Symmetric Linear Dynamical Systems are Learnable from Few Observations

2025-12-04

AI Summary

This paper addresses the problem of estimating the dynamics matrix of an $N$-dimensional symmetric linear stochastic system from a single short trajectory of length $T = O(log N)$, under both full and partial observation settings. We propose a regularization-free moment-based estimator that directly constructs unbiased, high-accuracy matrix estimates by analytically leveraging low-order temporal autocorrelation moments of the single trajectory. Our method breaks the conventional sample-length barrier, achieving element-wise minimax-optimal recovery of the symmetric dynamics matrix for the first time with only $T = O(log N)$ observations. It is robust to both sparse and dense matrix structures. Theoretically, the estimation error converges at the optimal rate; empirically, it demonstrates significant superiority over existing methods in finite-sample regimes. This work establishes a new paradigm for high-dimensional dynamical modeling from ultra-short time series.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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.

Problem

• Learning parameters of stochastic linear dynamics from limited observations
• Achieving accurate recovery of symmetric dynamic matrices with few samples
• Enabling structure discovery without problem-specific regularization

Innovation

• Method of moments for symmetric linear dynamics
• Logarithmic observations independent of sparsity
• No problem-specific regularization for structure discovery