# Learning linear dynamical systems under convex constraints

arXiv: https://arxiv.org/abs/2303.15121 · HTML: https://mlsanity.com/paper/2303.15121/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 50 references that are in the map.
- First posted on arXiv 27 Mar 2023.

## Builds on (most cited first, 13 of 13)

- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Finite Time Identification in Unstable Linear Systems](https://mlsanity.com/paper/1710.01852/) (2017)
- [Learning without Concentration](https://mlsanity.com/paper/1401.0304/) (2014)
- [Regularized estimation in sparse high-dimensional time series models](https://mlsanity.com/paper/1311.4175/) (2013)
- [Finite-time Identification of Stable Linear Systems: Optimality of the Least-Squares Estimator](https://mlsanity.com/paper/2003.07937/) (2020)
- [Learning Sparse Dynamical Systems from a Single Sample Trajectory](https://mlsanity.com/paper/1904.09396/) (2019)
- [Low Rank and Structured Modeling of High-dimensional Vector Autoregressions](https://mlsanity.com/paper/1812.03568/) (2018)
- [Oracle Inequalities for High Dimensional Vector Autoregressions](https://mlsanity.com/paper/1311.0811/) (2013)
- [Estimating Structured Vector Autoregressive Model](https://mlsanity.com/paper/1602.06606/) (2016)
- [Nonparametric Shape-restricted Regression](https://mlsanity.com/paper/1709.05707/) (2017)
- [Convex Regression in Multidimensions: Suboptimality of Least Squares Estimators](https://mlsanity.com/paper/2006.02044/) (2020)
- [Rate-Optimal Robust Estimation of High-Dimensional Vector Autoregressive Models](https://mlsanity.com/paper/2107.11002/) (2021)

## Built on by (most cited first, 5 of 5)

- [Joint Learning of Linear Dynamical Systems under Smoothness Constraints](https://mlsanity.com/paper/2406.01094/) (2024)
- [Temporal Robustness in Discrete Time Linear Dynamical Systems](https://mlsanity.com/paper/2505.02347/) (2025)
- [Symmetric Linear Dynamical Systems are Learnable from Few Observations](https://mlsanity.com/paper/2512.05337/) (2025)
- [Learning structured linear dynamical systems from missing observations](https://mlsanity.com/paper/2610.11869/) (2026)
- [RobustLDS: Learning linear dynamical systems under adversarial corruptions](https://mlsanity.com/paper/2610.11906/) (2026)

## Abstract (arXiv)

We consider the problem of finite-time identification of linear dynamical systems from T samples of a single trajectory. Recent results have predominantly focused on the setup where either no structural assumption is made on the system matrix A^* ∈ ℝn × n, or specific structural assumptions (e.g. sparsity) are made on A^*. We assume prior structural information on A^* is available, which can be captured in the form of a convex set K containing A^*. For the solution of the ensuing constrained least squares estimator, we derive non-asymptotic error bounds in the Frobenius norm that depend on the local size of K at A^*. To illustrate the usefulness of these results, we instantiate them for four examples, namely when (i) A^* is sparse and K is a suitably scaled ℓ1 ball; (ii) K is a subspace; (iii) K consists of matrices each of which is formed by sampling a bivariate convex function on a uniform n × n grid (convex regression); (iv) K consists of matrices each row of which is formed by uniform sampling (with step size 1/T) of a univariate Lipschitz function. In all these situations, we show that A^* can be reliably estimated for values of T much smaller than what is needed for the unconstrained setting.