RobustLDS: Learning linear dynamical systems under adversarial corruptions

Aravinda Kanchana Ruwanpathirana; Hemant Tyagi

stat.MLOct 8, 2026

RobustLDS: Learning linear dynamical systems under adversarial corruptions

Authors: Aravinda Kanchana Ruwanpathirana, Hemant Tyagi

Organizations: Division of Mathematical Sciences, SPMS, NTU Singapore 637371

Abstract

We consider the problem of learning linear dynamical systems under adversarial contamination from a single trajectory of length $T$ . While identification of linear dynamical systems itself is well-studied, the problem of robust system identification under adversarial contamination is relatively less explored. In this work, we study the setting where a fraction of the $T$ observations are contaminated by adversarial outliers. We propose different estimators based on relaxations of least-trimmed squares along with an alternating minimization algorithm. Furthermore, we also propose two estimators which exploit the group-sparsity (through penalization/hard-constraints) of the outliers. For the estimator with group-sparse penalty, we derive non-asymptotic error bounds which establish its robustness to outliers. We also show empirically that the proposed estimators work well in practice.

Figures & tables

Panel of Figure 1 — Figure 1: Frobenius norm error, $\big\|{\widehat{A}-A^{*}}\big\|_{F}/{\big\|{A^{*}}\big\|_{F}}$ , for $T\in\{50,100,200,400,800\}$ with (a) $\epsilon=0$ , (b) $\epsilon=0.02$ , and (c) $\epsilon=0.1$ , under the random heavy-tailed contamination model.

Panel of Figure 2 — Figure 2: Frobenius norm error, $\big\|{\widehat{A}-A^{*}}\big\|_{F}/{\big\|{A^{*}}\big\|_{F}}$ , for $\epsilon\in\{0,0.01,0.02,0.05,0.1\}$ with (a) $T=100$ , (b) $T=400$ , and (c) $T=800$ , under the random heavy-tailed contamination model.

Sphere-AM	Sphere Relaxation.
SDP-AM	Semi-definite program Relaxation.
Biconvex-AM	Bi-convex relaxation.
ConvexQ(A)-AM	Convexifying $Q(A)$ to estimate $z$ .
LS-BSP	LS with block-sparse penalty.
LS-BSHC	LS with block-sparse constraints.
OLS	$\mathcal{A}$ -constrained least squares.

Table 3

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 3 — Figure 3: Frobenius norm error, $\big\|{\widehat{A}-A^{*}}\big\|_{F}/{\big\|{A^{*}}\big\|_{F}}$ , for $T\in\{50,100,200,400,800\}$ with (a) $\epsilon=0$ , (b) $\epsilon=0.02$ , and (c) $\epsilon=0.1$ , for block burst contamination model.

Panel of Figure 4 — Figure 4: Frobenius norm error, $\big\|{\widehat{A}-A^{*}}\big\|_{F}/{\big\|{A^{*}}\big\|_{F}}$ , for $\epsilon=\{0,0.01,0.02,0.05,0.1\}$ with (a) $T=100$ , (b) $T=400$ , and (c) $T=800$ , for block burst contamination model.

Panel of Figure 5 — Figure 5: Frobenius norm error, $\big\|{\widehat{A}-A^{*}}\big\|_{F}/{\big\|{A^{*}}\big\|_{F}}$ , for $T\in\{50,100,200,400,800\}$ with (a) $\epsilon=0$ , (b) $\epsilon=0.02$ , and (c) $\epsilon=0.1$ , under the sign flip contamination model.

Panel of Figure 6 — Figure 6: Frobenius norm error, $\big\|{\widehat{A}-A^{*}}\big\|_{F}/{\big\|{A^{*}}\big\|_{F}}$ , for $\epsilon=\{0,0.01,0.02,0.05,0.1\}$ with (a) $T=100$ , (b) $T=400$ , and (c) $T=800$ , under the sign flip contamination model.

Panel of Figure 7 — Figure 7: Frobenius norm error, $\big\|{\widehat{A}-A^{*}}\big\|_{F}/{\big\|{A^{*}}\big\|_{F}}$ , for $T\in\{50,100,200,400,800\}$ with (a) $\epsilon=0$ , (b) $\epsilon=0.02$ , and (c) $\epsilon=0.1$ , under the high-leverage contamination model.

Panel of Figure 8 — Figure 8: Frobenius norm error, $\big\|{\widehat{A}-A^{*}}\big\|_{F}/{\big\|{A^{*}}\big\|_{F}}$ , for $\epsilon=\{0,0.01,0.02,0.05,0.1\}$ with (a) $T=100$ , (b) $T=400$ , and (c) $T=800$ , under the high-leverage contamination model.

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Learning structured linear dynamical systems from missing observations

Aravinda Kanchana Ruwanpathirana, Hemant Tyagi, Sunny G. W. Wang

Division of Mathematical Sciences, SPMS, NTU Singapore 637371

System Identification Learning with Missing Data Stochastic Linear Dynamical Systems Statistical Learning Theory

Open paper AskPDF arXiv

We consider the problem of learning structured linear dynamical systems over convex sets

\mathcal{K}

, where only a small subset of the observations are available at each time point. An estimator which minimizes a bias-corrected, potentially non-convex objective function is proposed. Non-asymptotic bounds are obtained for the statistical error, which depend on the local complexity of

\mathcal{K}

, the trajectory length

T

, and the sub-sampling probability

p

. Convergence of the projected gradient descent algorithm is also established. The general theory is applied to settings where (i)

\mathcal{K}

is a subspace, (ii)

\mathcal{K}

is the set of bi-isotonic matrices, and (iii)

\mathcal{K}

is the set of matrices whose rows are formed by sampling Lipschitz functions. We show meaningful recovery of the transition matrix is possible for values of

T

much smaller than what is required in the unconstrained case, and for

p = o(1)

Abstract

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Learning structured linear dynamical systems from missing observations

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