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{{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\qquad{\bm{\mathbf{\Lambda}}}=\begin{bmatrix}\lambda&{\lambda_{\mathrm{ST}}}&0&{\lambda_{\mathrm{DS}}}&0&0&\cdots&{\lambda_{\mathrm{DS}}}&0&0\\\
{\lambda_{\mathrm{ST}}}&\lambda&{\lambda_{\mathrm{ST}}}&0&{\lambda_{\mathrm{DS}}}&0&\cdots&0&{\lambda_{\mathrm{DS}}}&0\\\
0&{\lambda_{\mathrm{ST}}}&\lambda&0&0&{\lambda_{\mathrm{DS}}}&\cdots&0&0&{\lambda_{\mathrm{DS}}}\\\
{\lambda_{\mathrm{DS}}}&0&0&\lambda&{\lambda_{\mathrm{ST}}}&0&\cdots&{\lambda_{\mathrm{DS}}}&0&0\\\
0&{\lambda_{\mathrm{DS}}}&0&{\lambda_{\mathrm{ST}}}&\lambda&{\lambda_{\mathrm{ST}}}&\cdots&0&{\lambda_{\mathrm{DS}}}&0\\\
0&0&{\lambda_{\mathrm{DS}}}&0&{\lambda_{\mathrm{ST}}}&\lambda&\cdots&0&0&{\lambda_{\mathrm{DS}}}\\\
\raisebox{3.0pt}{$\scalebox{0.75}{$\vdots$}$}&\raisebox{3.0pt}{$\scalebox{0.75}{$\vdots$}$}&\raisebox{3.0pt}{$\scalebox{0.75}{$\vdots$}$}&\raisebox{3.0pt}{$\scalebox{0.75}{$\vdots$}$}&\raisebox{3.0pt}{$\scalebox{0.75}{$\vdots$}$}&\raisebox{3.0pt}{$\scalebox{0.75}{$\vdots$}$}&\raisebox{3.0pt}{$\scalebox{0.75}{$\ddots$}$}&\raisebox{3.0pt}{$\scalebox{0.75}{$\vdots$}$}&\raisebox{3.0pt}{$\scalebox{0.75}{$\vdots$}$}&\raisebox{3.0pt}{$\scalebox{0.75}{$\vdots$}$}\\\
{\lambda_{\mathrm{DS}}}&0&0&{\lambda_{\mathrm{DS}}}&0&0&\cdots&\lambda&{\lambda_{\mathrm{ST}}}&0\\\
0&{\lambda_{\mathrm{DS}}}&0&0&{\lambda_{\mathrm{DS}}}&0&\cdots&{\lambda_{\mathrm{ST}}}&\lambda&{\lambda_{\mathrm{ST}}}\\\
0&0&{\lambda_{\mathrm{DS}}}&0&0&{\lambda_{\mathrm{DS}}}&\cdots&0&{\lambda_{\mathrm{ST}}}&\lambda\end{bmatrix}$

The optimal penalties were found to be $\lambda^{\diamond}=2.2$ for the ridge
penalty, $\lambda^{\diamond}_{\mathrm{DS}}=0.0022$ for the data set fusion
penalty, and $\lambda^{\diamond}_{\mathrm{ST}}=6.8\times 10^{-4}$ for the
subtype fusion penalty, respectively.

Figure 4.  Summary of the estimated precision matrices for the NF-$\kappa$B
pathway. _Top row_ : Heat maps of the estimated precision matrices pooled
across data sets for each genetic subtype. _Middle row from left to right:_
The pooled target matrix for ABC, the difference between the pooled ABC and
GCB estimates, and the pooled target matrix for GCB. _Bottom:_ The color key
for the heat maps.

To summarize and visualize the 33 class precision estimates they were pooled
within DLBCL subtype. Panels A–C of Figure 4 visualizes the 3 pooled estimates
as heat maps. Panels D and F visualize the constructed target matrices for the
ABC and GCB subtypes, respectively. Panel E then gives the difference between
the pooled ABC and GCB estimates, indicating that they harbor differential
signals to some degree. We would like to capture the commonalities and
differences with a differential network representation.

The estimated class-specific precision matrices were subsequently scaled to
partial correlation matrices. Each precision matrix was then sparsified using
the lFDR procedure of Section 4.3. Given the class an edge was selected
whenever $1-\widehat{\mathrm{lFDR}}\geq 0.999$. To compactly visualize the the
multiple GGMs we obtained _signed edge-weighted total networks_ mentioned in
Section 4.4. Clearly, for inconsistent connections the weight would vary
around zero, while edges that are consistently selected as positive (negative)
will have a large positive (negative) weight. These meta-graphs are plotted in
Figure 5. Panels A–C give the signed edge-weighted total networks for each
subtype across the data sets. They show that (within DLBCL subtypes) there are
a number of edges that are highly concordant across all data sets. To evaluate
the greatest differences between the ABC and GCB subtypes, the signed edge-
weighted total network of the latter was subtracted from the former. The
resulting graph $\mathcal{G}_{\mathrm{ABC}-\mathrm{GCB}}$ can be found in
Panel D. Edges that are more stably present in the ABC subtype are represented
in orange and the edges more stably present in the GCB subtype are represented
in blue. Panel F represents the graph from panel D with only those edges
retained whose absolute weight exceeds $2$. In a sense, the graph of panel F
then represents the stable differential network. The strongest connections
here should suggest places of regulatory deregulation gained or lost across
the two subtypes. Interestingly, this differential network summary shows
relatively large connected subgraphs suggesting differing regulatory
mechanisms.

The graph in panel F of Figure 5 then conveys the added value of the
integrative fusion approach. Certain members of the CCL, CXCL, and TNF gene
families who were highly central in the analysis of Section 6.1 are still
considered to be central here. However, it is also seen that certain genes
that garnered high centrality measures in the single data set analyzed in
Section 6.1 do not behave stably _across_ data sets, such as CXCL2. In
addition, the integrative analysis appoints the BCL2 gene family a central
role, especially in relation to the ABC subtype. This contrasts with Section
6.1, where the BCL2 gene family was not considered central and appeared to be
connected mostly in the non-ABC classes. Moreover, whereas the analysis of the
single data set could not identify a signal for MYD88, the integrative
analysis identifies MYD88 to be stably connected across data sets. Especially
the latter two observations are in line with current knowledge on deregulation
in the NF-$\kappa$B pathway in DLBCL patients. Also in accordance with
litterature is the known interaction of LTA with LTB seen in panel F of Figure
5 [45, 10] which here appear to be differential between ABC/GCB. Thus,
borrowing information across classes enables a meta-analytic approach that can
uncover information otherwise unobtainable through the analysis of single data
sets.

Figure 5.  Summary of estimated GGMs for the NF-$\kappa$B pathway. _Panels
A–C_ : Graphs obtained by adding the signed adjacency matrices for each
subtype across the data sets. The edge widths are drawn proportional to the
absolute edge weight. _Panel D_ : Graph obtained by subtracting the summarized
signed adjacency matrix of GCB (panel A) from that of ABC (panel C). Edge
widths are drawn proportional to the absolute weight and colored according to
the sign. Orange implies edges more present in ABC and blue implies edges more
present in GCB. _Panel E_ : As the graph in panel D, however only edges with
absolute weight $>2$ are drawn. _Panel F_ : As the graph in panel E, but with
an alternative layout. _Far right-panel:_ EID key and corresponding HGNC
curated gene names of the NF-$\kappa$B pathway genes. Genes that are connected
in panel F are shown bold.

##  7\. Discussion and conclusion

We considered the problem of jointly estimating multiple inverse covariance
matrices from high-dimensional data consisting of distinct classes. A fused
ridge estimator was proposed that generalizes previous contributions in two
principal directions. First, we introduced the use of targets in fused ridge
precision estimation. The targeted approach helps to stabilize the estimation
procedure and allows for the incorporation of prior knowledge. It also
juxtaposes itself with various alternative penalized precision matrix
estimators that pull the estimates towards the edge of the parameter space,
i.e., who shrink towards the non-interpretable null matrix. Second, instead of
using a single ridge penalty and a single fusion penalty parameter for all
classes, the approach grants the use of _class-specific_ ridge penalties and
_class-pair-specific_ fusion penalties. This results in a flexible shrinkage
framework that (i) allows for class-specific tuning, that (ii) supports
analyzes when a factorial design underlies the available classes, and that
(iii) supports the appropriate handling of situations where some classes are
high-dimensional whilst others are low-dimensional. Targeted shrinkage and
usage of a flexible penalty matrix might also benefit other procedures for
precision matrix estimation such as the fused graphical lasso [13].

The targeted fused ridge estimator was combined with post-hoc support
determination, which serves as a basis for integrative or meta-analytic
Gaussian graphical modeling. This combination thus has applications in meta-,
integrative-, and differential network analysis of multiple data sets or
classes of data. This meta-approach to network analysis has multiple
motivations. First, by combining data it can effectively increase the sample
size in settings where samples are relatively scarce or expensive to produce.
In a sense it refocuses the otherwise declining attention to obtaining a
sufficient amount of data—a tendency we perceive to be untenable. Second,
aggregation across multiple data sets decreases the likelihood of capturing
idiosyncratic features (of individual data sets), thereby preventing over-
fitting of the data.

Insightful summarization of the results is important for the feasibility of
our approach to fused graphical modeling. To this end we have proposed various
basic tools to summarize commonalities and differences over multiple graphs.
These tools were subsequently used in a differential network analysis of the
NF-$\kappa$B signaling pathway in DLBCL subtypes over multiple GEP data sets.
This application is not without critique, as it experiences a problem present
in many GEP studies: The classification of the DLBCL subtypes (ABC and GBC) is
performed on the basis of the same GEP data on which the network analysis is
executed. This may be deemed methodologically undesirable. However, we justify
this double use of data as (a) the pathway of interest involves a selection of
genes whereas the classification uses all genes, and (b) the analysis
investigates partial correlations and differential networks whereas the
classification, in a sense, considers only differential expression.
Furthermore, as in all large-scale genetic screenings, the analyzes should be
considered ‘tentative’ and findings need to be validated in independent
experiments. Notwithstanding, the analyzes show that the fusion approach to
network integration has merit in uncovering class-specific information on
pathway deregulation. Moreover, they exemplify the exploratory _hypothesis
generating_ thrust of the framework we offer.

We see various inroad for further research. With regard to estimation one
could think of extending the framework to incorporate a fused version of the
elastic net. Mixed fusion, in the sense that one could do graphical lasso
estimation with ridge fusion or ridge estimation with lasso fusion, might also
be of interest. From an applied perspective the desire is to expand the
toolbox for insightful (visual) summarization of commonalities and differences
over multiple graphs. Moreover, it is of interest to explore improved ways for
support determination. The lFDR procedure, for example, could be expanded by
considering all classes jointly. Instead of applying the lFDR procedure to
each class-specific precision matrix, one would then be interested in
determining the proper mixture of a grand common null-distribution and
multiple class-specific non-null distributions. These inroads were out of the
scope of current work, but we hope to explore them elsewhere.

###  7.1. Software implementation

The fused ridge estimator and its accompanying estimation procedure is
implemented in the rags2ridges-package [31] for the statistical language R.
This package has many supporting functions for penalty parameter selection,
graphical modeling, as well as network analysis. We will report on its full
functionality elsewhere. The package is freely available from the
Comprehensive R Archive Network: http://cran.r-project.org/.

## Acknowledgements

Anders E. Bilgrau was supported by a grant from the Karen Elise Jensen Fonden,
a travel grant from the Danish Cancer Society, and a visitor grant by the
Dept. of Mathematics of the VU University Amsterdam. Carel F.W. Peeters
received funding from the European Community’s Seventh Framework Programme
(FP7, 2007-2013), Research Infrastructures action, under grant agreement No.
FP7-269553 (EpiRadBio project). The authors would also like to thank Karen
Dybkær of the Dept. of Haematology at Aalborg University Hospital, for her
help on the biological interpretations in the DLBCL application.

##  Appendix A Geometric interpretation of the fused ridge penalty

Some intuition behind the fused ridge is provided by pointing to the
equivalence of penalized and constrained optimization. To build this intuition
we study the geometric interpretation of the fused ridge penalty in the
special case of (6) with $\bm{\mathbf{T}}=\bm{\mathbf{0}}$. In this case
$\lambda_{gg}=\lambda$ for all $g$, and $\lambda_{g_{1}g_{2}}=\lambda_{f}$ for
all $g_{1}\neq g_{2}$. Clearly, the penalty matrix then amounts to
${\bm{\mathbf{\Lambda}}}=\lambda\bm{\mathbf{I}}_{p}+\lambda_{f}(\bm{\mathbf{J}}_{p}-\bm{\mathbf{I}}_{p})$.
Matters are simplified further by considering $G=2$ classes and by focusing on
a specific entry in the precision matrix, say
$({\bm{\mathbf{\Omega}}}_{g})_{jj^{\prime}}=\omega_{jj^{\prime}}^{(g)}$, for
$g=1,2$. By doing so we ignore the contribution of other precision elements to
the penalty. Now, the fused ridge penalty may be rewritten as:

$\displaystyle\frac{\lambda}{2}\Big{(}\big{\|}{\bm{\mathbf{\Omega}}}_{1}\big{\|}_{F}^{2}+\big{\|}{\bm{\mathbf{\Omega}}}_{2}\big{\|}_{F}^{2}\Big{)}+\frac{\lambda_{f}}{4}\sum_{g_{1}=1}^{2}\sum_{g_{2}=1}^{2}\big{\|}{\bm{\mathbf{\Omega}}}_{g_{1}}-{\bm{\mathbf{\Omega}}}_{g_{2}}\big{\|}_{F}^{2}$
$\displaystyle=\frac{\lambda}{2}\Big{(}\big{\|}{\bm{\mathbf{\Omega}}}_{1}\big{\|}_{F}^{2}+\big{\|}{\bm{\mathbf{\Omega}}}_{2}\big{\|}_{F}^{2}\Big{)}+\frac{\lambda_{f}}{2}\big{\|}{\bm{\mathbf{\Omega}}}_{1}-{\bm{\mathbf{\Omega}}}_{2}\big{\|}_{F}^{2}.$

Subsequently considering only the contribution of the
$\omega_{jj^{\prime}}^{(g)}$ entries implies this expression can be further
reduced to:

$\displaystyle\frac{\lambda}{2}\left[\big{(}\omega_{jj^{\prime}}^{(1)}\big{)}^{2}+\big{(}\omega_{jj^{\prime}}^{(2)}\big{)}^{2}\right]+\frac{\lambda_{f}}{2}\big{(}\omega_{jj^{\prime}}^{(1)}-\omega_{jj^{\prime}}^{(2)}\big{)}^{2}=\frac{\lambda+\lambda_{f}}{2}\left[\big{(}\omega_{jj^{\prime}}^{(1)}\big{)}^{2}+\big{(}\omega_{jj^{\prime}}^{(2)}\big{)}^{2}\right]-\lambda_{f}\omega_{jj^{\prime}}^{(1)}\omega_{jj^{\prime}}^{(2)}.$

It follows immediately that this penalty imposes constraints on the parameters
$\omega_{jj^{\prime}}^{(1)}$ and $\omega_{jj^{\prime}}^{(2)}$, amounting to
the set:

(20)
$\displaystyle\biggl{\\{}\big{(}\omega_{jj^{\prime}}^{(1)},\omega_{jj^{\prime}}^{(2)}\big{)}\in\mathbb{R}^{2}:\frac{\lambda+\lambda_{f}}{2}\Bigl{[}\bigl{(}\omega_{jj^{\prime}}^{(1)}\bigr{)}^{2}+\bigl{(}\omega_{jj^{\prime}}^{(2)}\bigr{)}^{2}\Bigr{]}-\lambda_{f}\omega_{jj^{\prime}}^{(1)}\omega_{jj^{\prime}}^{(2)}\leq
c\biggr{\\}},$

for some $c\in\mathbb{R}_{+}$. It implies that the fused ridge penalty can be
understood by the implied constraints on the parameters. Figure 6 shows the
boundary of the set for selected values.

Figure 6. Visualization of the effects of the fused ridge penalty in terms of
constraints. The left panel shows the effect of $\lambda_{f}$ for fixed
$\lambda$. Here, $\lambda_{f}=0$ is the regular ridge penalty. The right panel
shows the effect of $\lambda$ while keeping $\lambda_{f}$ fixed.

Panel 6A reveals the effect of the fused, inter-class penalty parameter
$\lambda_{f}$ (while keeping $\lambda$ fixed). At $\lambda_{f}=0$, the
constraint coincides with the regular ridge penalty. As $\lambda_{f}$
increases, the ellipsoid shrinks along the minor principal axis $x=-y$ with no
shrinkage along $x=y$. In the limit $\lambda_{f}\to\infty$ the ellipsoid
collapses onto the identity line. Hence, the parameters
$\omega_{jj^{\prime}}^{(1)}$ and $\omega_{jj^{\prime}}^{(2)}$ are shrunken
towards each other and while their differences vanish, their sum is not
affected. Hence, the fused penalty parameter primarily shrinks the ‘sum of the
parameters’, but also fuses them as a bound on their sizes implies a bound on
their difference.

Panel 6B shows the effect of the intra-class $\lambda$ penalty (while keeping
$\lambda_{f}$ fixed). When the penalty vanishes for $\lambda\to 0$ the domain
becomes a degenerated ellipse (i.e. cylindrical for more than 2 classes) and
parameters $\omega_{jj^{\prime}}^{(1)}$ and $\omega_{jj^{\prime}}^{(2)}$ may
assume any value as long as their difference is less than
$\sqrt{2c/\lambda_{f}}$. For any $\lambda>0$, the parameter-constraint is
ellipsoidal. As $\lambda$ increases the ellipsoid is primarily shrunken along
the principal axis formed by the identity line and along the orthogonal
principal axis $(y=-x)$. In the limit $\lambda\to\infty$ the ellipsoid
collapses onto the point $(0,0)$. It is clear that the shape of the domain in
(20) is only determined by the ratio of $\lambda$ and $\lambda_{f}$.

The effect of the penalties on the domain of the obtainable estimates can be
further understood by noting that the fused ridge penalty (4) can be rewritten
as

(21)
$\tilde{\lambda}\sum_{g_{1},g_{2}}\big{\lVert}({\bm{\mathbf{\Omega}}}_{g_{1}}\\!-\bm{\mathbf{T}}_{g_{1}})+({\bm{\mathbf{\Omega}}}_{g_{2}}\\!-\bm{\mathbf{T}}_{g_{2}})\big{\rVert}_{F}^{2}+\tilde{\lambda}_{f}\sum_{g_{1},g_{2}}\big{\lVert}({\bm{\mathbf{\Omega}}}_{g_{1}}\\!-\bm{\mathbf{T}}_{g_{1}})-({\bm{\mathbf{\Omega}}}_{g_{2}}\\!-\bm{\mathbf{T}}_{g_{2}})\big{\rVert}_{F}^{2},$

for some penalties $\tilde{\lambda}$ and $\tilde{\lambda}_{f}$. The details of
this derivation can be found in Section A.1 below. The first and second
summand of the rewritten penalty (21) respectively shrink the sum and
difference of the parameters of the precision matrices. Their contributions
thus coincide with the principal axes along which two penalty parameters
shrink the domain of the parameters.

###  A.1. Alternative form for the fused ridge penalty

This section shows that the alternative form (21) for the ridge penalty can be
written in the form (4). We again assume a common ridge penalty
$\lambda_{gg}=\lambda$ and a common fusion penalty
$\lambda_{g_{1}g_{2}}=\lambda_{f}$ for all classes and pairs thereof. To
simplify the notation, let
$\bm{\mathbf{A}}_{g}={\bm{\mathbf{\Omega}}}_{g}\\!-\bm{\mathbf{T}}_{g}$. Now,

$\displaystyle
f^{\mathrm{FR^{\prime}}}\bigl{(}\\{{\bm{\mathbf{\Omega}}}_{g}\\};\tilde{\lambda},\tilde{\lambda}_{f},\\{\bm{\mathbf{T}}_{g}\\}\bigr{)}$
$\displaystyle=\tilde{\lambda}\sum_{g_{1},g_{2}}\big{\|}({\bm{\mathbf{\Omega}}}_{g_{1}}\\!-\bm{\mathbf{T}}_{g_{1}})+({\bm{\mathbf{\Omega}}}_{g_{2}}\\!-\bm{\mathbf{T}}_{g_{2}})\big{\|}_{F}^{2}+\tilde{\lambda}_{f}\sum_{g_{1},g_{2}}\big{\|}({\bm{\mathbf{\Omega}}}_{g_{1}}\\!-\bm{\mathbf{T}}_{g_{1}})-({\bm{\mathbf{\Omega}}}_{g_{2}}\\!-\bm{\mathbf{T}}_{g_{2}})\big{\|}_{F}^{2}$
$\displaystyle=\tilde{\lambda}\sum_{g_{1},g_{2}}\big{\|}\bm{\mathbf{A}}_{g_{1}}\\!+\bm{\mathbf{A}}_{g_{2}}\big{\|}_{F}^{2}+\tilde{\lambda}_{f}\sum_{g_{1},g_{2}}\big{\|}\bm{\mathbf{A}}_{g_{1}}\\!-\bm{\mathbf{A}}_{g_{2}}\big{\|}_{F}^{2}$
$\displaystyle=\tilde{\lambda}\sum_{g_{1},g_{2}}\Big{(}\big{\|}\bm{\mathbf{A}}_{g_{1}}\big{\|}_{F}^{2}+\big{\|}\bm{\mathbf{A}}_{g_{2}}\big{\|}_{F}^{2}+2\langle\bm{\mathbf{A}}_{g_{1}},\bm{\mathbf{A}}_{g_{2}}\rangle\Big{)}+\tilde{\lambda}_{f}\sum_{g_{1},g_{2}}\big{\|}\bm{\mathbf{A}}_{g_{1}}\\!-\bm{\mathbf{A}}_{g_{2}}\big{\|}_{F}^{2}$
$\displaystyle=\tilde{\lambda}\sum_{g_{1},g_{2}}\Big{(}2\big{\|}\bm{\mathbf{A}}_{g_{1}}\big{\|}_{F}^{2}+2\big{\|}\bm{\mathbf{A}}_{g_{2}}\big{\|}_{F}^{2}-\big{\|}\bm{\mathbf{A}}_{g_{1}}-\bm{\mathbf{A}}_{g_{2}}\big{\|}_{F}^{2}\Big{)}+\tilde{\lambda}_{f}\sum_{g_{1},g_{2}}\big{\|}\bm{\mathbf{A}}_{g_{1}}\\!-\bm{\mathbf{A}}_{g_{2}}\big{\|}_{F}^{2}$
$\displaystyle=4\tilde{\lambda}G\sum_{g}\big{\|}\bm{\mathbf{A}}_{g}\big{\|}_{F}^{2}-\tilde{\lambda}\sum_{g_{1},g_{2}}\big{\|}\bm{\mathbf{A}}_{g_{1}}\\!-\bm{\mathbf{A}}_{g_{2}}\big{\|}_{F}^{2}+\tilde{\lambda}_{f}\sum_{g_{1},g_{2}}\big{\|}\bm{\mathbf{A}}_{g_{1}}\\!-\bm{\mathbf{A}}_{g_{2}}\big{\|}_{F}^{2}$
$\displaystyle=4\tilde{\lambda}G\sum_{g}\big{\|}\bm{\mathbf{A}}_{g}\big{\|}_{F}^{2}+(\tilde{\lambda}_{f}-\tilde{\lambda})\sum_{g_{1},g_{2}}\big{\|}\bm{\mathbf{A}}_{g_{1}}-\bm{\mathbf{A}}_{g_{2}}\big{\|}_{F}^{2}$
$\displaystyle=4\tilde{\lambda}G\sum_{g}\big{\|}({\bm{\mathbf{\Omega}}}_{g}\\!-\bm{\mathbf{T}}_{g})\big{\|}_{F}^{2}+(\tilde{\lambda}_{f}-\tilde{\lambda})\sum_{g_{1},g_{2}}\big{\|}({\bm{\mathbf{\Omega}}}_{g_{1}}\\!-\bm{\mathbf{T}}_{g_{1}})-({\bm{\mathbf{\Omega}}}_{g_{2}}\\!-\bm{\mathbf{T}}_{g_{2}})\big{\|}_{F}^{2}.$

Hence, the alternative penalty (21) is also of the form (4) and thus the fused
ridge of (21) is equivalent to (4) for appropriate choices of the penalties.

##  Appendix B Results and proofs

Section B.1 contains supporting results from other sources and results in
support of Algorithm 1. Section B.2 contains proofs of the results stated in
the main text as well as additional results conducive in those proofs.

###  B.1. Supporting results

######  Lemma 1 (van Wieringen and Peeters [42]).

Amend the log-likelihood (1) with the $\ell_{2}$-penalty

$\frac{\lambda}{2}\big{\lVert}{\bm{\mathbf{\Omega}}}\\!-\bm{\mathbf{T}}\big{\rVert}_{F}^{2},$

with $\mathbf{T}\in\mathcal{S}_{+}^{p}$ denoting a fixed symmetric p.s.d.
target matrix, and where $\lambda\in(0,\infty)$ denotes a penalty parameter.
The zero gradient equation w.r.t. the precision matrix then amounts to

(22)
$\hat{{\bm{\mathbf{\Omega}}}}^{-1}-(\bm{\mathbf{S}}-\lambda\bm{\mathbf{T}})-\lambda\hat{{\bm{\mathbf{\Omega}}}}=\bm{\mathbf{0}},$

whose solution gives a penalized ML ridge estimator of the precision matrix:

$\hat{\mathbf{\Omega}}(\lambda)=\left\\{\left[\lambda\mathbf{I}_{p}+\frac{1}{4}(\mathbf{S}-\lambda\mathbf{T})^{2}\right]^{1/2}+\frac{1}{2}(\mathbf{S}-\lambda\mathbf{T})\right\\}^{-1}.$

######  Lemma 2 (van Wieringen and Peeters [42]).

Consider $\hat{{\bm{\mathbf{\Omega}}}}(\lambda)$ from Lemma 1 and define
$[\hat{{\bm{\mathbf{\Omega}}}}(\lambda)]^{-1}\equiv\hat{{\bm{\mathbf{\Sigma}}}}(\lambda)$.
The following identity then holds:

$\bm{\mathbf{S}}-\lambda\bm{\mathbf{T}}=\hat{{\bm{\mathbf{\Sigma}}}}(\lambda)-\lambda\hat{{\bm{\mathbf{\Omega}}}}(\lambda).$

######  Lemma 3.

Let ${\bm{\mathbf{\Lambda}}}\in\mathcal{S}^{G}$ be a matrix of fixed penalty
parameters such that ${\bm{\mathbf{\Lambda}}}\geq\bm{\mathbf{0}}$. Moreover,
let $\\{\bm{\mathbf{T}}_{g}\\}\in\mathcal{S}_{+}^{p}$. Then if
$\operatorname{diag}({\bm{\mathbf{\Lambda}}})>\bm{\mathbf{0}}$, the problem of
(5) is strictly concave.

###### Proof of Lemma 3.

By $\operatorname{diag}({\bm{\mathbf{\Lambda}}})>\bm{\mathbf{0}}$, it is clear
that the fused ridge penalty (4) is strictly convex as it is a conical
combination of strictly convex and convex functions. Hence, the negative fused
ridge penalty is strictly concave. The log-likelihood of (3) is a conical
combination of concave functions and is thus also concave. Therefore, the
penalized log-likelihood is strictly concave. ∎

###  B.2. Proofs and additional results

###### Proof of Proposition 1.

To find the maximizing argument for a specific class of the general fused
ridge penalized log-likelihood problem (5) we must obtain its first-order
derivative w.r.t. that class and solve the resulting zero gradient equation.
To this end we first rewrite the ridge penalty (4) into a second alternative
form. Using that ${\bm{\mathbf{\Lambda}}}={{\bm{\mathbf{\Lambda}}}}^{\top}$,
and keeping in mind the cyclic property of the trace as well as properties of
${\bm{\mathbf{\Omega}}}_{g}$ and $\bm{\mathbf{T}}_{g}$ stemming from their
symmetry, we may find:

$\displaystyle
f^{\mathrm{FR^{\prime\prime}}}\bigl{(}\\{{\bm{\mathbf{\Omega}}}_{g}\\};{\bm{\mathbf{\Lambda}}},\\{\bm{\mathbf{T}}_{g}\\}\bigr{)}$
$\displaystyle\qquad=\sum_{g}\frac{\lambda_{gg}}{2}\big{\lVert}{\bm{\mathbf{\Omega}}}_{g}{-}\bm{\mathbf{T}}_{g}\big{\rVert}_{F}^{2}+\sum_{g_{1},g_{2}}\frac{\lambda_{g_{1}g_{2}}}{4}\big{\lVert}({\bm{\mathbf{\Omega}}}_{g_{1}}{-}\bm{\mathbf{T}}_{g_{1}})-({\bm{\mathbf{\Omega}}}_{g_{2}}{-}\bm{\mathbf{T}}_{g_{2}})\big{\rVert}_{F}^{2}$
(23)
$\displaystyle\qquad=\sum_{g}\frac{\lambda_{g\bullet}}{2}\operatorname*{tr}\left[({\bm{\mathbf{\Omega}}}_{g}{-}\bm{\mathbf{T}}_{g})^{\top}({\bm{\mathbf{\Omega}}}_{g}{-}\bm{\mathbf{T}}_{g})\right]-\sum_{\mathclap{\begin{subarray}{c}g_{1},g_{2}\\\
g_{1}\neq
g_{2}\end{subarray}}}\frac{\lambda_{g_{1}g_{2}}}{2}\operatorname*{tr}\left[({\bm{\mathbf{\Omega}}}_{g_{1}}{-}\bm{\mathbf{T}}_{g_{1}})^{\top}({\bm{\mathbf{\Omega}}}_{g_{2}}{-}\bm{\mathbf{T}}_{g_{2}})\right],$

where $\lambda_{g\bullet}=\sum_{g^{\prime}}\lambda_{gg^{\prime}}$ denotes the
sum over the g$th$ row (or column) of ${\bm{\mathbf{\Lambda}}}$. Taking the
first-order partial derivative of (23) w.r.t. ${\bm{\mathbf{\Omega}}}_{g_{0}}$
yields:

$\displaystyle\frac{\partial}{\partial{\bm{\mathbf{\Omega}}}_{g_{0}}}f^{\mathrm{FR^{\prime\prime}}}\bigl{(}\\{{\bm{\mathbf{\Omega}}}_{g}\\};{\bm{\mathbf{\Lambda}}},\\{\bm{\mathbf{T}}_{g}\\}\bigr{)}$
(24)
$\displaystyle\qquad=\lambda_{g_{0}\bullet}\left[2({\bm{\mathbf{\Omega}}}_{g_{0}}{-}\bm{\mathbf{T}}_{g_{0}})-({\bm{\mathbf{\Omega}}}_{g_{0}}{-}\bm{\mathbf{T}}_{g_{0}})\circ\bm{\mathbf{I}}_{p}\right]-\sum_{g\neq
g_{0}}\lambda_{gg_{0}}\left[2({\bm{\mathbf{\Omega}}}_{g}{-}\bm{\mathbf{T}}_{g})-({\bm{\mathbf{\Omega}}}_{g}{-}\bm{\mathbf{T}}_{g})\circ\bm{\mathbf{I}}_{p}\right].$

The first-order partial derivative of (3) w.r.t.
${\bm{\mathbf{\Omega}}}_{g_{0}}$ results in:

$\displaystyle\frac{\partial}{\partial{\bm{\mathbf{\Omega}}}_{g_{0}}}\mathcal{L}(\\{{\bm{\mathbf{\Omega}}}_{g}\\};\\{\bm{\mathbf{S}}_{g}\\})$
$\displaystyle=\frac{\partial}{\partial{\bm{\mathbf{\Omega}}}_{g_{0}}}\sum_{g}n_{g}\big{\\{}\ln|{\bm{\mathbf{\Omega}}}_{g}|-\operatorname*{tr}(\bm{\mathbf{S}}_{g}{\bm{\mathbf{\Omega}}}_{g})\big{\\}},$
(25)
$\displaystyle=n_{g_{0}}\left[2({\bm{\mathbf{\Omega}}}_{g_{0}}^{-1}\\!-\bm{\mathbf{S}}_{g_{0}})-({\bm{\mathbf{\Omega}}}_{g_{0}}^{-1}\\!-\bm{\mathbf{S}}_{g_{0}})\circ\bm{\mathbf{I}}_{p}\right].$

Subtracting (24) from (25) yields

(26)
$\left[n_{g_{0}}({\bm{\mathbf{\Omega}}}_{g_{0}}^{-1}\\!-\bm{\mathbf{S}}_{g_{0}})-\lambda_{g_{0}\bullet}({\bm{\mathbf{\Omega}}}_{g_{0}}{-}\bm{\mathbf{T}}_{g_{0}})+\sum_{g\neq
g_{0}}\lambda_{gg_{0}}({\bm{\mathbf{\Omega}}}_{g}{-}\bm{\mathbf{T}}_{g})\right]\circ(2\bm{\mathbf{J}}_{p}-\bm{\mathbf{I}}_{p}),$

which, clearly, is $\bm{\mathbf{0}}$ only when
$n_{g_{0}}({\bm{\mathbf{\Omega}}}_{g_{0}}^{-1}\\!-\bm{\mathbf{S}}_{g_{0}})-\lambda_{g_{0}\bullet}({\bm{\mathbf{\Omega}}}_{g_{0}}{-}\bm{\mathbf{T}}_{g_{0}})+\sum_{g\neq
g_{0}}\lambda_{gg_{0}}({\bm{\mathbf{\Omega}}}_{g}{-}\bm{\mathbf{T}}_{g})=\bm{\mathbf{0}}$.
From (26) we may then find our (conveniently scaled) zero gradient equation to
be:

(27)
$\hat{{\bm{\mathbf{\Omega}}}}_{g_{0}}^{-1}\\!-\bm{\mathbf{S}}_{g_{0}}-\frac{\lambda_{g_{0}\bullet}}{n_{g_{0}}}(\hat{{\bm{\mathbf{\Omega}}}}_{g_{0}}{-}\bm{\mathbf{T}}_{g_{0}})+\sum_{g\neq
g_{0}}\frac{\lambda_{gg_{0}}}{n_{g_{0}}}({\bm{\mathbf{\Omega}}}_{g}{-}\bm{\mathbf{T}}_{g})=\bm{\mathbf{0}}.$

Now, rewrite (27) to

(28)
$\hat{{\bm{\mathbf{\Omega}}}}_{g_{0}}^{-1}-\bar{\bm{\mathbf{S}}}_{g_{0}}-\bar{\lambda}_{g_{0}}(\hat{{\bm{\mathbf{\Omega}}}}_{g_{0}}\\!-\bar{\bm{\mathbf{T}}}_{g_{0}})=\bm{\mathbf{0}},$

where $\bar{\bm{\mathbf{S}}}_{g_{0}}=\bm{\mathbf{S}}_{g_{0}}-\sum_{g\neq
g_{0}}\frac{\lambda_{gg_{0}}}{n_{g_{0}}}({\bm{\mathbf{\Omega}}}_{g}\\!-\bm{\mathbf{T}}_{g})$,
$\bar{\bm{\mathbf{T}}}_{g_{0}}=\bm{\mathbf{T}}_{g_{0}}$, and
$\bar{\lambda}_{g_{0}}=\lambda_{g_{0}\bullet}/n_{g_{0}}$. It can be seen that
(28) is of the form (22). Lemma 1 may then be applied to obtain the solution
(7). ∎

######  Corollary 1.

Consider the estimator (7). Let
${\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigl{(}{\bm{\mathbf{\Lambda}}},\\{{\bm{\mathbf{\Omega}}}_{g^{\prime}}\\}_{g^{\prime}{\neq}g}\bigr{)}$
be the precision matrix estimate of the $g$th class. Also, let
$\operatorname{diag}({\bm{\mathbf{\Lambda}}})>\bm{\mathbf{0}}$ and assume that
all off-diagonal elements of ${\bm{\mathbf{\Lambda}}}$ are zero. Then
${\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigl{(}{\bm{\mathbf{\Lambda}}},\\{{\bm{\mathbf{\Omega}}}_{g^{\prime}}\\}_{g^{\prime}{\neq}g}\bigr{)}$
reduces to the non-fused ridge estimate of class $g$:

(29)
${\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigl{(}{\bm{\mathbf{\Lambda}}},\\{{\bm{\mathbf{\Omega}}}_{g^{\prime}}\\}_{g^{\prime}{\neq}g}\bigr{)}={\hat{\bm{\mathbf{\Omega}}}}{}_{g}(\lambda_{gg})=\left\\{\left[\frac{\lambda_{gg}}{n_{g}}\bm{\mathbf{I}}_{p}+\frac{1}{4}\left(\bm{\mathbf{S}}_{g}-\frac{\lambda_{gg}}{n_{g}}\bm{\mathbf{T}}_{g}\right)^{2}\right]^{1/2}+\frac{1}{2}\left(\bm{\mathbf{S}}_{g}-\frac{\lambda_{gg}}{n_{g}}\bm{\mathbf{T}}_{g}\right)\right\\}^{-1}.$

###### Proof of Corollary 1.

The result follows directly from equations (7) and (8) by using that
$\sum_{g^{\prime}\neq g}\lambda_{gg^{\prime}}=\sum_{g^{\prime}\neq
g}\lambda_{g^{\prime}g}=0$ for all $g$. ∎

######  Lemma 4.

Let $\\{\bm{\mathbf{T}}_{g}\\}\in\mathcal{S}_{+}^{p}$ and assume
$\lambda_{gg}\in\mathbb{R}_{++}$ in addition to
$0\leq\lambda_{gg^{\prime}}<\infty$ for all $g^{\prime}\neq g$. Then

$\lim_{\lambda_{gg}\rightarrow\infty^{-}}\left\|{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigl{(}{\bm{\mathbf{\Lambda}}},\\{{\bm{\mathbf{\Omega}}}_{g^{\prime}}\\}_{g^{\prime}{\neq}g}\bigr{)}\right\|_{F}<\infty.$

###### Proof of Lemma 4.

The result is shown through proof by contradiction. Hence, suppose

$\lim_{\lambda_{gg}\rightarrow\infty^{-}}\|{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigl{(}{\bm{\mathbf{\Lambda}}},\\{{\bm{\mathbf{\Omega}}}_{g^{\prime}}\\}_{g^{\prime}{\neq}g}\bigr{)}\|_{F}$

is unbounded. Let $d[\cdot]_{jj}$ denote the $j$th largest eigenvalue. Then,
as

$\left\|{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigl{(}{\bm{\mathbf{\Lambda}}},\\{{\bm{\mathbf{\Omega}}}_{g^{\prime}}\\}_{g^{\prime}{\neq}g}\bigr{)}\right\|_{F}=\left\\{\sum_{j=1}^{p}d\left[{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigl{(}{\bm{\mathbf{\Lambda}}},\\{{\bm{\mathbf{\Omega}}}_{g^{\prime}}\\}_{g^{\prime}{\neq}g}\bigr{)}\right]_{jj}^{2}\right\\}^{1/2},$

at least one eigenvalue must tend to infinity along with $\lambda_{gg}$.
Assume without loss of generality that this is only the first (and largest)
eigenvalue:

(30)
$\lim_{\lambda_{gg}\rightarrow\infty^{-}}d\left[{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigl{(}{\bm{\mathbf{\Lambda}}},\\{{\bm{\mathbf{\Omega}}}_{g^{\prime}}\\}_{g^{\prime}{\neq}g}\bigr{)}\right]_{11}=\mathcal{O}(\lambda_{gg}^{\gamma}),$

for some $\gamma>0$. Now, for any $\lambda_{gg}$, the precision can be written
as an eigendecomposition:

(31)
${\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigl{(}{\bm{\mathbf{\Lambda}}},\\{{\bm{\mathbf{\Omega}}}_{g^{\prime}}\\}_{g^{\prime}{\neq}g}\bigr{)}=d_{11}\mathbf{v}_{1}\mathbf{v}_{1}^{\top}+\sum_{j=2}^{p}d_{jj}\mathbf{v}_{j}\mathbf{v}_{j}^{\top},$

where the dependency of the eigenvalues and eigenvectors on the target
matrices and penalty parameters has been suppressed (for notational brevity
and clarity). It is the first summand on the right-hand side that dominates
the precision for large $\lambda_{gg}$. Furthermore, this ridge ML precision
estimate of the $g$th group satisfies, by (26), the following gradient
equation:

$n_{g}({\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{-1}\\!-\bm{\mathbf{S}}_{g})-\lambda_{gg}({\hat{\bm{\mathbf{\Omega}}}}{}_{g}{-}\bm{\mathbf{T}}_{g})-\sum_{g^{\prime}\neq
g}\lambda_{g^{\prime}g}({\hat{\bm{\mathbf{\Omega}}}}{}_{g}{-}\bm{\mathbf{T}}_{g})+\sum_{g^{\prime}\neq
g}\lambda_{g^{\prime}g}({\bm{\mathbf{\Omega}}}_{g^{\prime}}{-}\bm{\mathbf{T}}_{g^{\prime}})=\bm{\mathbf{0}}.$

We now make three observations: (i) From item (i) of Proposition 2 it follows
that
${\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigl{(}{\bm{\mathbf{\Lambda}}},\\{{\bm{\mathbf{\Omega}}}_{g^{\prime}}\\}_{g^{\prime}{\neq}g}\bigr{)}$
is always p.d. for $\lambda_{gg}\in\mathbb{R}_{++}$. Consequently,
$\lim_{\lambda_{gg}\rightarrow\infty^{-}}\|{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigl{(}{\bm{\mathbf{\Lambda}}},\\{{\bm{\mathbf{\Omega}}}_{g^{\prime}}\\}_{g^{\prime}{\neq}g}\bigr{)}^{-1}\|_{F}<\infty$;
(ii) The target matrices do not depend on $\lambda_{gg}$; and (iii) The finite
$\lambda_{gg^{\prime}}$ ensure that the norms of
${\bm{\mathbf{\Omega}}}_{g^{\prime}}$ can only exceed the norm of
${\hat{\bm{\mathbf{\Omega}}}}{}_{g}$ by a function (independent of
$\lambda_{gg}$) of the constant $\lambda_{gg^{\prime}}$. Hence, in the limit,
the norms of the ${\bm{\mathbf{\Omega}}}_{g^{\prime}}$ cannot exceed the norm
of ${\hat{\bm{\mathbf{\Omega}}}}{}_{g}$. These observations give that, as
$\lambda_{gg}$ tends towards infinity, the term
$\lambda_{gg}({\hat{\bm{\mathbf{\Omega}}}}{}_{g}{-}\bm{\mathbf{T}}_{g})$ will
dominate the gradient equation. In fact, the term
$\lambda_{gg}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}$ will dominate as, using (30)
and (31):

$\displaystyle\bm{\mathbf{0}}$ $\displaystyle\approx$
$\displaystyle-\lambda_{gg}({\hat{\bm{\mathbf{\Omega}}}}{}_{g}-\bm{\mathbf{T}}_{g})$
$\displaystyle\approx$
$\displaystyle-\lambda_{gg}d_{11}\mathbf{v}_{1}\mathbf{v}_{1}^{\top}+\lambda_{gg}\bm{\mathbf{T}}$
$\displaystyle\approx$
$\displaystyle-\lambda_{gg}^{1+\gamma}\mathbf{v}_{1}\mathbf{v}_{1}^{\top}+\lambda_{gg}\bm{\mathbf{T}}$
$\displaystyle\approx$
$\displaystyle-\lambda_{gg}^{1+\gamma}(\mathbf{v}_{1}\mathbf{v}_{1}^{\top}+\lambda_{gg}^{-\gamma}\bm{\mathbf{T}})$
$\displaystyle\approx$
$\displaystyle-\lambda_{gg}^{1+\gamma}\mathbf{v}_{1}\mathbf{v}_{1}^{\top}.$

This latter statement is contradictory as it can only be true if the first
eigenvalue tends to zero. This, in turn, contradicts the assumption of
unboundedness (in the Frobenius norm) of the precision estimate. Hence, the
fused ridge ML precision estimate must be bounded. ∎

###### Proof of Proposition 2.

(i) Note that (27) for class $g$ may be rewritten to

$\hat{{\bm{\mathbf{\Omega}}}}_{g}^{-1}\\!-\bm{\mathbf{S}}_{g}-\frac{\lambda_{g\bullet}}{n_{g}}\left\\{\hat{{\bm{\mathbf{\Omega}}}}_{g}-\left[\bm{\mathbf{T}}_{g}+\sum_{g^{\prime}\neq
g}\frac{\lambda_{gg^{\prime}}}{\lambda_{g\bullet}}({\bm{\mathbf{\Omega}}}_{g^{\prime}}{-}\bm{\mathbf{T}}_{g^{\prime}})\right]\right\\}=\bm{\mathbf{0}},$

implying that (7) can be obtained under the following alternative updating
scheme to (8):

$\bar{\bm{\mathbf{S}}}_{g}=\bm{\mathbf{S}}_{g},\quad\bar{\bm{\mathbf{T}}}_{g}=\bm{\mathbf{T}}_{g}+\sum_{g^{\prime}\neq
g}\frac{\lambda_{gg^{\prime}}}{\lambda_{g\bullet}}({\bm{\mathbf{\Omega}}}_{g^{\prime}}\\!-\bm{\mathbf{T}}_{g^{\prime}}),\quad\text{and}\quad\bar{\lambda}_{g}=\frac{\lambda_{g\bullet}}{n_{g}}.$

Now, let $d[{\;\cdot\;}]_{jj}$ denote the $j$th largest eigenvalue. Then

$\displaystyle
d\left\\{[\hat{{\bm{\mathbf{\Omega}}}}_{g}]^{-1}\right\\}_{jj}=d\left[\frac{1}{2}(\bm{\mathbf{S}}_{g}-\bar{\lambda}_{g}\bar{\bm{\mathbf{T}}}_{g})\right]_{jj}+\sqrt{\left\\{d\left[\frac{1}{2}(\bm{\mathbf{S}}_{g}-\bar{\lambda}_{g}\bar{\bm{\mathbf{T}}}_{g})\right]_{jj}\right\\}^{2}+\bar{\lambda}_{g}}>0,$

when $\bar{\lambda}_{g}>0$. As
$\bar{\lambda}_{g}=\sum_{g^{\prime}}(\lambda_{g^{\prime}g}/n_{g})$ and as
$\lambda_{g^{\prime}g}$ may be $0$ for all $g^{\prime}\neq g$,
$\hat{{\bm{\mathbf{\Omega}}}}_{g}$ is guaranteed to be p.d. whenever
$\lambda_{gg}\in\mathbb{R}_{++}$.

(ii) Note that $\sum_{g^{\prime}\neq
g}\lambda_{gg^{\prime}}=\sum_{g^{\prime}\neq g}\lambda_{g^{\prime}g}=0$
implies that $\hat{{\bm{\mathbf{\Omega}}}}_{g}$ reduces to the non-fused class
estimate (29) by way of Corollary 1. The stated right-hand limit is then
immediate by using $\lambda_{gg}=0$ in (29). Under the distributional
assumptions this limit exists with probability 1 when $p\leq n_{g}$.

(iii) Consider the zero gradient equation (27) for the $g$th class. Multiply
it by $n_{g}/\lambda_{g\bullet}$ to factor out the dominant term:

(32)
$\frac{n_{g}}{\lambda_{g\bullet}}\hat{{\bm{\mathbf{\Omega}}}}_{g}^{-1}\\!-\frac{n_{g}}{\lambda_{g\bullet}}\bm{\mathbf{S}}_{g}-(\hat{{\bm{\mathbf{\Omega}}}}_{g}\\!-\bm{\mathbf{T}}_{g})+\sum_{g^{\prime}\neq
g}\frac{\lambda_{g^{\prime}g}}{\lambda_{g\bullet}}({\bm{\mathbf{\Omega}}}_{g^{\prime}}\\!-\bm{\mathbf{T}}_{g^{\prime}})=\bm{\mathbf{0}}.$

When $\lambda_{gg}\to\infty^{-}$,
$\lambda_{g\bullet}=\sum_{g^{\prime}}\lambda_{gg^{\prime}}\to\infty^{-}$,
implying that the first two terms of (32) vanish. Under the assumption that
$\lambda_{gg^{\prime}}<\infty$ for all $g^{\prime}\neq g$ we have that
$\lambda_{g^{\prime}g}/\lambda_{g\bullet}\to 0$ when
$\lambda_{gg}\to\infty^{-}$ for all $g^{\prime}\neq g$. Thus, all terms of the
sum also vanish as Lemma 4 implies that the
${\bm{\mathbf{\Omega}}}_{g^{\prime}}$ are all bounded. Hence, when
$\lambda_{gg}\to\infty^{-}$ and $\lambda_{gg^{\prime}}<\infty$ for all
$g^{\prime}\neq g$, the zero gradient equation reduces to
$\hat{{\bm{\mathbf{\Omega}}}}_{g}\\!-\bm{\mathbf{T}}_{g}=\bm{\mathbf{0}}$,
implying the stated left-hand limit.

(iv) The proof strategy follows the proof of item iii. Multiply the zero
gradient equation (27) for the $g_{1}$th class with
$n_{g_{1}}/\lambda_{g_{1}g_{2}}$ to obtain:

(33)
$\frac{n_{g_{1}}}{\lambda_{g_{1}g_{2}}}\hat{{\bm{\mathbf{\Omega}}}}_{g_{1}}^{-1}\\!-\frac{n_{g_{1}}}{\lambda_{g_{1}g_{2}}}\bm{\mathbf{S}}_{g_{1}}-\frac{\lambda_{g_{1}\bullet}}{\lambda_{g_{1}g_{2}}}(\hat{{\bm{\mathbf{\Omega}}}}_{g_{1}}\\!-\bm{\mathbf{T}}_{g_{1}})+\sum_{g^{\prime}\neq
g_{1}}\frac{\lambda_{g^{\prime}g_{1}}}{\lambda_{g_{1}g_{2}}}({\bm{\mathbf{\Omega}}}_{g^{\prime}}\\!-\bm{\mathbf{T}}_{g^{\prime}})=\bm{\mathbf{0}}.$

The first two terms are immediately seen to vanish when
$\lambda_{g_{1}g_{2}}\to\infty^{-}$. Under the assumption that all penalties
except $\lambda_{g_{1}g_{2}}$ are finite, we have that
$\lambda_{g_{1}\bullet}/\lambda_{g_{1}g_{2}}\to 1$ for
$\lambda_{g_{1}g_{2}}\to\infty^{-}$. Similarly, all elements of the sum term
in (33) vanish except the element where $g^{\prime}=g_{2}$. Hence, when
$\lambda_{g_{1}g_{2}}\to\infty^{-}$ and when
$\lambda_{g_{1}^{\prime}g_{2}^{\prime}}<\infty$ for all
$\\{g_{1}^{\prime},g_{2}^{\prime}\\}\neq\\{g_{1},g_{2}\\}$, the zero gradient
equation for class $g_{1}$ reduces to:

(34)
$-(\hat{{\bm{\mathbf{\Omega}}}}_{g_{1}}\\!-\bm{\mathbf{T}}_{g_{1}})+({\bm{\mathbf{\Omega}}}_{g_{2}}\\!-\bm{\mathbf{T}}_{g_{2}})=\bm{\mathbf{0}}.$

Conversely, by multiplying the zero gradient equation (27) for the $g_{2}$th
class with $n_{g_{2}}/\lambda_{g_{1}g_{2}}$ one obtains, through the same
development as above, that the zero gradient equation for class $g_{2}$
reduces to the $\hat{{\bm{\mathbf{\Omega}}}}_{g_{2}}$-analogy of equation
(34). The result (34) then immediately implies the stated limiting result. ∎

######  Corollary 2.

Consider item iv of Proposition 2. When, in addition,
$\bm{\mathbf{T}}_{g_{1}}=\bm{\mathbf{T}}_{g_{2}}$, we have that

$\lim\limits_{\lambda_{g_{1}g_{2}}\to\infty^{-}}({\hat{\bm{\mathbf{\Omega}}}}{}_{g_{1}}-\bm{\mathbf{T}}_{g_{1}})=\lim\limits_{\lambda_{g_{1}g_{2}}\to\infty^{-}}({\hat{\bm{\mathbf{\Omega}}}}{}_{g_{2}}-\bm{\mathbf{T}}_{g_{2}})\qquad\Longrightarrow\qquad\hat{{\bm{\mathbf{\Omega}}}}_{g_{1}}=\hat{{\bm{\mathbf{\Omega}}}}_{g_{2}}.$

###### Proof of Corollary 2.

The implication follows directly by using
$\bm{\mathbf{T}}_{g_{1}}=\bm{\mathbf{T}}_{g_{2}}$ in (34). ∎

###### Proof of Proposition 3.

The result follows directly from Proposition 1 and Lemma 2. ∎

###### Proof of Proposition 4.

Note that line 8 of Algorithm 1 implies that the initializing estimates are
p.d. Moreover, regardless of the value of the fused penalties (in the feasible
domain), the estimate in line 11 of Algorithm 1 is p.d. as a consequence of
Proposition 2. ∎

## References

  * Alizadeh et al. [2000] A. A. Alizadeh, M. B. Eisen, R. E. Davis, C. Ma, I. S. Lossos, A. Rosenwald, J. C. Boldrick, H. Sabet, T. Tran, X. Yu, J. I. Powell, L. Yang, G. E. Marti, T. Moore, J. Hudson, L. Lu, D. B. Lewis, R. Tibshirani, G. Sherlock, W. C. Chan, T. C. Greiner, D. D. Weisenburger, J. O. Armitage, R. Warnke, R. Levy, W. Wilson, M. R. Grever, J. C. Byrd, D. Botstein, P. O. Brown, and L. M. Staudt.  Distinct types of diffuse large B-cell lymphoma identified by gene expression profiling.  _Nature_ , 403(6769):503–511, 2000.
  * Banerjee et al. [2008] O. Banerjee, L. El Ghaoui, and A. D’Aspremont.  Model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data.  _The Journal of Machine Learning Research_ , 9:485–516, 2008.
  * Barabási [2009] A. L. Barabási.  Scale-free networks: A decade and beyond.  _Science_ , 325(5939):412–413, 2009.
  * Barabási and Albert [1999] A. L. Barabási and R. Albert.  Emergence of scaling in random networks.  _Science_ , 286(5439):509–512, 1999.
  * Barrett et al. [2013] T. Barrett, S. E. Wilhite, P. Ledoux, C. Evangelista, I. F. Kim, M. Tomashevsky, K. A. Marshall, K. H. Phillippy, P. M. Sherman, M. Holko, A. Yefanov, H. Lee, N. Zhang, C. L. Robertson, N. Serova, S Davis, and A. Soboleva.  NCBI GEO: Archive for functional genomics data sets–update.  _Nucleic Acids Research_ , 41(D1):D991–D995, 2013\.
  * Bera and Bilias [2001] A. K. Bera and Y. Bilias.  Rao’s score, Neyman’s $c(\alpha)$ and Silvey’s LM tests: An essay on historical developments and some new results.  _Journal of Statistical Planning and Inference_ , 97(1):9–44, 2001.
  * Bidère et al. [2009] N. Bidère, V. N. Ngo, J. Lee, C. Collins, L. Zheng, F. Wan, R. E. Davis, G. Lenz, D. E. Anderson, D. Arnoult, A. Vazquez, K. Sakai, J. Zhang, Z. Meng, T. D. Veenstra, L. M. Staudt, and M. J. Lenardo.  Casein kinase 1$\alpha$ governs antigen-receptor-induced NF-$\kappa$B activation and human lymphoma cell survival.  _Nature_ , 458(7234):92–96, 2009.
  * Bilgrau and Falgreen [2014] A. E. Bilgrau and S. Falgreen.  _DLBCLdata : Automated and Reproducible Download and Preprocessing of DLBCL Data_, 2014.  URL http://github.com/AEBilgrau/DLBCLdata.  R package version 0.9.
  * Bilgrau et al. [2015] A. E. Bilgrau, P. S. Eriksen, K. Dybkær, and M. Bøgsted.  Estimation of a common covariance matrix for multiple classes with applications in meta- and discriminant analysis.  _Submitted to Annals of Applied Statistics, arXiv:1503.07990_ , 269553, 2015.
  * Browning et al. [1997] J. L. Browning, I. D. Sizing, P. Lawton, P. R. Bourdon, P. D. Rennert, G. R. Majeau, C. M. Ambrose, C. Hession, K. Miatkowski, D. A. Griffiths, Ngam ek A., Meier W., Benjamin C. D., and Hochman P. S.  Characterization of lymphotoxin-$\alpha\beta$ complexes on the surface of mouse lymphocytes.  _The Journal of Immunology_ , 159(7):3288–3298, 1997.
  * Care et al. [2013] M. A. Care, S. Barrans, L. Worrillow, A. Jack, D. R. Westhead, and R. M. Tooze.  A microarray platform-independent classification tool for cell of origin class allows comparative analysis of gene expression in diffuse large B-cell lymphoma.  _PLoS One_ , 8(2):e55895, 2013.
  * Dai et al. [2005] M. Dai, P. Wang, A. D. Boyd, G. Kostov, B. Athey, E. G. Jones, W. E. Bunney, R. M. Myers, T. P. Speed, H. Akil, S. J. Watson, and F. Meng.  Evolving gene/transcript definitions significantly alter the interpretation of GeneChip data.  _Nucleic Acids Research_ , 33(20):e175, 2005.
  * Danaher et al. [2014] P. Danaher, P. Wang, and D. M. Witten.  The joint graphical lasso for inverse covariance estimation across multiple classes.  _Journal of the Royal Statistical Society, Series B_ , 76(2):373–397, 2014.
  * Dybkær et al. [2015] K. Dybkær, M. Bøgsted, S. Falgreen, J. S. Bødker, M. K. Kjeldsen, A. Schmitz, A. E. Bilgrau, Z. Y. Xu-Monette, L. Li, K. S. Bergkvist, M. B. Laursen, M. Rodrigo-Domingo, S. C. Marques, S. B. Rasmussen, M. Nyegaard, M. Gaihede, M. B. Møller, R. J. Samworth, R. D. Shah, P. Johansen, T. C. El-Galaly, K. H. Young, and H. E. Johnsen.  A diffuse large B-cell lymphoma classification system that associates normal B-cell subset phenotypes with prognosis.  _Journal Of Clinical Oncology_ , 33(12):1379–1388, 2015.
  * Eddelbuettel [2013] D. Eddelbuettel.  _Seamless R and C++ Integration with Rcpp_.  Springer-Verlag, New York, 2013.
  * Eddelbuettel and François [2011] D. Eddelbuettel and R. François.  Rcpp: Seamless R and C++ integration.  _Journal of Statistical Software_ , 40(8), 2011.
  * Efron et al. [2001] B. Efron, R. Tibshirani, J. D. Storey, and V. Tusher.  Empirical Bayes analysis of a microarray experiment.  _Journal of the American Statistical Association_ , 96:1151–1160, 2001.
  * François et al. [2012] R. François, D. Eddelbuettel, and D. Bates.  _RcppArmadillo : Rcpp Integration for Armadillo Templated Linear Algebra Library_, 2012.  URL http://CRAN.R-project.org/package=RcppArmadillo.  R package version 0.3.6.1.
  * Friedman et al. [2008] J. Friedman, T. Hastie, and R. Tibshirani.  Sparse inverse covariance estimation with the graphical lasso.  _Biostatistics_ , 9(3):432–41, 2008.
  * Gautier et al. [2004] L. Gautier, L. Cope, B. M. Bolstad, and R. A. Irizarry.  affy—analysis of Affymetrix GeneChip data at the probe level.  _Bioinformatics_ , 20(3):307–315, 2004.
  * Guo et al. [2011] Y. Guo, E. Levina, G. Michailidis, and J. Zhu.  Joint estimation of multiple graphical models.  _Biometrika_ , 98(1):1–15, 2011.
  * Jones and West [2005] B. Jones and M. West.  Covariance decomposition in undirected Gaussian graphical models.  _Biometrika_ , 92:779–786, 2005.
  * Kanehisa and Goto [2000] M. Kanehisa and S. Goto.  KEGG: Kyoto Encyclopedia of Genes and Genomes.  _Nucleic Acids Research_ , 28(1):27–30, 2000\.
  * Lauritzen [1996] S. L. Lauritzen.  _Graphical models_.  Clarendon Press, Oxford, 1996.
  * Liu et al. [2010] H. Liu, K. Roeder, and L. Wasserman.  Stability approach to regularization selection (StARS) for high dimensional graphical models.  In J.D. Lafferty, C.K.I. Williams, J. Shawe-Taylor, R.S. Zemel, and A. Culotta, editors, _Advances in Neural Information Processing Systems 23_ , pages 1432–1440. Curran Associates, Inc., 2010.
  * Lu and Zhang [2006] X. Lu and X. Zhang.  The effect of GeneChip gene definitions on the microarray study of cancers.  _Bioessays_ , 28(7):739–46, 2006.
  * Mei et al. [2011] S. Mei, X. Zhang, and M. Cao.  _Power Grid Complexity_.  Tsinghua University Press, Beijing and Springer-Verlag Berlin, 2011.
  * Mersmann [2014] O. Mersmann.  _microbenchmark : Accurate Timing Functions_, 2014.  URL http://CRAN.R-project.org/package=microbenchmark.  R package version 1.4-2.
  * Newman [2010] M. E. J. Newman.  _Networks: An Introduction_.  Oxford University Press, Oxford, 2010.
  * Nowakowski et al. [2015] G. S. Nowakowski, B. LaPlant, W. R. Macon, C. B. Reeder, J. M. Foran, G. D. Nelson, C. A. Thompson, C. E. Rivera, D. J. Inwards, I. N. Micallef, P. B. Johnston, L. F. Porrata, S. M. Ansell, R. D. Gascoyne, T. M. Habermann, and T. E. Witzig.  Lenalidomide combined with R-CHOP overcomes negative prognostic impact of non-germinal center B-cell phenotype in newly diagnosed diffuse large B-cell lymphoma: A phase II study.  _Journal of Clinical Oncology_ , 33(3):251–257, 2015.
  * Peeters et al. [forthcoming] C. F. W. Peeters, A. E. Bilgrau, and W. N. van Wieringen.  _rags2ridges : Ridge Estimation of Precision Matrices from High-Dimensional Data_, forthcoming.  R package version 2.0.
  * Peterson et al. [2015] C. Peterson, F. C. Stingo, and M. Vannucci.  Bayesian inference of multiple Gaussian graphical models.  _Journal of the American Statistical Association_ , 110(509):159–174, 2015.
  * Price et al. [2015] B. S. Price, C. J. Geyer, and A. J. Rothman.  Ridge fusion in statistical learning.  _Journal of Computational and Graphical Statistics_ , 24(2):439–454, 2015.
  * R Core Team [2012] R Core Team.  _R : A Language and Environment for Statistical Computing_.  R Foundation for Statistical Computing, Vienna, Austria, 2012.  URL http://www.R-project.org/.
  * Roschewski et al. [2014] M. Roschewski, L. M. Staudt, and W. H. Wilson.  Diffuse large B-cell lymphoma-treatment approaches in the molecular era.  _Nature Reviews Clinical Oncology_ , 11(1):12–23, 2014.
  * Ruan et al. [2011] J. Ruan, P. Martin, R. R. Furman, S. M. Lee, K. Cheung, J. M. Vose, A. LaCasce, J. Morrison, R. Elstrom, S. Ely, A. Chadburn, E. Cesarman, M. Coleman, and J. P. Leonard.  Bortezomib plus CHOP-rituximab for previously untreated diffuse large B-cell lymphoma and mantle cell lymphoma.  _Journal of Clinical Oncology_ , 29(6):690–697, 2011.
  * Sandberg and Larsson [2007] R. Sandberg and O. Larsson.  Improved precision and accuracy for microarrays using updated probe set definitions.  _BMC Bioinformatics_ , 8(1):48, 2007.
  * Sanderson [2010] C. Sanderson.  _Armadillo : An Open Source C++ Linear Algebra Library for Fast Prototyping and Computationally Intensive Experiments._ Technical Report, NICTA, 2010.  URL http://arma.sourceforge.net.
  * Schäfer and Strimmer [2005] J. Schäfer and K. Strimmer.  A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics.  _Statistical Applications in Genetics and Molecular Biology_ , 4:art. 32, 2005.
  * Schuetz et al. [2012] J. M. Schuetz, N. A. Johnson, R. D. Morin, D. W. Scott, K. Tan, S Ben-Nierah, M Boyle, G. W. Slack, M. A. Marra, J. M. Connors, A. R. Brooks-Wilson, and R. D. Gascoyne.  BCL2 mutations in diffuse large B-cell lymphoma.  _Leukemia_ , 26(6):1383–90, 2012.
  * The Non-Hodgkin’s Lymphoma Classification Project [1997] The Non-Hodgkin’s Lymphoma Classification Project.  A clinical evaluation of the international lymphoma study group classification of non-Hodgkin’s lymphoma.  _Blood_ , 89(11):3909–3918, 1997.
  * van Wieringen and Peeters [2015] W. N. van Wieringen and C. F. W. Peeters.  Ridge estimation of inverse covariance matrices from high-dimensional data.  __Submitted to_ Computational Statistics & Data Analysis, arXiv:1403.0904v3_, 2015.
  * Vujačić et al. [2015] I. Vujačić, A. Abbruzzo, and E. Wit.  A computationally fast alternative to cross-validation in penalized gaussian graphical models.  _Journal of Statistical Computation and Simulation_ , (ahead-of-print):1–13, 2015.  doi: 10.1080/00949655.2014.992020.
  * Watts and Strogatz [1998] D. J. Watts and S. H. Strogatz.  Collective dynamics of ‘small-world’ networks.  _Nature_ , 393(6684):440–442, 1998.
  * Williams-Abbott et al. [1997] L. Williams-Abbott, B. N. Walter, T. C. Cheung, C. R. Goh, A. G. Porter, and C. F. Ware.  The lymphotoxin-$\alpha$ (lt$\alpha$) subunit is essential for the assembly, but not for the receptor specificity, of the membrane-anchored lt$\alpha 1\beta 2$ heterotrimeric ligand.  _The Journal of Biological Chemistry_ , 271(31):19451–6, 1997.
  * Witten and Tibshirani [2009] D. M. Witten and R. Tibshirani.  Covariance-regularized regression and classification for high-dimensional problems.  _Journal of the Royal Statistical Society, Series B_ , 71:615–636, 2009.
  * Yang et al. [2012] Y. Yang, A. L. Shaffer, N. C. T. Emre, M. Ceribelli, M. Zhang, G. Wright, W. Xiao, J. Powell, J. Platig, H. Kohlhammer, Young R. M., H. Zhao, Y. Yang, W. Xu, J. J. Buggy, S. Balasubramanian, L. A. Mathews, P. Shinn, R. Guha, M. Ferrer, C. Thomas, T. A. Waldmann, and L. M. Staudt.  Exploiting synthetic lethality for the therapy of ABC diffuse large B cell lymphoma.  _Cancer cell_ , 21(6):723–737, 2012.
  * Yuan and Lin [2007] M. Yuan and Y. Lin.  Model selection and estimation in the Gaussian graphical model.  _Biometrika_ , 94:19–35, 2007.
  * Yuan [2008] Y. Yuan.  Efficient computation of $\ell_{1}$ regularized estimates in Gaussian graphical models.  _Journal of Computational and Graphical Statistics_ , 17:809–826, 2008.
  * Zhang and Wiemann [2009] J. D. Zhang and S. Wiemann.  KEGGgraph: A graph approach to KEGG pathway in R and Bioconductor.  _Bioinformatics_ , 25(11):1470–1471, 2009.

SUPPLEMENTARY MATERIAL

## 1\. Alternative fused ridge solutions

This section derives two equivalent (in terms of (7)) alternative updating
schemes to (8). The motivation for the exploration of these alternative
recursive estimators is twofold. First, alternative recursions can exhibit
differing numerical (in)stability for extreme values of the penalty matrix
${\bm{\mathbf{\Lambda}}}=[\lambda_{g_{1}g_{2}}]$. Second, they provide
additional intuition and understanding of the targeted fused ridge estimator.

The general strategy to finding the alternatives is to rewrite the gradient
equation (27) into the non-fused form (28), which we will repeat here:

(S1)
$\hat{{\bm{\mathbf{\Omega}}}}_{g_{0}}^{-1}-\bar{\bm{\mathbf{S}}}_{g_{0}}-\bar{\lambda}_{g_{0}}(\hat{{\bm{\mathbf{\Omega}}}}_{g_{0}}\\!-\bar{\bm{\mathbf{T}}}_{g_{0}})=\bm{\mathbf{0}},$

where $\bar{\lambda}_{g_{0}}$, $\bar{\bm{\mathbf{T}}}_{g_{0}}$, and
$\bar{\bm{\mathbf{S}}}_{g_{0}}$ do not depend on
${\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}$. Note that an explicit closed-form
solution to (S1) exists in the form of (7).

### 1.1. First alternative

The first alternative scheme is straightforward. Rewrite (27) to:

(S2) $\displaystyle\bm{\mathbf{0}}$
$\displaystyle=n_{g_{0}}{\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}^{-1}\\!-n_{g_{0}}\bm{\mathbf{S}}_{g_{0}}-\lambda_{g_{0}\bullet}({\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}\\!-\bm{\mathbf{T}}_{g_{0}})+\sum_{g\neq
g_{0}}\lambda_{gg_{0}}({\bm{\mathbf{\Omega}}}_{g}\\!-\bm{\mathbf{T}}_{g})$
$\displaystyle=n_{g_{0}}{\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}^{-1}\\!-n_{g_{0}}\bm{\mathbf{S}}_{g_{0}}-\lambda_{g_{0}\bullet}\left\\{{\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}\\!-\bigg{[}\bm{\mathbf{T}}_{g_{0}}+\sum_{g\neq
g_{0}}\frac{\lambda_{gg_{0}}}{\lambda_{g_{0}\bullet}}({\bm{\mathbf{\Omega}}}_{g}\\!-\bm{\mathbf{T}}_{g})\bigg{]}\right\\},$

where $\lambda_{g_{0}\bullet}=\sum_{g}\lambda_{gg_{0}}$. In terms of (S1), we
thus have the updating scheme given in equation (9). As stated in the main
text, it has the intuitive interpretation that a fused class target is used
which is a combination of the class-specific target and the ‘target corrected’
estimates of remaining classes.

### 1.2. Second alternative

We now derive a second alternative recursion scheme. Add and subtract
$\lambda_{g_{0}\bullet}\sum_{g\neq
g_{0}}\lambda_{gg_{0}}{\bm{\mathbf{\Omega}}}_{g}$, to (S2) and rewrite such
that:

$\displaystyle\bm{\mathbf{0}}$
$\displaystyle=n_{g_{0}}{\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}^{-1}\\!-n_{g_{0}}\bm{\mathbf{S}}_{g_{0}}-\lambda_{g_{0}\bullet}({\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}\\!-\bm{\mathbf{T}}_{g_{0}})+\lambda_{g_{0}\bullet}\sum_{g\neq
g_{0}}\lambda_{gg_{0}}{\bm{\mathbf{\Omega}}}_{g}+\sum_{g\neq
g_{0}}\lambda_{gg_{0}}({\bm{\mathbf{\Omega}}}_{g}\\!-\bm{\mathbf{T}}_{g})-\lambda_{g_{0}\bullet}\sum_{g\neq
g_{0}}\lambda_{gg_{0}}{\bm{\mathbf{\Omega}}}_{g}$
$\displaystyle=n_{g_{0}}{\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}^{-1}\\!-n_{g_{0}}\bm{\mathbf{S}}_{g_{0}}-\lambda_{g_{0}\bullet}\left[{\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}\\!-\bigg{(}\bm{\mathbf{T}}_{g_{0}}+\sum_{g\neq
g_{0}}\lambda_{gg_{0}}{\bm{\mathbf{\Omega}}}_{g}\bigg{)}\right]+\sum_{g\neq
g_{0}}\lambda_{gg_{0}}{\bm{\mathbf{\Omega}}}_{g}-\sum_{g\neq
g_{0}}\lambda_{gg_{0}}\bm{\mathbf{T}}_{g}-\lambda_{g_{0}\bullet}\sum_{g\neq
g_{0}}\lambda_{gg_{0}}{\bm{\mathbf{\Omega}}}_{g}$
$\displaystyle=n_{g_{0}}{\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}^{-1}\\!-n_{g_{0}}\bm{\mathbf{S}}_{g_{0}}-\lambda_{g_{0}\bullet}\left[{\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}\\!-\bigg{(}\bm{\mathbf{T}}_{g_{0}}+\sum_{g\neq
g_{0}}\lambda_{gg_{0}}{\bm{\mathbf{\Omega}}}_{g}\bigg{)}\right]-\sum_{g\neq
g_{0}}\lambda_{gg_{0}}\bm{\mathbf{T}}_{g}-(\lambda_{g_{0}\bullet}{-}1)\sum_{g\neq
g_{0}}\lambda_{gg_{0}}{\bm{\mathbf{\Omega}}}_{g}$
$\displaystyle=n_{g_{0}}{\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}^{-1}-n_{g_{0}}\left[\bm{\mathbf{S}}_{g_{0}}+\frac{\lambda_{g_{0}\bullet}{-}1}{n_{g_{0}}}\sum_{g\neq
g_{0}}\lambda_{gg_{0}}{\bm{\mathbf{\Omega}}}_{g}+\sum_{g\neq
g_{0}}\frac{\lambda_{gg_{0}}}{n_{g_{0}}}\bm{\mathbf{T}}_{g}\right]-\lambda_{g_{0}\bullet}\left[{\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}\\!-\bigg{(}\bm{\mathbf{T}}_{g_{0}}+\sum_{g\neq
g_{0}}\lambda_{gg_{0}}{\bm{\mathbf{\Omega}}}_{g}\bigg{)}\right].$

Dividing by $n_{g_{0}}$ gives

$\displaystyle\bm{\mathbf{0}}={\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}^{-1}-\left[\bm{\mathbf{S}}_{g_{0}}+\frac{\lambda_{g_{0}\bullet}{-}1}{n_{g_{0}}}\sum_{g\neq
g_{0}}\lambda_{gg_{0}}{\bm{\mathbf{\Omega}}}_{g}+\sum_{g\neq
g_{0}}\frac{\lambda_{gg_{0}}}{n_{g_{0}}}\bm{\mathbf{T}}_{g}\right]-\frac{\lambda_{g_{0}\bullet}}{n_{g_{0}}}\left[{\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}\\!-\bigg{(}\bm{\mathbf{T}}_{g_{0}}+\sum_{g\neq
g_{0}}\lambda_{gg_{0}}{\bm{\mathbf{\Omega}}}_{g}\bigg{)}\right],$

which brings the expression to the desired form (S1) with the updating scheme

$\displaystyle\bar{\bm{\mathbf{S}}}_{g_{0}}=\bm{\mathbf{S}}_{g_{0}}+\frac{\lambda_{g_{0}\bullet}{-}1}{n_{g_{0}}}\sum_{g\neq
g_{0}}\lambda_{gg_{0}}{\bm{\mathbf{\Omega}}}_{g}+\sum_{g\neq
g_{0}}\frac{\lambda_{gg_{0}}}{n_{g_{0}}}\bm{\mathbf{T}}_{g},\quad\bar{\bm{\mathbf{T}}}_{g_{0}}=\bm{\mathbf{T}}_{g_{0}}+\sum_{g\neq
g_{0}}\lambda_{gg_{0}}{\bm{\mathbf{\Omega}}}_{g},\quad\text{and}\quad\bar{\lambda}_{g_{0}}=\frac{\lambda_{g_{0}\bullet}}{n_{g_{0}}}.$

Again, a solution for ${\hat{\bm{\mathbf{\Omega}}}}{}_{g_{0}}$ with fixed
${\bm{\mathbf{\Omega}}}_{g}$ for all $g\neq g_{0}$, is available through Lemma
1 [42] and is given in (7).

### 1.3. Motivation

Though seemingly more complicated, these alternative updating schemes can be
numerically more stable for extreme penalties. In both alternatives, we see
that $\bar{\bm{\mathbf{S}}}_{g_{0}}$ is p.s.d. for (nearly) all very large and
very small penalties. Likewise, $\bar{\bm{\mathbf{T}}}_{g_{0}}$ is always
positive definite. Compare the alternative expressions to the updating scheme
given by (8) which can be seen to be numerically unstable for very large
penalties: For very large $\lambda_{gg}$ or $\lambda_{g_{1}g_{2}}$ the
$\bar{\bm{\mathbf{S}}}_{g_{0}}$ in (8) may be a matrix with numerically
extreme values. This implies ill-conditioning and numerical instability under
finite computer precision. On the other hand, ‘updating’ the target matrix
will generally lead to updates for which the resulting estimator is not
rotationally equivariant. This implies a reduction in computational speed.

## 2\. Estimation in special cases

Here we explore scenarios for which we arrive at explicit targeted fused ridge
estimators. These explicit solutions further insight into the behavior of the
general estimator and they can provide computational speed-ups in certain
situations. Three special cases are covered:

  1. I.

$\lambda_{gg^{\prime}}=0$ for all $g\neq g^{\prime}$ or equivalently
$\sum_{g^{\prime}}\lambda_{gg^{\prime}}=\lambda_{g\bullet}=\lambda_{gg}$ for
all $g$;

  2. II.

${\bm{\mathbf{\Omega}}}_{1}=\cdots={\bm{\mathbf{\Omega}}}_{G}$ and
$\bm{\mathbf{T}}_{g}=\bm{\mathbf{T}}$ for all $g$;

  3. III.

$\bm{\mathbf{T}}_{g}=\bm{\mathbf{T}}$ for all $g$, $\lambda_{gg}=\lambda$ for
all $g$, $\lambda_{g_{1}g_{2}}=\lambda_{f}$ for all $g_{1}\neq g_{2}$, and
$\lambda_{f}\to\infty^{-}$.

### 2.1. Special case I

When $\sum_{g^{\prime}}\lambda_{gg^{\prime}}=\lambda_{g\bullet}=\lambda_{gg}$
for all $g$, we have that $\sum_{g^{\prime}\neq
g}\lambda_{gg^{\prime}}=\sum_{g^{\prime}\neq g}\lambda_{g^{\prime}g}=0$ for
all $g$. Hence, all fusion penalties are zero. The zero gradient equation (27)
for class $g$ then no longer hinges upon information from the remaining
classes $g^{\prime}$. The targeted fused precision estimate for class $g$ then
reduces to (29) of Corollary 1. This case thus coincides, as expected, with
obtaining $G$ decoupled non-fused ridge precision estimates. A special case
that results in the same estimates occurs when considering
$\lambda_{g_{1}g_{2}}=\lambda_{f}$ for all $g_{1}\neq g_{2}$ and $\lambda_{f}$
is taken to be $0$.

### 2.2. Special case II

Suppose ${\bm{\mathbf{\Omega}}}_{g}={\bm{\mathbf{\Omega}}}$ and
$\bm{\mathbf{T}}_{g}=\bm{\mathbf{T}}$ for all $g$. Consequently, the fusion
penalty term vanishes irrespective of the values of the
$\lambda_{g_{1}g_{2}}$, $g_{1}\neq g_{2}$. The zero gradient equation (27)
then reduces to

$\bm{\mathbf{0}}=n_{g}{\hat{\bm{\mathbf{\Omega}}}}{}^{-1}-n_{g}\bm{\mathbf{S}}_{g}-\lambda_{gg}({\hat{\bm{\mathbf{\Omega}}}}{}-\bm{\mathbf{T}}),$

for each class $g$. Adding all $G$ equations implies:

$\displaystyle\bm{\mathbf{0}}$
$\displaystyle=\sum_{g=1}^{G}n_{g}{\hat{\bm{\mathbf{\Omega}}}}{}^{-1}-\sum_{g=1}^{G}n_{g}\bm{\mathbf{S}}_{g}-\left(\sum_{g=1}^{G}\lambda_{gg}\right)({\hat{\bm{\mathbf{\Omega}}}}{}-\bm{\mathbf{T}})$
$\displaystyle=n_{\bullet}{\hat{\bm{\mathbf{\Omega}}}}{}^{-1}-n_{\bullet}\bm{\mathbf{S}}_{\bullet}-\operatorname*{tr}({\bm{\mathbf{\Lambda}}})({\hat{\bm{\mathbf{\Omega}}}}{}-\bm{\mathbf{T}})$
(S3)
$\displaystyle={\hat{\bm{\mathbf{\Omega}}}}{}^{-1}-\left[\bm{\mathbf{S}}_{\bullet}-\frac{\operatorname*{tr}({\bm{\mathbf{\Lambda}}})}{n_{\bullet}}\bm{\mathbf{T}}\right]-\frac{\operatorname*{tr}({\bm{\mathbf{\Lambda}}})}{n_{\bullet}}{\hat{\bm{\mathbf{\Omega}}}}{}.$

We recognize that (S3) is of the form (22). Lemma 1 may then be directly
applied to obtain the solution:

(S4)
$\displaystyle{\hat{\bm{\mathbf{\Omega}}}}{}({\bm{\mathbf{\Lambda}}})=\left\\{\left[\lambda^{\ast}\bm{\mathbf{I}}_{p}+\frac{1}{4}(\bm{\mathbf{S}}_{\bullet}-\lambda^{\ast}\bm{\mathbf{T}})^{2}\right]^{1/2}+\frac{1}{2}(\bm{\mathbf{S}}_{\bullet}-\lambda^{\ast}\bm{\mathbf{T}})\right\\}^{-1},$

where
$\lambda^{\ast}=\operatorname*{tr}({\bm{\mathbf{\Lambda}}})/n_{\bullet}$.
Hence, this second special case gives a non-fused penalized estimate that uses
the pooled covariance matrix. It can be interpreted as an averaged penalized
estimator. It is of importance in testing equality of the class precision
matrices (see Section 4.1 of the main text).

### 2.3. Special case III

Suppose that $\bm{\mathbf{T}}_{g}=\bm{\mathbf{T}}$ for all $g$, that
$\lambda_{gg}=\lambda$ for all $g$, and that
$\lambda_{g_{1}g_{2}}=\lambda_{f}$ for all $g_{1}\neq g_{2}$. The main
optimization problem then reduces to (6). Clearly, for
$\lambda_{f}\to\infty^{-}$ the fused penalty

$\displaystyle
f^{\text{FR}}(\\{\mathbf{{\bm{\mathbf{\Omega}}}_{g}}\\};\lambda,\lambda_{f},\bm{\mathbf{T}})=\frac{\lambda}{2}\sum_{g}\big{\|}{\bm{\mathbf{\Omega}}}_{g}\\!-\bm{\mathbf{T}}\big{\|}_{F}^{2}+\frac{\lambda_{f}}{4}\sum_{g_{1},g_{2}}\big{\|}({\bm{\mathbf{\Omega}}}_{g_{1}}\\!-{\bm{\mathbf{\Omega}}}_{g_{2}})\big{\|}_{F}^{2}$

is minimized when
${\bm{\mathbf{\Omega}}}_{1}={\bm{\mathbf{\Omega}}}_{2}=\cdots={\bm{\mathbf{\Omega}}}_{G}$.
This is also implied, more rigorously, by Corollary 2. Hence, the problem
reduces to the special case of section 2.2 considered above. The solution to
the penalized ML problem when $\lambda_{f}=\infty$ is then given by (S4) where
$\operatorname*{tr}({\bm{\mathbf{\Lambda}}})$ now implies $G\lambda$.

## 3\. Fused Kullback-Leibler approximate cross-validation

### 3.1. Motivation

In $\ell_{1}$-penalized estimation of the precision matrix, penalty selection
implies (graphical) model selection: Regularization results in automatic
selection of conditional dependencies. One then seeks to select an optimal
value for the penalty parameter in terms of model selection consistency. To
this end, the Bayesian information criterion (BIC), the extended BIC (EBIC),
and the stability approach to regularization selection (StARS) are appropriate
[25]. The (fused) $\ell_{2}$-penalty will not directly induce sparsity in
precision matrix estimates. Hence, in $\ell_{2}$-penalized problems it is
natural to choose the penalty parameters on the basis of efficiency loss. Of
interest are then estimators of the Kullback-Leibler (KL) divergence, such as
LOOCV, generalized approximate cross-validation (GACV), and Akaike’s
information criterion (AIC). While superior in terms of predictive accuracy
due to its data-driven nature, the LOOCV is computationally very expensive.
Vujačić et al. [43] proposed a KL-based CV loss with superior performance to
both AIC and GACV. The proposed method has closed-form solutions and thus
provides a fast approximation to LOOCV. Here, we extend this method to provide
a computationally friendly approximation of the fused LOOCV score.

### 3.2. Formulation

Following Vujačić et al. [43], we now restate the KL approximation to LOOCV in
the fused ridge setting. Let the true precision matrix for class $g$ be
denoted by ${\bm{\mathbf{\Omega}}}_{g}$. Its estimate, shorthanded by
${\hat{\bm{\mathbf{\Omega}}}}{}_{g}$ can be obtained through Algorithm 1. The
KL divergence between the multivariate normal distributions
$\mathcal{N}_{p}(\bm{\mathbf{0}},{\bm{\mathbf{\Omega}}}_{g}^{-1})$ and
$\mathcal{N}_{p}(\bm{\mathbf{0}},{\hat{\bm{\mathbf{\Omega}}}}{}{}_{g}^{-1})$
can be shown to be:

$\operatorname{KL}({\bm{\mathbf{\Omega}}}_{g},{\hat{\bm{\mathbf{\Omega}}}}{}_{g})=\frac{1}{2}\Bigl{\\{}\operatorname*{tr}({\bm{\mathbf{\Omega}}}_{g}^{-1}{\hat{\bm{\mathbf{\Omega}}}}{}_{g})-\ln\lvert{\bm{\mathbf{\Omega}}}_{g}^{-1}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\rvert-p\Bigr{\\}}.$

For each $g$ we wish to minimize this divergence. In the fused case we
therefore consider the _fused Kullback-Leibler_ (FKL) divergence which,
motivated by the LOOCV score, is taken to be a weighted average of KL
divergences:

$\displaystyle\operatorname{FKL}(\\{{\bm{\mathbf{\Omega}}}_{g}\\},\\{{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\\})$
(S5)
$\displaystyle\quad=\frac{1}{n_{\bullet}}\sum_{g=1}^{G}n_{g}\operatorname{KL}({\bm{\mathbf{\Omega}}}_{g},{\hat{\bm{\mathbf{\Omega}}}}{}_{g})=\frac{1}{n_{\bullet}}\sum_{g=1}^{G}\frac{n_{g}}{2}\Bigl{\\{}\operatorname*{tr}({\bm{\mathbf{\Omega}}}_{g}^{-1}{\hat{\bm{\mathbf{\Omega}}}}{}_{g})-\ln\lvert{\bm{\mathbf{\Omega}}}_{g}^{-1}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\rvert-p\Bigr{\\}}.$

The FKL divergence (S5) can, using the likelihood $\eqref{eq:loglik}$, be
rewritten as

$\displaystyle\operatorname{FKL}=-\frac{1}{n_{\bullet}}\mathcal{L}\bigl{(}\\{{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\\};\\{\bm{\mathbf{S}}_{g}\\}\bigr{)}+\mathrm{bias},\quad\text{where}\quad\mathrm{bias}=\frac{1}{2n_{\bullet}}\sum_{g=1}^{G}n_{g}\operatorname*{tr}\bigl{[}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}({\bm{\mathbf{\Omega}}}_{g}^{-1}-\bm{\mathbf{S}}_{g})\bigr{]},$

and where the equality holds up to the addition of a constant. It is clear
that the $\mathrm{bias}$ term depends on the unknown true precision matrices
and thus needs to be estimated. The fused analogue to the proposal of Vujačić
et al. [43], called the _fused Kullback-Leibler approximate cross-validation_
score or simply _approximate fused LOOCV_ score, then is

(S6)
$\operatorname{\widehat{\operatorname{FKL}}}\bigl{(}{\bm{\mathbf{\Lambda}}}\bigr{)}=-\frac{1}{n_{\bullet}}\mathcal{L}\bigl{(}\\{{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\\};\\{\bm{\mathbf{S}}_{g}\\}\bigr{)}+\widehat{\mathrm{bias}},$

with

(S7)
$\widehat{\mathrm{bias}}=\frac{1}{2n_{\bullet}}\sum_{g=1}^{G}\sum_{i=1}^{n_{g}}\Bigl{\\{}{\bm{\mathrm{y}}}_{ig}^{\top}({\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{2}-{\hat{\bm{\mathbf{\Omega}}}}{}_{g}){\bm{\mathrm{y}}}_{ig}+\bar{\lambda}_{g}{\bm{\mathrm{y}}}_{ig}^{\top}({\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{4}-{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{3}){\bm{\mathrm{y}}}_{ig}\Bigr{\\}},$

and where $\bar{\lambda}_{g}=\tfrac{\lambda_{g\bullet}}{n_{g}}$. The
derivation of this estimate is given in Section 3.3. Derivation below. One
would then choose ${\bm{\mathbf{\Lambda}}}^{\star}$ such that the FKL
approximate cross-validation score is minimized:

(S8)
${\bm{\mathbf{\Lambda}}}^{\star}=\operatorname*{arg\,min}_{\mathclap{{\bm{\mathbf{\Lambda}}}}}\operatorname{\widehat{\operatorname{FKL}}}\bigl{(}{\bm{\mathbf{\Lambda}}}\bigr{)},\quad\text{subject
to:}\quad{\bm{\mathbf{\Lambda}}}\geq\bm{\mathbf{0}}\wedge\operatorname{diag}({\bm{\mathbf{\Lambda}}})>\bm{\mathbf{0}}.$

The closed form expression in (S6) implies that
${\bm{\mathbf{\Lambda}}}^{\star}$ is more rapidly determined than
${\bm{\mathbf{\Lambda}}}^{*}$. As seen in the derivation,
${\bm{\mathbf{\Lambda}}}^{*}\approx{\bm{\mathbf{\Lambda}}}^{\star}$ for large
sample sizes.

### 3.3. Derivation

Here we give, borrowing some ideas from Vujačić et al. [43], the derivation of
the estimate (S6). Let observation $i$ in class $g$ be denoted by
${\bm{\mathrm{y}}}_{ig}$ and let
$\bm{\mathbf{S}}=\bm{\mathbf{S}}_{ig}={\bm{\mathrm{y}}}_{ig}{\bm{\mathrm{y}}}_{ig}^{\top}$
be the sample covariance or scatter matrix of that observation. As before, the
singularly indexed
$\bm{\mathbf{S}}_{g}=\frac{1}{n_{g}}\sum_{i=1}^{n_{g}}\bm{\mathbf{S}}_{ig}$ is
the class-specific sample covariance matrix. Throughout this section we will
conveniently drop (some of) the explicit notation.

The $\operatorname{FKL}$ divergence reframes the $\operatorname{LOOCV}$ score
in terms of a likelihood evaluation and a bias term when $\bm{\mathbf{S}}$ is
_not_ left out of class $g$. We thus study the change in the estimate as
function of the single scatter matrix $\bm{\mathbf{S}}$. Let
${\hat{\bm{\mathbf{\Omega}}}}{}_{g}(\bm{\mathbf{S}})={\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{\neg
ig}$ be the estimate in class $g$ when $\bm{\mathbf{S}}$ is omitted. That is,
${\hat{\bm{\mathbf{\Omega}}}}{}_{g}(\bm{\mathbf{S}})$ is part of the solution
to the system

(S9)
${\bm{\mathbf{\Omega}}}{}^{-1}_{a}+\mu_{aa}{\bm{\mathbf{\Omega}}}_{a}+\mathds{1}[a{=}g]\bm{\mathbf{S}}+\sum_{b\not=a}\mu_{ab}{\bm{\mathbf{\Omega}}}_{b}+\bm{\mathbf{A}}_{a}=\bm{\mathbf{0}},\quad\text{for
all}\quad a=1,\ldots,G,$

where $\mu_{aa}=-\tfrac{\lambda_{a\bullet}}{n_{a}}$,
$\mu_{ab}=\tfrac{\lambda_{ab}}{n_{a}}$, and where $\bm{\mathbf{A}}_{a}$ is a
matrix determined by the remaining data, penalty parameters and targets. Note
that the penalized MLE can be denoted
${\hat{\bm{\mathbf{\Omega}}}}{}_{g}={\hat{\bm{\mathbf{\Omega}}}}{}_{g}(\bm{\mathbf{0}})$,
which corresponds to the ‘full’ estimate resulting from the full gradient
equation (27).

We wish to approximate ${\hat{\bm{\mathbf{\Omega}}}}{}_{g}(\bm{\mathbf{S}})$
by a Taylor expansion around
${\hat{\bm{\mathbf{\Omega}}}}{}_{g}(\bm{\mathbf{0}})$, i.e.:

${\hat{\bm{\mathbf{\Omega}}}}{}_{a}(\bm{\mathbf{S}})\approx{\hat{\bm{\mathbf{\Omega}}}}{}_{a}(\bm{\mathbf{0}})+\sum_{j,j^{\prime}}\frac{\partial{\hat{\bm{\mathbf{\Omega}}}}{}_{a}}{\partial
S_{jj^{\prime}}}S_{jj^{\prime}}.$

Differentiating (S9) w.r.t. $S_{jj^{\prime}}$, the $(j,j^{\prime})$th entry in
$\bm{\mathbf{S}}$, and equating to zero yields

$\displaystyle\bm{\mathbf{0}}$
$\displaystyle=-{\hat{\bm{\mathbf{\Omega}}}}{}{}^{-1}_{a}\frac{\partial{\hat{\bm{\mathbf{\Omega}}}}{}_{a}}{\partial
S_{jj^{\prime}}}{\hat{\bm{\mathbf{\Omega}}}}{}{}^{-1}_{a}+\mu_{aa}\frac{\partial{\hat{\bm{\mathbf{\Omega}}}}{}_{a}}{\partial
S_{jj^{\prime}}}+\mathds{1}[a{=}g]\bm{\mathbf{E}}_{jj^{\prime}}+\sum_{b\not=a}\mu_{ab}\frac{\partial{\hat{\bm{\mathbf{\Omega}}}}{}_{b}}{\partial
S_{jj^{\prime}}}$ (S10)
$\displaystyle=-{\hat{\bm{\mathbf{\Omega}}}}{}{}^{-1}_{a}\frac{\partial{\hat{\bm{\mathbf{\Omega}}}}{}_{a}}{\partial
S_{jj^{\prime}}}{\hat{\bm{\mathbf{\Omega}}}}{}{}^{-1}_{a}+\sum_{b}\mu_{ab}\frac{\partial{\hat{\bm{\mathbf{\Omega}}}}{}_{b}}{\partial
S_{jj^{\prime}}}+\mathds{1}[a{=}g]\bm{\mathbf{E}}_{jj^{\prime}},\quad\text{for
all}\quad j,j^{\prime},$

where $\bm{\mathbf{E}}_{jj^{\prime}}$ is the null matrix except for unity in
entries $(j,j^{\prime})$ and $(j^{\prime},j)$. The third term is obtained as
$\partial\bm{\mathbf{S}}/\partial
S_{jj^{\prime}}=\bm{\mathbf{E}}_{jj^{\prime}}$ by the symmetric structure of
$\bm{\mathbf{S}}$. This is also seen from the fact that
$\bm{\mathbf{S}}=\sum_{jj^{\prime}}S_{jj^{\prime}}\bm{\mathbf{E}}_{jj^{\prime}}$.
Let

$\bm{\mathbf{V}}(\bm{\mathbf{S}})_{a}=\sum_{j,j^{\prime}}\frac{\partial{\hat{\bm{\mathbf{\Omega}}}}{}_{a}}{\partial
S_{jj^{\prime}}}S_{jj^{\prime}},$

and multiply (S10) by $S_{jj^{\prime}}$ and sum over all $j,j^{\prime}$ to
obtain

(S11)
${\hat{\bm{\mathbf{\Omega}}}}{}_{a}^{-1}\bm{\mathbf{V}}(\bm{\mathbf{S}})_{a}{\hat{\bm{\mathbf{\Omega}}}}{}_{a}^{-1}-\sum_{b}\mu_{ab}\bm{\mathbf{V}}(\bm{\mathbf{S}})_{b}=\mathds{1}[a{=}g]\bm{\mathbf{S}},\quad\text{for
all}\quad a=1,\ldots,G.$

We seek the solution vector
$\bm{\mathbf{V}}=\bigl{\\{}\bm{\mathbf{V}}(\bm{\mathbf{S}})_{a}\bigr{\\}}{}_{a=1}^{G}$
of square matrices for the system of equations in (S11) which can be rewritten
in the following way. Introduce and consider the linear operator (or block
matrix):

$\displaystyle\bm{\mathbf{N}}=\bigl{\\{}\bm{\mathbf{N}}_{ab}\bigr{\\}}{}_{a,b=1}^{G}\quad\text{where}\quad\bm{\mathbf{N}}_{ab}=\begin{cases}{\hat{\bm{\mathbf{\Omega}}}}{}{}^{-1}_{a}\otimes{\hat{\bm{\mathbf{\Omega}}}}{}{}^{-1}_{a}-\mu_{aa}\bm{\mathbf{I}}_{p}\otimes\bm{\mathbf{I}}_{p}&\text{if}\quad
a=b\\\ -\mu_{ab}\bm{\mathbf{I}}_{p}\otimes\bm{\mathbf{I}}_{p}&\text{if}\quad
a\neq b\end{cases}.$

Then $\bm{\mathbf{V}}$ can be verified to be the solution to the system (S10)
as

$\displaystyle\bm{\mathbf{N}}(\bm{\mathbf{V}})_{a}$
$\displaystyle=\sum_{b}\bm{\mathbf{N}}_{ab}\bm{\mathbf{V}}(\bm{\mathbf{S}})_{b}=\bm{\mathbf{0}}\quad\text{for}\quad
a\neq g,\quad\text{and}\quad$
$\displaystyle\bm{\mathbf{N}}(\bm{\mathbf{V}})_{g}$
$\displaystyle=\sum_{b}\bm{\mathbf{N}}_{gb}\bm{\mathbf{V}}(\bm{\mathbf{S}})_{b}=\bm{\mathbf{S}}\quad\text{for}\quad
a=g.$

Hence we need to invert $\bm{\mathbf{N}}$ to solve for $\bm{\mathbf{V}}$. The
structure of $\bm{\mathbf{N}}$ is relatively simple, but there seems to be no
(if any) simple inverse. Note that
$\bm{\mathbf{N}}=\bm{\mathbf{D}}-\bm{\mathbf{M}}$ is the difference of a
(block) diagonal matrix $\bm{\mathbf{D}}$ and a matrix $\bm{\mathbf{M}}$
depending on the $\mu$’s:

$\displaystyle\bm{\mathbf{D}}_{aa}$
$\displaystyle={\hat{\bm{\mathbf{\Omega}}}}{}{}^{-1}_{a}\otimes{\hat{\bm{\mathbf{\Omega}}}}{}{}^{-1}_{a},$
$\displaystyle\bm{\mathbf{M}}_{ab}$
$\displaystyle=\mu_{ab}\bm{\mathbf{I}}_{p}\otimes\bm{\mathbf{I}}_{p}.$

In terms of the $\mu$’s we obtain to first order that

$\bm{\mathbf{N}}^{-1}=(\bm{\mathbf{D}}-\bm{\mathbf{M}})^{-1}\approx\bm{\mathbf{D}}^{-1}+\bm{\mathbf{D}}^{-1}\bm{\mathbf{M}}\bm{\mathbf{D}}^{-1},$

yielding the approximation

$\displaystyle{\hat{\bm{\mathbf{\Omega}}}}{}_{g}(\bm{\mathbf{S}})$
$\displaystyle\approx{\hat{\bm{\mathbf{\Omega}}}}{}_{g}+({\hat{\bm{\mathbf{\Omega}}}}{}_{g}\otimes{\hat{\bm{\mathbf{\Omega}}}}{}_{g}+\mu_{gg}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{2}\otimes{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{2})(\bm{\mathbf{S}})$
(S12)
$\displaystyle={\hat{\bm{\mathbf{\Omega}}}}{}_{g}+{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}+\mu_{gg}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{2}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{2},$

where
${\hat{\bm{\mathbf{\Omega}}}}{}_{g}={\hat{\bm{\mathbf{\Omega}}}}{}(\bm{\mathbf{0}})$.
To a first order in $\mu_{gg}$ this is the same as the approximation

${\hat{\bm{\mathbf{\Omega}}}}{}_{g}(\bm{\mathbf{S}})\approx{\hat{\bm{\mathbf{\Omega}}}}{}_{g}+({\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{-1}\otimes{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{-1}-\mu_{gg}\bm{\mathbf{I}}_{p}\otimes\bm{\mathbf{I}}_{p})^{-1}(\bm{\mathbf{S}}).$

We also need an approximation for
$\ln\lvert{\hat{\bm{\mathbf{\Omega}}}}{}_{g}(\bm{\mathbf{S}})\rvert$. By
first-order Taylor expansion around $\bm{\mathbf{S}}=\bm{\mathbf{0}}$ we have

$\displaystyle\ln\lvert{\hat{\bm{\mathbf{\Omega}}}}{}_{g}(\bm{\mathbf{S}})\rvert$
$\displaystyle\approx\ln\lvert{\hat{\bm{\mathbf{\Omega}}}}{}_{g}(\bm{\mathbf{0}})\rvert+\sum_{j,j^{\prime}}\operatorname*{tr}\Bigl{[}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{-1}(\bm{\mathbf{0}})\frac{\partial{\hat{\bm{\mathbf{\Omega}}}}{}_{g}}{\partial
S_{jj^{\prime}}}\Bigr{]}S_{jj^{\prime}}$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\eqref{eq:OmegaSapprox1}}}}{{\approx}}\ln\lvert{\hat{\bm{\mathbf{\Omega}}}}{}_{g}(\bm{\mathbf{0}})\rvert+\operatorname*{tr}\Bigl{[}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{-1}({\hat{\bm{\mathbf{\Omega}}}}{}_{g}\otimes{\hat{\bm{\mathbf{\Omega}}}}{}_{g}+\mu_{gg}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{2}\otimes{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{2})(\bm{\mathbf{S}})\Bigr{]}$
(S13)
$\displaystyle=\ln\lvert{\hat{\bm{\mathbf{\Omega}}}}{}_{g}(\bm{\mathbf{0}})\rvert+\operatorname*{tr}\bigl{(}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}+\mu_{gg}{\hat{\bm{\mathbf{\Omega}}}}{}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{2}\bigr{)},$

where we have used that
$\tfrac{d}{dt}\ln\lvert\bm{\mathbf{A}}(t)\rvert=\operatorname*{tr}\bigl{[}\bm{\mathbf{A}}(t)^{-1}\tfrac{d\bm{\mathbf{A}}}{dt}\bigr{]}$
and $\frac{\partial{\bm{\mathbf{\Omega}}}_{g}}{\partial
S_{jj^{\prime}}}\approx({\hat{\bm{\mathbf{\Omega}}}}{}_{g}\otimes{\hat{\bm{\mathbf{\Omega}}}}{}_{g}+\mu_{gg}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{2}\otimes{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{2})(\bm{\mathbf{E}}_{jj^{\prime}}).$
We now have the necessary equations to derive the $\operatorname{FKL}$
approximate cross-validation score.

Define

(S14) $\displaystyle
f(\bm{\mathbf{A}},\bm{\mathbf{B}})=\ln|\bm{\mathbf{B}}|-\operatorname*{tr}(\bm{\mathbf{B}}\bm{\mathbf{A}})$

by which the identity

(S15)
$\displaystyle\sum_{i=1}^{n_{g}}f(\bm{\mathbf{S}}_{ig},{\bm{\mathbf{\Omega}}}_{g})=n_{g}f(\bm{\mathbf{S}}_{g},{\bm{\mathbf{\Omega}}}_{g})$

holds for all $g$. The full likelihood (3) in terms of $f$ is given by

(S16)
$\displaystyle\mathcal{L}(\\{{\bm{\mathbf{\Omega}}}_{g}\\};\\{\bm{\mathbf{S}}_{g}\\})\propto\sum_{g=1}^{G}\frac{n_{g}}{2}\Bigl{\\{}\ln|{\bm{\mathbf{\Omega}}}_{g}|-\operatorname*{tr}({\bm{\mathbf{\Omega}}}_{g}\bm{\mathbf{S}}_{g})\Bigr{\\}}=\sum_{g=1}^{G}\frac{n_{g}}{2}f(\bm{\mathbf{S}}_{g},{\bm{\mathbf{\Omega}}}_{g}),$

while the likelihood of a single $\bm{\mathbf{S}}_{ig}$ is

(S17)
$\displaystyle\mathcal{L}_{ig}({\bm{\mathbf{\Omega}}}_{g};\bm{\mathbf{S}}_{ig})\propto\frac{1}{2}\Bigl{\\{}\ln|{\bm{\mathbf{\Omega}}}_{g}|-\operatorname*{tr}({\bm{\mathbf{\Omega}}}_{g}\bm{\mathbf{S}}_{ig})\Bigr{\\}}=\frac{1}{2}f(\bm{\mathbf{S}}_{ig},{\bm{\mathbf{\Omega}}}_{g}).$

In our setting, the fused LOOCV score is given by:

$\displaystyle\operatorname{LOOCV}$
$\displaystyle=-\frac{1}{n_{\bullet}}\sum_{g=1}^{G}\sum_{i=1}^{n_{g}}\mathcal{L}_{ig}\big{(}{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{\neg
ig};\bm{\mathbf{S}}_{ig}\big{)}$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\eqref{eq:singleloglik}}}}{{=}}-\frac{1}{n_{\bullet}}\sum_{g=1}^{G}\sum_{i=1}^{n_{g}}\frac{1}{2}f(\bm{\mathbf{S}}_{ig},{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{\neg
ig})$
$\displaystyle=-\frac{1}{n_{\bullet}}\sum_{g=1}^{G}\frac{1}{2}\sum_{i=1}^{n_{g}}\Bigl{[}f\bigl{(}\bm{\mathbf{S}}_{ig},{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigr{)}+f\bigl{(}\bm{\mathbf{S}}_{ig},{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{\neg
ig}\bigr{)}-f\bigl{(}\bm{\mathbf{S}}_{ig},{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigr{)}\Bigr{]}$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\eqref{eq:fidentity}}}}{{=}}-\frac{1}{n_{\bullet}}\sum_{g=1}^{G}\frac{n_{g}}{2}f\bigl{(}\bm{\mathbf{S}}_{g},{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigr{)}-\frac{1}{n_{\bullet}}\sum_{g=1}^{G}\frac{1}{2}\sum_{i=1}^{n_{g}}\Bigl{[}f\bigl{(}\bm{\mathbf{S}}_{ig},{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{\neg
ig}\bigr{)}-f\bigl{(}\bm{\mathbf{S}}_{ig},{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigr{)}\Bigr{]}$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\eqref{eq:loglikidentity}}}}{{=}}-\frac{1}{n_{\bullet}}\mathcal{L}(\\{{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\\};\\{\bm{\mathbf{S}}_{g}\\})-\frac{1}{2n_{\bullet}}\sum_{g=1}^{G}\sum_{i=1}^{n_{g}}\Bigl{[}f\bigl{(}\bm{\mathbf{S}}_{ig},{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{\neg
ig}\bigr{)}-f\bigl{(}\bm{\mathbf{S}}_{ig},{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bigr{)}\Bigr{]}$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\eqref{eq:fdefinition}}}}{{=}}-\frac{1}{n_{\bullet}}\mathcal{L}(\\{{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\\};\\{\bm{\mathbf{S}}_{g}\\})-\frac{1}{2n_{\bullet}}\sum_{g=1}^{G}\sum_{i=1}^{n_{g}}\Bigl{[}\ln\lvert{\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{\neg
ig}\rvert-\operatorname*{tr}({\hat{\bm{\mathbf{\Omega}}}}{}_{g}^{\neg
ig}\bm{\mathbf{S}}_{ig})-\ln\lvert{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\rvert+\operatorname*{tr}({\hat{\bm{\mathbf{\Omega}}}}{}_{g}\bm{\mathbf{S}}_{ig})\Bigr{]}.$

Now, substitution of (S12) and (S13) gives the $\operatorname{FKL}$
approximate cross-validation score as an approximation to the fused LOOCV
score:

$\displaystyle\operatorname{LOOCV}\approx\widehat{\operatorname{FKL}}=-\frac{1}{n_{\bullet}}\mathcal{L}(\\{{\hat{\bm{\mathbf{\Omega}}}}{}_{g}\\};\\{\bm{\mathbf{S}}_{g}\\})+\frac{1}{2n_{\bullet}}\sum_{g=1}^{G}\sum_{i=1}^{n_{g}}{\bm{\mathbf{\zeta}}}_{ig},$

where

$\displaystyle{\bm{\mathbf{\zeta}}}_{ig}$
$\displaystyle=\operatorname*{tr}\bigl{(}{\hat{\bm{\mathbf{\Omega}}}}{}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}+\mu_{gg}{\hat{\bm{\mathbf{\Omega}}}}{}^{2}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}^{2}\bigr{)}-\operatorname*{tr}\bigl{(}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}+\mu_{gg}{\hat{\bm{\mathbf{\Omega}}}}{}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}^{2}\bigr{)}$
$\displaystyle=\operatorname*{tr}\bigl{(}{\hat{\bm{\mathbf{\Omega}}}}{}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}\bigr{)}+\mu_{gg}\operatorname*{tr}\bigl{(}{\hat{\bm{\mathbf{\Omega}}}}{}^{2}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}^{2}\bigr{)}-\operatorname*{tr}\bigl{(}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}\bigr{)}-\mu_{gg}\operatorname*{tr}\bigl{(}{\hat{\bm{\mathbf{\Omega}}}}{}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}^{2}\bigr{)}$
$\displaystyle=\operatorname*{tr}\bigl{(}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}^{2}\bigr{)}+\mu_{gg}\operatorname*{tr}\bigl{(}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}^{4}\bigr{)}-\operatorname*{tr}\bigl{(}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}\bigr{)}-\mu_{gg}\operatorname*{tr}\bigl{(}\bm{\mathbf{S}}{\hat{\bm{\mathbf{\Omega}}}}{}^{3}\bigr{)}$
$\displaystyle=\operatorname*{tr}\bigl{[}\bm{\mathbf{S}}({\hat{\bm{\mathbf{\Omega}}}}{}^{2}-{\hat{\bm{\mathbf{\Omega}}}}{})\bigr{]}+\mu_{gg}\operatorname*{tr}\bigl{[}\bm{\mathbf{S}}({\hat{\bm{\mathbf{\Omega}}}}{}^{4}-{\hat{\bm{\mathbf{\Omega}}}}{}^{3})\bigr{]}$
(S18)
$\displaystyle={\bm{\mathrm{y}}}_{ig}^{\top}({\hat{\bm{\mathbf{\Omega}}}}{}^{2}-{\hat{\bm{\mathbf{\Omega}}}}{}){\bm{\mathrm{y}}}_{ig}+\mu_{gg}{\bm{\mathrm{y}}}_{ig}^{\top}({\hat{\bm{\mathbf{\Omega}}}}{}^{4}-{\hat{\bm{\mathbf{\Omega}}}}{}^{3}){\bm{\mathrm{y}}}_{ig}.$

To arrive at (S18) we have used the linear and cyclic properties of the trace
operator. As
$\bm{\mathbf{S}}={\bm{\mathrm{y}}}_{ig}{\bm{\mathrm{y}}}_{ig}^{\top}$, the
cyclic property implies the final equality since
$\operatorname*{tr}(\bm{\mathbf{S}}\bm{\mathbf{A}})=\operatorname*{tr}({\bm{\mathrm{y}}}_{ig}{\bm{\mathrm{y}}}_{ig}^{\top}\bm{\mathbf{A}})=\operatorname*{tr}({\bm{\mathrm{y}}}_{ig}^{\top}\bm{\mathbf{A}}{\bm{\mathrm{y}}}_{ig})={\bm{\mathrm{y}}}_{ig}^{\top}\bm{\mathbf{A}}{\bm{\mathrm{y}}}_{ig}$.
Equation (S18) is equivalent to the summand in (S7).