Joint learning of a network of linear dynamical systems via total variation penalization

TL;DR

The paper tackles joint identification of $m$ linear dynamical systems on a connected graph by enforcing graph-based total variation through a TV-penalized least-squares objective. It derives non-asymptotic, high-probability MSE bounds that depend on graph geometry (e.g., the Fiedler value), inverse scaling factors, and a dispersion functional $\Delta_G$, showing that in well-connected graphs the MSE can vanish as $m$ grows even when each trajectory is short. The analysis combines a restricted eigenvalue argument for a block-diagonal design with concentration results for dependent observations, and yields interpretable corollaries under Schur-stability of the dynamics. Empirically, graph-TV LDS demonstrates superior or competitive performance across synthetic topologies and real EPA data, validating gains from pooling information over the graph and preserving sharp edges in parameter structure. Overall, the work advances scalable, graph-aware multi-LDS identification with strong finite-sample guarantees and practical benefits for networks with short trajectories.

Abstract

We consider the problem of joint estimation of the parameters of $m$ linear dynamical systems, given access to single realizations of their respective trajectories, each of length $T$. The linear systems are assumed to reside on the nodes of an undirected and connected graph $G = ([m], \mathcal{E})$, and the system matrices are assumed to either vary smoothly or exhibit small number of ``jumps'' across the edges. We consider a total variation penalized least-squares estimator and derive non-asymptotic bounds on the mean squared error (MSE) which hold with high probability. In particular, the bounds imply for certain choices of well connected $G$ that the MSE goes to zero as $m$ increases, even when $T$ is constant. The theoretical results are supported by extensive experiments on synthetic and real data.

Figures (10)

• Figure 1: Examples of coefficients $\beta^*_l$ generated, overlaid on the graph. Here, the coefficients $(\beta^*_l)_{l=1}^m$ are chosen to be piecewise constant across the graph, and the maximal jump size is constrained to $\| \widetilde{D} a^*\|_{\infty} = 0.4$.
• Figure 2: Examples of coefficients $\beta_l$ generated, overlaid on the graph. Here, the coefficients $(\beta^*_l)_{l=1}^m$ are chosen to be smoothly varying across the graph, as per \ref{['eq:smooth']}.
• Figure 3: Performance of the TV regularizer and other competing methods as a function of the maximal jump size $\|D \beta^*\|_{\infty}$ on a piecewise constant signal (as per \ref{['eq:pw']}).
• Figure 4: Performance of the TV regularizer as a function of the maximal jump size $\|D \beta^*\|_{\infty}$ on smooth signal (defined as per \ref{['eq:smooth']}).
• Figure 5: Parameter and test prediction MSE as a function of training time length $T$ for different graph topologies.

Theorems & Definitions (36)

• Proposition 1: Bounds on inverse scaling factors
• proof
• Definition 1
• Proposition 2: Bounds on compatibility factor
• Theorem 1
• Remark 1
• Remark 2: Interpreting and controlling $\Delta_G$
• Corollary 1: Stable $A_l^*$
• Corollary 2: Two canonical choices of $S$
• Remark 3: Regime selection