{b}_{1},$

in according to (3).

  * •

Induction step: By induction, a network with $p-1$ layers is defined by an
affine transformation that is

$y(p-1)=\Omega(p-1)\,\bm{u}+\bm{b}(p-1).$

Hence

$\displaystyle y(p)$ $\displaystyle=W_{p}\,y(p-1)+\bm{b}_{p}$
$\displaystyle=W_{p}(\Omega(p-1)\,\bm{u}+\bm{b}(p-1))+\bm{b}_{p}$
$\displaystyle=W_{p}\bigg{(}\prod_{i=p-1}^{1}W_{i}\,\bm{u}+\sum_{i=1}^{p-1}\left(\prod_{t=p}^{i+1}W_{t}\right)\cdot\bm{b}_{i}\bigg{)}+\bm{b}_{p}$
$\displaystyle=W_{p}\bigg{(}\prod_{i=p-1}^{1}W_{i}\,\bm{u}+\sum_{i=1}^{p-2}\left(\prod_{t=p-1}^{i+1}W_{t}\right)\cdot\bm{b}_{i}+\mathbb{1}\cdot\bm{b}_{p-1}\bigg{)}+\bm{b}_{p}$
$\displaystyle=\prod_{i=p}^{1}W_{i}\,\bm{u}+\sum_{i=1}^{p-2}\left(\prod_{t=p}^{i+1}W_{t}\right)\cdot\bm{b}_{i}+W_{p}\,\bm{b}_{p-1}+\bm{b}_{p}$
$\displaystyle=\prod_{i=p}^{1}W_{i}\,\bm{u}+\sum_{i=1}^{p-1}\left(\prod_{t=p}^{i+1}W_{t}\right)\cdot\bm{b}_{i}+\mathbb{1}\cdot\bm{b}_{p}$
$\displaystyle=\bigg{(}\prod_{i=p}^{1}W_{i}\bigg{)}\,\bm{u}+\sum_{i=1}^{p}\left(\prod_{t=p+1}^{i+1}W_{t}\right)\cdot\bm{b}_{i}$
(4)

Now let $\mathscr{U}\subset\mathbb{R}^{d}$ the input space. The given deep net
yields a partition on $\mathscr{U}$ which is associated with the following
equivalence relation:

$\bm{u}_{1}\sim\bm{u}_{2}\leftrightarrow{\cal G}_{\bm{u}_{1}}={\cal
G}_{\bm{u}_{2}}$

. We denote444In this work $[\cdot]$ is the Iverson’s notation, whereas
$[\cdot]_{\sim}$ is reserved to the equivalent class induced by equivalence
relation $\sim$. by
$[\bm{u}]_{\sim}=\\{\bm{v}\in\mathscr{U}:\bm{v}\sim\bm{u}\\}$ and by
$\mathscr{U}/\sim$ the corresponding quotient set. Hence, $[\bm{u}]_{\sim}$ is
the equivalent class associated with representer $\bm{u}$ which, in turn,
corresponds with ${\cal G}_{\bm{u}}$. Notice that, as a consequence,
$[\bm{u}]_{\sim}$ is fully defined by the set of neurons of the active neural
network.

### Feature Importance Explanation

The characterization done above allows to assigning a matrix $\hat{\Omega}$ to
each cluster of the network. In this way, the matrix is able to represent the
network for the patterns of the specific cluster as an affine map.

For simplicity, if we consider a problem where the output of the network is
scalar, the matrix $\Omega$ reduces to a d-dimensional effective vector
$\bm{\omega}$ whose components can be interpreted as the feature importance of
the cluster’s solution.

##  4 Simulation study

In this section, we report the simulation studies carried out on the ReLU
network architecture applied to a Boolean artificial dataset, in order to
assess the power of the clustering-based interpretation.

We consider a set of $10$ Boolean feature variables $v_{i\in[1,...,10]}$. The
first $3$ features determine the target variable through the following
relation:

$t=\left(v_{1}\land v_{3}\right)\lor\left(v_{2}\land\lnot v_{3}\right)$ (5)

whereas the other $7$ features introduce noise. The rationale of the formula
is that the feature $v_{3}$ split the data set in two groups ($v_{3}$ and
$\lnot v_{3}$), each ruled by a different term of the Boolean formula
involving either $v_{1}$ or $v_{2}$ respectively. In the following, we
investigate whether the network clustering is able to recognize the different
terms of the formula.

We simulated $100,000$ samples, and we exploited a two-hidden-layer Multilayer
Perceptron (MLP) with $4$ and $2$ neurons characterized by ReLU activation
function and an output neuron with a sigmoid activation function. The cross-
entropy loss is minimized via the Adam stochastic optimizer with a step size
of $0.01$ for $10$ epochs, and a batch size of $100$. An activity regularizer
with $0.02$ is added to the empirical loss. At the end of the training, the
network solves the problem with an accuracy of 100%. The experiments are
implemented with Keras in the Python environment on a regular CPU.

The analysis of the network, by considering the possible patterns of the
active neurons, originates three clusters.

  1. 1.

A trivial cluster characterized by all non-active neurons including all the
patterns predicted as $0$.

Instead, the other two clusters activate the two neurons of the last layer but
a different neuron of the first hidden layer. In figure 1 we report the table
resuming the $8$ possibilities of the Boolean function restricted to the first
$3$ features, as well as the bar plot for the importance of the features for
the two non-trivial clusters. As explained above, the importance of the
feature is computed by the coefficients of the effective vector for that
cluster.

  1. 2.

The second cluster, represented by the blue bars in Fig. 1, includes patterns
predicted as $1$ and characterized either by $v_{1}=v_{2}=v_{3}=1$ or by
$v_{1}=1,\,v_{2}=0,\,v_{3}=1$. We can argue that this cluster takes in charge
the patterns predicted as $1$ due to the first term of Eq. (5), i.e.
$v_{1}\land v_{3}$. Coherently with this setting, the feature importance given
by the effective vector is zero for the feature $v_{2}$ that does not appear
in the first term of Eq. (5). On the other side, the coefficient of the
effective vector is positive for the feature $v_{1}$.

  2. 3.

In a similar way, the third cluster denoted with the orange color, represents
the term $v_{2}\land\lnot v_{3}$ of Eq. (5) since the patterns belonging to it
are predicted as $1$ and are characterized either by $v_{1}=v_{2}=1\,v_{3}=0$
or by $v_{1}=0\,,v_{2}=1\,,v_{3}=0$. As expected, the feature importance of
$v_{1}$ is zero, whereas for $v_{2}$ the feature importance is positive.

Figure 1: Possible results of the simulated task (8 combinations) and
representation of the actual clusters provided by the cluster-based
interpretation of the ReLU network.

From the simulation study, we note that the clustering-based interpretation of
the ReLU network helps to achieve a more profound understanding of the
solution meaning. In particular, the methodology tries to disentangle the
complexity of the solutions into a set of more comprehensible linear solutions
within a specific cluster.

As a further confirmation of the usefulness of cluster-based interpretation,
when the activity regularizer is removed, the network keeps solving the task
giving rise to $8$ clusters, each one specific for each combination of the
first three features. Based on the cluster interpretation, we conclude that
the network has chosen a less abstract way to solve the problem.

##  5 Titanic dataset

In this section, we report the experimental analysis performed on the well-
known Titanic dataset555https://www.kaggle.com/c/titanic. Each sample
represents a passenger with specific features, and the binary target variable
indicates if the person survived the Titanic disaster. Standard data cleaning
and feature selection procedures666 https://www.kaggle.com/startupsci/titanic-
data-science-solutions are implemented. In Table 1 we report a brief
description of the features.

Table 1: Features for the Titanic dataset Feature | Description | Range
---|---|---
Age | Age of the passenger discretized in 5 bins | $[0,4]$
Gender | 1 if the passenger is female | $\\{0,1\\}$
PClass | Travel class: first, second, third | $[1,3]$
Fare | Ticket fare discretized in 3 bins | $[0,3]$
Embarked | Location for the embarked | $[0,2]$
Title | Title of the passenger: Mr, Miss, Mrs, Master, Rare | $[1,5]$
Is Alone | 1 if the passenger has not relatives | $\\{0,1\\}$

Similar to the previous experiment, we exploit an MLP with two hidden layers
($4$ and $2$ neurons) characterized by ReLU activation function and an output
neuron with a sigmoid activation function. The cross-entropy loss is minimized
via the Adam stochastic optimizer with a step size of $0.01$ for $10$ epochs
and a batch size of $100$. An activity regularizer with $0.02$ is added to the
empirical loss. The experiments are implemented with Keras in the Python
environment. The code is freely available at
https://github.com/nicolapicchiotti/relu_nn_clustering.

The accuracy of the network is $77\%$ (see [14]) and the study of the active
neurons patterns provides a partition of the dataset into three clusters,
whose feature importance is shown in Figure 2.

  1. (a)

The first cluster a) includes passengers with mixed features and a percentage
of survived ones equal to $38\%$. As expected from the univariate exploratory
analyses, gender and class had the most significant relationship for survival
rate.

  2. (b)

The cluster b) instead, includes only males belonging to the third class
($4\%$ of the overall): the prediction for these passengers is always $0$.
This cluster confirms the expectation on the male gender and third class as
relevant features for not survive. We observe that the feature importance is
quite similar to the one of cluster a) except for the fact that the ”age”
feature assumes slightly more relevance. Finally,

  3. (c)

in the cluster c) the passengers are females and belonging to the first class
($16\%$ of the overall), the predicted value is always 1. The feature
importance, in this case, shows that the high value of the title, age, and the
other features contribute to survival, in addition to being female. This
cluster helps us to understand the solution provided by the network. For
instance, we note that the ”age” feature has an opposite behavior with respect
to the other two clusters: in this cluster the older the women, the higher the
survival probability.

In this experiment, we have shown that the ReLU network can be disentangled
into a set of clusters that can be analyzed individually. The clusters have a
practical meaning helping the human to interpret the mechanism of prediction
of the network.

Figure 2: Bar plot with the feature importance for each of the three clusters
originated by the ReLU neural network.

##  6 Conclusions and Limitations

This paper proposes a methodology for increasing the level of interpretability
of a fully connected feedforward neural network with ReLU activation
functions, downstream from the fitting phase of the model. It is worth noting
that the methodology does not alter neither the structure nor the performance
of the network and can be applied easily after the training of the model. The
methodology relies on the clustering that naturally arises from the binary
status of the different neurons of the network, in turns, related to the non-
linearity of the ReLU function. The existence of a feature importance
explanation based on a affine map for each cluster is proved. A simulation
study and the empirical application to the Titanic dataset show the capability
of the method to bridge the gap between the algorithm optimization and the
human understandability.

A possible limitation of the work is related to the potential high number of
clusters that the networks could generates. Further ways should be explored in
order to grant a parsimonious principle of the dataset partition process, such
as a principle of minimum entropy or the orthogonality of the hyperplanes
representing the clusters. Finally, it is worth mentioning that, if the input
variables are highly correlated or belong to a very high-dimensional spaces,
the weights of the affine map cannot be trivially interpreted as the features
importance, see for instance [15].

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