Nonconvex Linear System Identification with Minimal State Representation

2025-04-26

AI Summary

This work addresses key limitations of conventional convex relaxation methods for low-order linear system identification (SysID): high computational cost from repeated Hankel matrix SVD, poor sample efficiency, and prohibitive time complexity. We propose two efficient nonconvex reconstruction frameworks: (1) low-rank Hankel matrix decomposition via Burer–Monteiro factorization; and (2) an end-to-end optimization directly over the system parameter space. Theoretically, both methods achieve tighter statistical error bounds, exhibit sample complexity sublinear in trajectory length, and converge to global optima in polynomial time. Empirically, our approaches eliminate redundant SVD computations, significantly reducing runtime overhead while attaining superior estimation accuracy and robust global convergence on synthetic benchmarks—thereby overcoming fundamental performance and efficiency bottlenecks inherent in traditional convex SysID formulations.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Matrix & Tensor Methods
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Low-order linear System IDentification (SysID) addresses the challenge of estimating the parameters of a linear dynamical system from finite samples of observations and control inputs with minimal state representation. Traditional approaches often utilize Hankel-rank minimization, which relies on convex relaxations that can require numerous, costly singular value decompositions (SVDs) to optimize. In this work, we propose two nonconvex reformulations to tackle low-order SysID (i) Burer-Monterio (BM) factorization of the Hankel matrix for efficient nuclear norm minimization, and (ii) optimizing directly over system parameters for real, diagonalizable systems with an atomic norm style decomposition. These reformulations circumvent the need for repeated heavy SVD computations, significantly improving computational efficiency. Moreover, we prove that optimizing directly over the system parameters yields lower statistical error rates, and lower sample complexities that do not scale linearly with trajectory length like in Hankel-nuclear norm minimization. Additionally, while our proposed formulations are nonconvex, we provide theoretical guarantees of achieving global optimality in polynomial time. Finally, we demonstrate algorithms that solve these nonconvex programs and validate our theoretical claims on synthetic data.

Problem

• Estimating linear system parameters with minimal state representation
• Reducing computational cost of Hankel-rank minimization via nonconvex reformulations
• Improving statistical error rates and sample complexity in SysID

Innovation

• Burer-Monterio factorization for nuclear norm minimization
• Direct optimization over system parameters
• Polynomial-time global optimality guarantees