sparse undirected graph as a prior on the precision matrix of the hidden state
noise and a sparse directed graph for the transition matrix that models the
state dynamics. By bridging the gap between the static graphical Lasso model
and the dynamic state-space model, we provide a novel comprehensive framework
for interpretable inference with time-series data. The presented inference
method, based on an efficient block alternating majorization-minimization
algorithm, enables simultaneous estimation of both graphs and the construction
of the filtering/smoothing distribution for the time series. The established
convergence of our algorithm, departing from recent nonlinear analysis tools,
enhances the reliability and practicality of our approach. Through extensive
experimental validation on synthetic data, we have demonstrated the
effectiveness and potential of our proposed model and inference algorithm. The
results showcase its ability to uncover meaningful insights from time-series
data, contributing not only to better forecasting performance but also to a
better understanding of complex phenomena in various scientific and
engineering domains. Future research can further explore automatic
hyperparameter tuning, and extend the presented framework to tackle even more
challenging state-space models, and more complex graphical structures.

Acknowledgments

É.C. acknowledges support from the European Research Council Starting Grant
MAJORIS ERC-2019-STG-850925. The work of V. E. is supported by the _Agence
Nationale de la Recherche_ of France under PISCES (ANR-17-CE40-0031-01), the
Leverhulme Research Fellowship (RF-2021-593), and by ARL/ARO under grant
W911NF-22-1-0235.
The authors thank Gustau Camps-Valls and his team, for providing the dataset
from Runge et al. (2020).

##  Appendix A Proof of Theorem 1

For any initial state ${\mathbf{x}}_{0:K}$, with non zero probability, the
neg-log-likelihood
$({\mathbf{A}},{\mathbf{P}})\to\left(\mathcal{L}_{1:K}({\mathbf{A}},{\mathbf{P}})=-\log
p({\mathbf{y}}_{1:K}|{\mathbf{A}},{\mathbf{P}})\right)$ reads, according to
Bayes’ rule,

$(\forall{\mathbf{A}}\in\mathbb{R}^{N_{x}\times
N_{x}})(\forall{\mathbf{P}}\in\mathcal{S}_{N_{x}})\\\
\mathcal{L}_{1:K}({\mathbf{A}},{\mathbf{P}})=-\log
p({\mathbf{x}}_{0:K},{\mathbf{y}}_{1:K}|{\mathbf{A}},{\mathbf{P}})+\log
p({\mathbf{x}}_{0:K}|{\mathbf{y}}_{1:K},{\mathbf{A}},{\mathbf{P}}).$ (70)

Again, note that the l.h.s. does not depend on ${\mathbf{x}}_{0:K}$, so the
r.h.s. is valid for any arbitrary value of ${\mathbf{x}}_{0:K}$ with non-zero
probability under $p({\mathbf{x}}_{0:K})$, i.e., for all
${\mathbf{x}}_{0:K}\in\mathbb{R}^{(K+1)N_{x}}$. According to Eqs. (2)-(3)

$\log p({\mathbf{x}}_{0:K},{\mathbf{y}}_{1:K}|{\mathbf{A}},{\mathbf{P}})=\log
p({\mathbf{x}}_{0})+\sum_{k=1}^{K}\log
p({\mathbf{x}}_{k}|{\mathbf{x}}_{k-1},{\mathbf{A}},{\mathbf{P}})+\sum_{k=1}^{K}\log
p({\mathbf{y}}_{k}|{\mathbf{x}}_{k}).$ (71)

Moreover, using again Eq. (2) and the statistical model of the state noise,

$\displaystyle(\forall{\mathbf{A}}\in\mathbb{R}^{N_{x}\times N_{x}})$
$\displaystyle(\forall{\mathbf{P}}\in\mathcal{S}_{N_{x}})$ (72)
$\displaystyle\sum_{k=1}^{K}\log
p({\mathbf{x}}_{k}|{\mathbf{x}}_{k-1},{\mathbf{A}},{\mathbf{P}})$
$\displaystyle=-\frac{{1}}{2}\sum_{k=1}^{K}\left(\left({\mathbf{x}}_{k}-{\mathbf{A}}{\mathbf{x}}_{k-1}\right)^{\top}{\mathbf{P}}\left({\mathbf{x}}_{k}-{\mathbf{A}}{\mathbf{x}}_{k-1}\right)-\log\det({2\pi}{{\mathbf{P}}})\right),$
(73)
$\displaystyle=-\frac{{1}}{2}\sum_{k=1}^{K}\left(\left({\mathbf{x}}_{k}-{\mathbf{A}}{\mathbf{x}}_{k-1}\right)^{\top}{\mathbf{P}}\left({\mathbf{x}}_{k}-{\mathbf{A}}{\mathbf{x}}_{k-1}\right)\right)+\frac{K}{2}\log\det({2\pi}{{\mathbf{P}}}),$
(74)

which concludes the first part of the proof.
Let us now consider some $\widetilde{{\mathbf{A}}}\in\mathbb{R}^{N_{x}\times
N_{x}}$ and $\widetilde{{\mathbf{P}}}\in\mathcal{S}_{N_{x}}^{++}$. We start by
recalling some known results arising from the EM methodology (Dempster et al.,
1977; Wu, 1983), that we specify here for our context for the sake of clarity.
First, we notice that

$(\forall{\mathbf{A}}\in\mathbb{R}^{N_{x}\times
N_{x}})(\forall{\mathbf{P}}\in\mathcal{S}_{N_{x}})\\\
\mathcal{L}_{1:K}({\mathbf{A}},{\mathbf{P}})=-\int\log
p({\mathbf{y}}_{1:K}|{\mathbf{A}},{\mathbf{P}})p({\mathbf{x}}_{0:K}|{\mathbf{y}}_{1:K},\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})\rm{d}{\mathbf{x}}_{0:K},$
(75)

since $\log p({\mathbf{y}}_{1:K}|{\mathbf{A}},{\mathbf{P}})$ is constant with
respect to the integration variable, and the distribution
$p({\mathbf{x}}_{0:K}|{\mathbf{y}}_{1:K},\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})$
integrates to one. Then, according to (70) and (75), the expectation of the
neg-log-likelihood multiplied by
$p({\mathbf{x}}_{0:K}|{\mathbf{y}}_{1:K},\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})$
over all possible values of the unknown state reads:

$(\forall{\mathbf{A}}\in\mathbb{R}^{N_{x}\times
N_{x}})(\forall{\mathbf{P}}\in\mathcal{S}_{N_{x}})\quad\mathcal{L}_{1:K}({\mathbf{A}},{\mathbf{P}})=\overbrace{-\int
p({\mathbf{x}}_{0:K}|{\mathbf{y}}_{1:K},\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})\log
p({\mathbf{x}}_{0:K},{\mathbf{y}}_{1:K}|{\mathbf{A}},{\mathbf{P}})\rm{d}{\mathbf{x}}_{0:K}}^{q({\mathbf{A}},{\mathbf{P}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})}\\\
+\overbrace{\int
p({\mathbf{x}}_{0:K}|{\mathbf{y}}_{1:K},\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})\log
p({\mathbf{x}}_{0:K}|{\mathbf{y}}_{1:K},{\mathbf{A}},{\mathbf{P}})\rm{d}{\mathbf{x}}_{0:K}}^{h({\mathbf{A}},{\mathbf{P}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})}.$
(76)

In particular, for
$({\mathbf{A}},{\mathbf{P}})=(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})$,
$\mathcal{L}_{1:K}(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})=q(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})+h(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})$
so that

$(\forall{\mathbf{A}}\in\mathbb{R}^{N_{x}\times
N_{x}})(\forall{\mathbf{P}}\in\mathcal{S}_{N_{x}})\quad\mathcal{L}_{1:K}({\mathbf{A}},{\mathbf{P}})-\mathcal{L}_{1:K}(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})=q({\mathbf{A}},{\mathbf{P}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})-q(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})\\\
+h({\mathbf{A}},{\mathbf{P}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})-h(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}}).$
(77)

Using Gibbs’s inequality,
$h({\mathbf{A}},{\mathbf{P}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})\leq
h(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})$,
with equality at
$({\mathbf{A}},{\mathbf{P}})=(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})$.
Thus, using (77), that is

$(\forall{\mathbf{A}}\in\mathbb{R}^{N_{x}\times
N_{x}})(\forall{\mathbf{P}}\in\mathcal{S}_{N_{x}})\quad\mathcal{L}_{1:K}({\mathbf{A}},{\mathbf{P}})\leq
q({\mathbf{A}},{\mathbf{P}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})+\mathcal{L}_{1:K}(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})-q(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}}),$
(78)

where equality holds at
$({\mathbf{A}},{\mathbf{P}})=(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})$.
Notice that, for any function reading

$(\forall{\mathbf{A}}\in\mathbb{R}^{N_{x}\times
N_{x}})(\forall{\mathbf{P}}\in\mathcal{S}_{N_{x}})\quad\mathcal{Q}({\mathbf{A}},{\mathbf{P}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})=q({\mathbf{A}},{\mathbf{P}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})+{\mathrm{ct}_{/\mathbf{A},\mathbf{P}}},$
(79)

we obviously still have

$(\forall{\mathbf{A}}\in\mathbb{R}^{N_{x}\times
N_{x}})(\forall{\mathbf{P}}\in\mathcal{S}_{N_{x}})\quad\mathcal{L}_{1:K}({\mathbf{A}},{\mathbf{P}})\leq\mathcal{Q}({\mathbf{A}},{\mathbf{P}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})+\mathcal{L}_{1:K}(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})-\mathcal{Q}(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}}),$
(80)

where equality hereagain holds at
$({\mathbf{A}},{\mathbf{P}})=(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})$.
Using (80), (21), (23) and noticing that, for every
$(\theta_{A},\theta_{P})>0$,

$(\forall{\mathbf{A}}\in\mathbb{R}^{N_{x}\times
N_{x}})(\forall{\mathbf{P}}\in\mathcal{S}_{N_{x}})\quad\frac{1}{2\theta_{A}}\|{\mathbf{A}}-\widetilde{{\mathbf{A}}}\|^{2}_{F}\geq
0,\quad\frac{1}{2\theta_{P}}\|{\mathbf{P}}-\widetilde{{\mathbf{P}}}\|^{2}_{F}\geq
0,$ (81)

with equality holding at
$({\mathbf{A}},{\mathbf{P}})=(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})$,
we deduce the desired majorizing property

$(\forall{\mathbf{A}}\in\mathbb{R}^{N_{x}\times
N_{x}})(\forall{\mathbf{P}}\in\mathcal{S}_{N_{x}})\quad\mathcal{L}({\mathbf{A}},{\mathbf{P}})\leq\mathcal{Q}({\mathbf{A}},{\mathbf{P}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})+\mathcal{L}_{1:K}(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})-\mathcal{Q}(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})\\\
+\lambda_{A}\|{\mathbf{A}}\|_{1}+\lambda_{P}\|{\mathbf{P}}\|_{1}+\frac{1}{2\theta_{A}}\|{\mathbf{A}}-\widetilde{{\mathbf{A}}}\|^{2}_{F}+\frac{1}{2\theta_{P}}\|{\mathbf{P}}-\widetilde{{\mathbf{P}}}\|^{2}_{F},$
(82)

with equality holding at
$({\mathbf{A}},{\mathbf{P}})=(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})$.
The remaining of the proof amounts to expliciting the expression for
$({\mathbf{A}},{\mathbf{P}})\to\mathcal{Q}({\mathbf{A}},{\mathbf{P}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})$
satisfying (79) with function $q$ defined as in (76):

$(\forall{\mathbf{A}}\in\mathbb{R}^{N_{x}\times
N_{x}})(\forall{\mathbf{P}}\in\mathcal{S}_{N_{x}})\\\
q({\mathbf{A}},{\mathbf{P}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})=-\int
p({\mathbf{x}}_{0:K}|{\mathbf{y}}_{1:K},\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})\log
p({\mathbf{x}}_{0:K},{\mathbf{y}}_{1:K}|{\mathbf{A}},{\mathbf{P}})\rm{d}{\mathbf{x}}_{0:K}.$
(83)

Following (Sarkka, 2013, Theorem 12.4) (see also an alternative proof in
(Elvira and Chouzenoux, 2022, Sec. III-B)), (76)-(79) hold for

$(\forall{\mathbf{A}}\in\mathbb{R}^{N_{x}\times
N_{x}})(\forall{\mathbf{P}}\in\mathcal{S}_{N_{x}})\quad\mathcal{Q}({\mathbf{A}},{\mathbf{P}};\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})=\frac{K}{2}\text{tr}\left({\mathbf{P}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Psi}}}}-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}{\mathbf{A}}^{\top}-{\mathbf{A}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}^{\top}+{\mathbf{A}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Phi}}}}{\mathbf{A}}^{\top})\right)-\frac{K}{2}\log\det({2\pi}{{\mathbf{P}}}),$
(84)

and

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Psi}}}}$
$\displaystyle=\frac{1}{K}\sum_{k=1}^{K}\left({\bm{\Sigma}}^{\text{s}}_{k}+\bm{\mu}^{\text{s}}_{k}(\bm{\mu}^{\text{s}}_{k})^{\top}\right),$
(85)
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}$
$\displaystyle=\frac{1}{K}\sum_{k=1}^{K}\left({\bm{\Sigma}}^{\text{s}}_{k}{\mathbf{G}}_{k-1}^{\top}+\bm{\mu}^{\text{s}}_{k}(\bm{\mu}^{\text{s}}_{k-1})^{\top}\right),$
(86)
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Phi}}}}$
$\displaystyle=\frac{1}{K}\sum_{k=1}^{K}\left({\bm{\Sigma}}^{\text{s}}_{k-1}+\bm{\mu}^{\text{s}}_{k-1}(\bm{\mu}^{\text{s}}_{k-1})^{\top}\right).$
(87)

Hereabove, $(\bm{\mu}^{\text{s}}_{k},{\bm{\Sigma}}^{\text{s}}_{k})_{0\leq
k\leq K-1}$ denotes the mean and covariance of the smoothing distribution
obtained when running Algs. 1-2 using
$(\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}})$. Moreover, the matrix

${\mathbf{G}}_{k}={\bm{\Sigma}}_{k}\widetilde{{\mathbf{A}}}^{\top}\left(\widetilde{{\mathbf{A}}}{\bm{\Sigma}}_{k}\widetilde{{\mathbf{A}}}^{\top}+\widetilde{{\mathbf{P}}}^{-1}\right)$
(88)

is defined as in Algorithm 2. This concludes the proof.

##  Appendix B Proximal algorithms to solve the inner steps

We present Algorithms 4 and 5, that are proximal splitting algorithms to
solve, respectively, the inner problems (33) and (34). Specifically, both
algorithms are special instances of the Dykstra-like splitting algorithm from
Bauschke and Combettes (2008) (see also (Combettes and Pesquet, 2011, Sec.5)),
for the minimization of the sum of two convex but non-differentiable
functions. Sequence $({\mathbf{A}}_{n})_{n\in\mathbb{N}}$ (resp.
$({\mathbf{P}}_{n})_{n\in\mathbb{N}}$) is guaranteed to converge to the
solution of problem (33) (resp. (34)). The proximity steps involved in
Algorithms 4 and 5 have closed form expressions that can be found for instance
in Bauschke and Combettes (2017). We explicit them hereafter for the sake of
completeness.

##### Proximity of $\ell_{1}$.

Let $\gamma>0$ and $\widetilde{{\mathbf{V}}}\in\mathbb{R}^{N_{x}\times
N_{x}}$. Then,

$\displaystyle\operatorname{prox}_{\gamma\ell_{1}}(\widetilde{{\mathbf{V}}})$
(89)
$\displaystyle=\left(\text{sign}(\widetilde{V}(n,\ell)\max(0,\widetilde{V}(n,\ell)-\gamma)\right)_{1\leq
n,\ell\leq N_{x}},$ (90)

which amounts to applying the soft thresholding operator with weight $\gamma$
on every entry of the matrix input $\widetilde{{\mathbf{V}}}$.

##### Proximity of quadratic term.

Let $\gamma>0$ and $\widetilde{{\mathbf{W}}}\in\mathbb{R}^{N_{x}\times
N_{x}}$. Then, by definition,

$\displaystyle\widehat{{\mathbf{Z}}}$
$\displaystyle=\operatorname{prox}_{{\mathbf{W}}\to\gamma\rm{tr}\left(-\widetilde{{\mathbf{P}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}{\mathbf{W}}-\widetilde{{\mathbf{P}}}{\mathbf{W}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}^{\top}+\widetilde{{\mathbf{P}}}{\mathbf{W}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Psi}}}}{\mathbf{W}}^{\top}\right)}\left(\widetilde{{\mathbf{W}}}\right)$
(91)
$\displaystyle=\underset{\begin{subarray}{c}{{\mathbf{W}}\in\mathbb{R}^{N_{x}\times
N_{x}}}\end{subarray}}{\operatorname{argmin}}\;\;\gamma\rm{tr}\left(-\widetilde{{\mathbf{P}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}{\mathbf{W}}^{\top}-\widetilde{{\mathbf{P}}}{\mathbf{W}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}^{\top}+\widetilde{{\mathbf{P}}}{\mathbf{W}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Psi}}}}{\mathbf{W}}^{\top}\right)+\frac{1}{2}\|{\mathbf{W}}-\widetilde{{\mathbf{W}}}\|_{F}^{2}.$
(92)

The optimality condition for (92) gives

$-\gamma\widetilde{{\mathbf{P}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}-\gamma{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}^{\top}\widetilde{{\mathbf{P}}}+\gamma\widetilde{{\mathbf{P}}}\widehat{{\mathbf{Z}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Psi}}}}+\gamma\widetilde{{\mathbf{P}}}^{\top}\widehat{{\mathbf{Z}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(93)

Since $\widetilde{{\mathbf{P}}}\in\mathcal{S}_{N_{x}}^{++}$, and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Psi}}}}\in\mathcal{S}_{N_{x}}$
(by construction), we have equivalently,

$-\gamma{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}-\gamma\widetilde{{\mathbf{P}}}^{-1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}^{\top}\widetilde{{\mathbf{P}}}+2\gamma\widehat{{\mathbf{Z}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(94)

Thus,

$\widehat{{\mathbf{Z}}}=\operatorname{lyapunov}\left(\widetilde{{\mathbf{P}}}^{-1},2\gamma{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Psi}}}},\gamma({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}+\widetilde{{\mathbf{P}}}^{-1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}^{\top}\widetilde{{\mathbf{P}}})+\widetilde{{\mathbf{P}}}^{-1}\widetilde{{\mathbf{W}}}\right),$
(95)

where $\mathsf{A}=\operatorname{lyapunov}(\mathsf{X},\mathsf{Y},\mathsf{Z})$
provides the solution to the Lyapunov equation $\mathsf{XA+AY=Z}$.

##### Proximity of log-determinant term.

Let $\gamma>0$ and $\widetilde{{\mathbf{W}}}\in\mathcal{S}_{N_{x}}^{++}$.
Then, by definition,

$\displaystyle\widehat{{\mathbf{Z}}}$
$\displaystyle=\operatorname{prox}_{{\mathbf{W}}\to\gamma(-\log\det({\mathbf{W}})+\text{tr}({\mathbf{W}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Pi}}}}))}\left(\widetilde{{\mathbf{W}}}\right)$
(96)
$\displaystyle=\underset{\begin{subarray}{c}{{\mathbf{W}}\in\mathcal{S}_{N_{x}}}\end{subarray}}{\operatorname{argmin}}\;\;-\gamma\log\det({\mathbf{W}})+\gamma\text{tr}({\mathbf{W}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Pi}}}})+\frac{1}{2}\|{\mathbf{W}}-\widetilde{{\mathbf{W}}}\|_{F}^{2}.$
(97)

Using (Bauschke and Combettes, 2017, Chap. 24) (see also Benfenati et al.
(2020)), for every $\alpha>0$,

$\displaystyle\widehat{{\mathbf{Z}}}$
$\displaystyle={\mathbf{U}}\text{Diag}\left(\left(\frac{1}{2}(\omega(n)+\sqrt{\omega(n)^{2}+4\gamma})\right)_{1\leq
n\leq N_{x}}\right){\mathbf{U}}^{\top}$ (98)

where $\bm{\omega}=(\omega(n))_{1\leq n\leq N_{x}}$ gathers the eigenvalues of
$\widetilde{{\mathbf{W}}}-\gamma{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Pi}}}}\in\mathcal{S}_{N_{x}}$
and ${\mathbf{U}}\in\mathbb{R}^{N_{x}\times N_{x}}$ is an orthogonal matrix
such that

$\widetilde{{\mathbf{W}}}-\gamma{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Pi}}}}={\mathbf{U}}\text{Diag}(\bm{\omega}){\mathbf{U}}^{\top}.$
(99)

###### Algorithm 4

Proximal splitting method to solve (33) Inputs.
$\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Psi}}}},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Phi}}}}$.
Precision $\xi>0$. 1. Setting. Set stepsize $\vartheta\in(0,2)$. 2.
Initialization. Set ${\mathbf{V}}_{0}=\widetilde{{\mathbf{A}}}$. 3. Recursive
step. For $n=1,2,\ldots$:
$\begin{array}[]{l}{\mathbf{A}}_{n}=\operatorname{prox}_{\theta_{A}\lambda_{A}\ell_{1}}\left(\widetilde{{\mathbf{A}}}-{\mathbf{V}}_{n}\right)\\\
{\mathbf{W}}_{n}={\mathbf{V}}_{n}+\vartheta{\mathbf{A}}_{n}\\\
{\mathbf{Z}}_{n}=\operatorname{prox}_{{\mathbf{W}}\to\frac{\vartheta\theta_{A}K}{2}\rm{tr}\left(-\widetilde{{\mathbf{P}}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}{\mathbf{W}}-\widetilde{{\mathbf{P}}}{\mathbf{W}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}}^{\top}+\widetilde{{\mathbf{P}}}{\mathbf{W}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Psi}}}}{\mathbf{W}}^{\top}\right)}\left(\vartheta^{-1}{\mathbf{W}}_{n}\right)\\\
{\mathbf{V}}_{n+1}={\mathbf{W}}_{n}-\vartheta{\mathbf{Z}}_{n}.\end{array}$ If
$|\mathcal{C}_{1}({\mathbf{A}}_{n})-\mathcal{C}_{1}({\mathbf{A}}_{n-1})|\leq\xi$,
stop the recursion.  Output. Transition matrix
$\widehat{{\mathbf{A}}}={\mathbf{A}}_{n}$.
---

###### Algorithm 5

Proximal splitting method to solve (34) Inputs.
$\widetilde{{\mathbf{A}}},\widetilde{{\mathbf{P}}},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Psi}}}},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Delta}}}},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Phi}}}}$.
Precision $\xi>0$. 1. Setting. Set stepsize $\vartheta\in(0,2)$ and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Pi}}}}$
as in (35). 2. Initialization. Set
${\mathbf{V}}_{0}=\widetilde{{\mathbf{P}}}$. 3. Recursive step. For
$n=1,2,\ldots$:
$\begin{array}[]{l}{\mathbf{P}}_{n}=\operatorname{prox}_{\theta_{P}\lambda_{P}\ell_{1}}\left(\widetilde{{\mathbf{P}}}-{\mathbf{V}}_{n}\right)\\\
{\mathbf{W}}_{n}={\mathbf{V}}_{n}+\vartheta{\mathbf{P}}_{n}\\\
{\mathbf{Z}}_{n}=\operatorname{prox}_{{\mathbf{W}}\to\frac{\vartheta\theta_{P}K}{2}\left(-\log\det({\mathbf{W}})+\rm{tr}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{{\bm{\Pi}}}}{\mathbf{W}})\right)}\left(\vartheta^{-1}{\mathbf{W}}_{n}\right)\\\
{\mathbf{V}}_{n+1}={\mathbf{W}}_{n}-\vartheta{\mathbf{Z}}_{n}.\end{array}$ If
$|\mathcal{C}_{2}({\mathbf{A}}_{n})-\mathcal{C}_{2}({\mathbf{A}}_{n-1})|\leq\xi$,
stop the recursion.  Output. Precision matrix
$\widehat{{\mathbf{P}}}={\mathbf{P}}_{n}$.
---

##  Appendix C Additional experiments

We present here additional experimental results completing Section 5.1.

###  C.1 Robustness to initialization

DGLASSO algorithm amounts to minimizing a non-convex loss function. As such,
its results might be sensitive to the initialization of the algorithm. To
evaluate this aspect, we consider the computation of
$(\widehat{{\mathbf{A}}},\widehat{{\mathbf{P}}})$ given the observation of a
single time series generated by the ground truth LG-SSM, when using 50
different initializations of DGLASSO algorithm. To do so, we use the same
initialization strategy as discussed above, now with $(a,p)$ randomly selected
as $p\sim\mathcal{U}([0,1])$ and $a\sim\mathcal{U}([0,1])$. Figure 10 displays
the box plots for RMSE and F1 scores obtained for dataset A. One can notice
that the box plots are very concentrated, showing a good robustness of the
method to its initialization, with the wider spreading observed for the F1
score on ${\mathbf{A}}$. Similar behavior was observed for the other three
datasets.

---
Figure 10: Box plots for the RMSE (left) and F1 (right) scores for retrieving
$({\mathbf{A}},{\mathbf{P}},{\mathbf{Q}})$ matrices, when running DGLASSO on
one single LG-SSM time series using dataset A, and 50 random initializations
$({\mathbf{A}}^{(0)},{\mathbf{P}}^{(0)})$. Noticeably, low variability is
observed for all metrics.

###  C.2 Influence of sparsity level

We evaluate here the performance of DGLASSO, as well as the benchmarks, when
varying the sparsity level of the ground truth matrices. To do so, we perform
slight changes in the dataset, to vary the sparsity pattern of the matrices
(i.e., the edge structure of the graphs). First, we modify the ground truth
matrix ${\mathbf{A}}^{*}$ by keeping $s_{A}\in\\{27,15,10,5\\}$ entries of it,
within the $27$ block diagonal ones, to be non-zero, the others being set to
zero. We then rescaled the matrix to keep a spectral norm equal to $0.99$.
Matrix ${\mathbf{Q}}^{*}$ is taken from dataset A. The results are reported in
Table 4.

As we can observe, the performance of MLEM, in terms of RMSE and F1 score,
drop dramatically when the sparsity level on ${\mathbf{A}}$ increases. This is
expected as this approach does not promote any sparsity prior on matrix
${\mathbf{A}}$. The best AUC are either obtained by MLEM, DGLASSO or rGLASSO
(for ${\mathbf{P}}$), depending on the test cases. GLASSO/rGLASSO metrics
slightly improve when ${\mathbf{A}}^{*}$ gets sparser, which is expected, as
their assumption of a zero transition matrix gets more realistic. Hereagain,
DGLASSO outperforms the benchmarks in most cases.

Table 4: Results for sparsity levels $s_{A}\in\\{27,15,10,5\\}$ of ${\mathbf{A}}^{*}$, using $({\mathbf{P}}^{*},{\mathbf{Q}}^{*})$ from dataset A. The first set of results, with $s_{A}=27$ identifies with the first row block of Tab. 1. |  Estimation of ${\mathbf{A}}$ |  Estimation of ${\mathbf{P}}$ |  Estim. ${\mathbf{Q}}$ | State distrib. | Predictive distrib.
---|---|---|---|---|---
| Method | RMSE | AUC | F1 | RMSE | AUC | F1 | RMSE |  cNMSE$(\bm{\mu}^{*},\widehat{\bm{\mu}})$ |  cNMSE$(\bm{\mu}^{\text{s}*},\widehat{\bm{\mu}}^{\text{s}})$ |  cNMSE$({\bm{\nu}}^{*},\widehat{{\bm{\nu}}})$ | $\mathcal{L}_{1:K}(\widehat{{\mathbf{A}}},\widehat{{\mathbf{P}}})$
$s_{A}=27$ | DGLASSO | $0.060\,525$ | $0.842\,55$ | $0.640\,57$ | $0.081\,927$ | $0.778\,01$ | $0.698\,12$ | $0.082\,601$ | $6.393\,5\text{\times}{10}^{-08}$ | $1.050\,4\text{\times}{10}^{-07}$ | $2.983\,7\text{\times}{10}^{-4}$ | $12\,307.168\,7$
MLEM | $0.076\,063$ | $0.816\,93$ | $0.5$ | $0.105\,28$ | $0.857\,39$ | $0.5$ | $0.101\,52$ | $1.094\,7\text{\times}{10}^{-07}$ | $1.802\,5\text{\times}{10}^{-07}$ | $4.842\,5\text{\times}{10}^{-4}$ | $12\,341.204\,5$
GLASSO | NA | NA | NA | $0.818\,38$ | $0.803\,94$ | $0.496\,16$ | $1\,073.510\,4$ | $4.485\,3\text{\times}{10}^{-06}$ | $7.180\,4\text{\times}{10}^{-06}$ | $1$ | $28\,459.293\,5$
rGLASSO | NA | NA | NA | $0.763\,76$ | $0.923\,77$ | $0.597\,59$ | $31.689\,1$ | $2.825\,6\text{\times}{10}^{-06}$ | $5.492\,4\text{\times}{10}^{-06}$ | $1$ | $22\,957.692\,8$
GRAPHEM | $0.044\,631$ | $0.894\,81$ | $0.846\,54$ | NA | NA | NA | NA | $4.364\,2\text{\times}{10}^{-06}$ | $6.944\,3\text{\times}{10}^{-06}$ | $2.98\text{\times}{10}^{-4}$ | $29\,035.029\,6$
$s_{A}=15$ | DGLASSO | $0.108\,45$ | $0.916\,44$ | $0.781\,16$ | $0.077\,396$ | $0.883\,35$ | $0.855\,43$ | $0.075\,47$ | $8.267\,5\text{\times}{10}^{-07}$ | $1.157\,4\text{\times}{10}^{-06}$ | $5.183\text{\times}{10}^{-3}$ | $11\,801.220\,2$
MLEM | $0.158\,94$ | $0.933\,43$ | $0.312\,5$ | $0.105\,01$ | $0.899\,48$ | $0.5$ | $0.100\,88$ | $1.297\,5\text{\times}{10}^{-06}$ | $2.109\,5\text{\times}{10}^{-06}$ | $6.051\,7\text{\times}{10}^{-3}$ | $11\,806.483\,3$
GLASSO | NA | NA | NA | $0.463\,09$ | $0.702\,22$ | $0.575\,79$ | $7.087\,5$ | $2.063\,8\text{\times}{10}^{-05}$ | $3.722\,9\text{\times}{10}^{-05}$ | $1$ | $14\,258.009\,5$
rGLASSO | NA | NA | NA | $0.424\,59$ | $0.722\,55$ | $0.597\,72$ | $4.562\,4$ | $1.378\,9\text{\times}{10}^{-05}$ | $2.768\,4\text{\times}{10}^{-05}$ | $1$ | $14\,432.765\,3$
GRAPHEM | $0.126\,8$ | $0.927\,66$ | $0.720\,61$ | NA | NA | NA | NA | $5.076\,6\text{\times}{10}^{-05}$ | $6.454\,7\text{\times}{10}^{-05}$ | $3.801\,5\text{\times}{10}^{-3}$ | $29\,034.715\,4$
$s_{A}=10$ | DGLASSO | $0.089\,227$ | $0.981\,44$ | $0.938\,49$ | $0.083\,783$ | $0.767\,86$ | $0.685\,62$ | $0.084\,488$ | $9.385\,6\text{\times}{10}^{-07}$ | $1.414\,5\text{\times}{10}^{-06}$ | $9.747\,5\text{\times}{10}^{-3}$ | $12\,290.789\,1$
MLEM | $0.167\,31$ | $0.999\,94$ | $0.219\,78$ | $0.105\,36$ | $0.899\,59$ | $0.5$ | $0.101\,03$ | $2.383\,1\text{\times}{10}^{-06}$ | $3.917\,9\text{\times}{10}^{-06}$ | $2.591\,7\text{\times}{10}^{-2}$ | $11\,819.738\,9$
GLASSO | NA | NA | NA | $0.364\,81$ | $0.666\,67$ | $0.5$ | $0.468\,89$ | $3.025\,8\text{\times}{10}^{-05}$ | $5.958\text{\times}{10}^{-05}$ | $1$ | $13\,173.462\,1$
rGLASSO | NA | NA | NA | $0.338\,57$ | $0.749\,42$ | $0.698\,61$ | $0.632\,11$ | $1.970\,2\text{\times}{10}^{-05}$ | $4.374\,6\text{\times}{10}^{-05}$ | $1$ | $13\,497.060\,9$
GRAPHEM | $0.117\,48$ | $0.999\,86$ | $0.744\,78$ | NA | NA | NA | NA | $8.955\,6\text{\times}{10}^{-05}$ | $1.124\,4\text{\times}{10}^{-4}$ | $1.225\,5\text{\times}{10}^{-2}$ | $29\,034.726\,8$
$s_{A}=5$ | DGLASSO | $0.099\,734$ | $0.822\,79$ | $0.773\,69$ | $0.084\,239$ | $0.764\,21$ | $0.682\,26$ | $0.085\,378$ | $9.217\,4\text{\times}{10}^{-07}$ | $1.205\,5\text{\times}{10}^{-06}$ | $9.981\,3\text{\times}{10}^{-3}$ | $12\,268.667\,1$
MLEM | $0.233\,8$ | $0.893\,05$ | $0.116\,28$ | $0.105\,81$ | $0.856\,02$ | $0.5$ | $0.101\,82$ | $2.231\,8\text{\times}{10}^{-06}$ | $3.537\,5\text{\times}{10}^{-06}$ | $5.610\,4\text{\times}{10}^{-2}$ | $12\,312.748\,7$
GLASSO | NA | NA | NA | $0.242\,41$ | $0.666\,67$ | $0.5$ | $0.305\,84$ | $1.406\,3\text{\times}{10}^{-05}$ | $3.095\,2\text{\times}{10}^{-05}$ | $1$ | $12\,892.143\,9$
rGLASSO | NA | NA | NA | $0.261\,81$ | $0.703\,54$ | $0.566\,27$ | $0.451\,22$ | $1.195\,3\text{\times}{10}^{-05}$ | $2.884\,2\text{\times}{10}^{-05}$ | $1$ | $12\,834.736\,7$
GRAPHEM | $0.127\,23$ | $0.930\,53$ | $0.573\,28$ | NA | NA | NA | NA | $1.060\,3\text{\times}{10}^{-4}$ | $1.224\,5\text{\times}{10}^{-4}$ | $1.477\text{\times}{10}^{-2}$ | $29\,034.456\,3$

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