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Symmetric Linear Dynamical Systems are Learnable from Few Observations

Minh Vu, Andrey Y. Lokhov, Marc Vuffray

TL;DR

This work tackles learning the parameters of a symmetric, stable linear dynamical system from trajectory data under both full and partial observations. It introduces a moment-based estimator that recovers powers of the dynamic matrix with finite, element-wise maximum-norm guarantees using only $T=\mathcal{O}(\log N)$ observations, without problem-specific regularization. The authors provide asymptotic unbiasedness results, finite-sample concentration bounds, and a corollary extending the approach to partial observations, showing logarithmic sample complexity for the observed block and higher costs for unobserved cross-terms. Numerical experiments confirm the theoretical scaling in both sparse and dense settings and illustrate the practical viability for structure discovery. The method advances learnability of dynamical networks by enabling accurate, regularization-free recovery from minimal data, with clear pathways to extend to marginally stable and non-symmetric systems.

Abstract

We consider the problem of learning the parameters of a $N$-dimensional stochastic linear dynamics under both full and partial observations from a single trajectory of time $T$. We introduce and analyze a new estimator that achieves a small maximum element-wise error on the recovery of symmetric dynamic matrices using only $T=\mathcal{O}(\log N)$ observations, irrespective of whether the matrix is sparse or dense. This estimator is based on the method of moments and does not rely on problem-specific regularization. This is especially important for applications such as structure discovery.

Symmetric Linear Dynamical Systems are Learnable from Few Observations

TL;DR

This work tackles learning the parameters of a symmetric, stable linear dynamical system from trajectory data under both full and partial observations. It introduces a moment-based estimator that recovers powers of the dynamic matrix with finite, element-wise maximum-norm guarantees using only observations, without problem-specific regularization. The authors provide asymptotic unbiasedness results, finite-sample concentration bounds, and a corollary extending the approach to partial observations, showing logarithmic sample complexity for the observed block and higher costs for unobserved cross-terms. Numerical experiments confirm the theoretical scaling in both sparse and dense settings and illustrate the practical viability for structure discovery. The method advances learnability of dynamical networks by enabling accurate, regularization-free recovery from minimal data, with clear pathways to extend to marginally stable and non-symmetric systems.

Abstract

We consider the problem of learning the parameters of a -dimensional stochastic linear dynamics under both full and partial observations from a single trajectory of time . We introduce and analyze a new estimator that achieves a small maximum element-wise error on the recovery of symmetric dynamic matrices using only observations, irrespective of whether the matrix is sparse or dense. This estimator is based on the method of moments and does not rely on problem-specific regularization. This is especially important for applications such as structure discovery.

Paper Structure

This paper contains 14 sections, 8 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Methods and Results
  4. Learning fully observed linear dynamical systems
  5. Learning partially observed linear dynamical systems
  6. Analysis
  7. Proof of Theorem \ref{['Theorem: expectation']}
  8. Auxiliary Lemmas
  9. Proof of Theorem \ref{['Theorem: main']}
  10. Proof of Corollary \ref{['corollary1']}
  11. Numerical Experiments
  12. Fully Observed Systems
  13. Partially Observed Systems
  14. Conclusion

Key Result

Theorem 1

(Asymptotic Analysis): Consider a dynamical system defined in Eq. eq: model, where $A \in \mathbb{R}^{N\times N}$ is a symmetric matrix with spectral radius $\rho(A) < 1$, and with initial state $x_0 = 0$. Suppose ${\xi_t}$ are i.i.d. Gaussian random variables with $\mathbb{E}[\xi_t] = 0$ and $\math where the bias term is equal to $h_m(T)=\frac{1}{T-m} \sigma^2A^m(A^{2(T-m)}-I)(I-A^2)^{-2} - \fra

Figures (2)

  • Figure 1: Sample complexity comparison for fully observed systems. (Left) Sparse graph (degree 2). (Right) Dense graph with structured first row/column. The curves represent the best fit of the empirical points. Our estimator and LS+L1 both achieve $\mathcal{O}(\log N)$ scaling in the sparse case, while only our estimator maintains logarithmic scaling in the dense regime.
  • Figure 2: Performance under partial observability. The curves represent the best fit of the empirical points. The observable sub-matrix $B$ of the system matrix is recovered with $\mathcal{O}(\log N)$ complexity, while recovery of Markov parameters mixing other other sub-blocks involving hidden dimensions show an empirical scale of $\mathcal{O}(N \log N)$.

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • proof
  • Lemma 1: Quadratic Representation
  • proof
  • Lemma 2: Trajectory Distribution
  • proof
  • Lemma 3: Spectral Radius Bound
  • proof
  • ...and 6 more