# Efficient learning of hidden state LTI state space models of unknown order

arXiv: https://arxiv.org/abs/2202.01625 · HTML: https://mlsanity.com/paper/2202.01625/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 54 references that are in the map.
- First posted on arXiv 3 Feb 2022.

## Builds on (most cited first, 9 of 9)

- [On the Sample Complexity of the Linear Quadratic Regulator](https://mlsanity.com/paper/1710.01688/) (2017)
- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Low-rank Solutions of Linear Matrix Equations via Procrustes Flow](https://mlsanity.com/paper/1507.03566/) (2015)
- [Finite Sample Analysis of Stochastic System Identification](https://mlsanity.com/paper/1903.09122/) (2019)
- [Finite Time Identification in Unstable Linear Systems](https://mlsanity.com/paper/1710.01852/) (2017)
- [Non-Asymptotic Analysis of Robust Control from Coarse-Grained Identification](https://mlsanity.com/paper/1707.04791/) (2017)
- [Robust recovery of complex exponential signals from random Gaussian projections via low rank Hankel matrix reconstruction](https://mlsanity.com/paper/1503.02893/) (2015)
- [Finite-Data Performance Guarantees for the Output-Feedback Control of an Unknown System](https://mlsanity.com/paper/1803.09186/) (2018)
- [Finite-time System Identification and Adaptive Control in Autoregressive Exogenous Systems](https://mlsanity.com/paper/2108.11959/) (2021)

## Built on by (most cited first, 7 of 7)

- [Statistical Learning Theory for Control: A Finite Sample Perspective](https://mlsanity.com/paper/2209.05423/) (2022)
- [A New Approach to Learning Linear Dynamical Systems](https://mlsanity.com/paper/2301.09519/) (2023)
- [Finite Sample Identification of Bilinear Dynamical Systems](https://mlsanity.com/paper/2208.13915/) (2022)
- [Learning Linear Dynamics from Bilinear Observations](https://mlsanity.com/paper/2409.16499/) (2024)
- [Finite Sample Identification of Partially Observed Bilinear Dynamical Systems](https://mlsanity.com/paper/2501.07652/) (2025)
- [Learning Low-dimensional Latent Dynamics from High-dimensional Observations: Non-asymptotics and Lower Bounds](https://mlsanity.com/paper/2405.06089/) (2024)
- [Near-optimal Rank Adaptive Inference of High Dimensional Matrices](https://mlsanity.com/paper/2510.08117/) (2025)

## Abstract (arXiv)

The aim of this paper is to address two related estimation problems arising in the setup of hidden state linear time invariant (LTI) state space systems when the dimension of the hidden state is unknown. Namely, the estimation of any finite number of the system's Markov parameters and the estimation of a minimal realization for the system, both from the partial observation of a single trajectory. For both problems, we provide statistical guarantees in the form of various estimation error upper bounds, rank recovery conditions, and sample complexity estimates. Specifically, we first show that the low rank solution of the Hankel penalized least square estimator satisfies an estimation error in Sp-norms for p ∈ [1,2] that captures the effect of the system order better than the existing operator norm upper bound for the simple least square. We then provide a stability analysis for an estimation procedure based on a variant of the Ho-Kalman algorithm that improves both the dependence on the dimension and the least singular value of the Hankel matrix of the Markov parameters. Finally, we propose an estimation algorithm for the minimal realization that uses both the Hankel penalized least square estimator and the Ho-Kalman based estimation procedure and guarantees with high probability that we recover the correct order of the system and satisfies a new fast rate in the S2-norm with a polynomial reduction in the dependence on the dimension and other parameters of the problem.