# Spectral Filtering for General Linear Dynamical Systems

arXiv: https://arxiv.org/abs/1802.03981 · HTML: https://mlsanity.com/paper/1802.03981/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Feb 2018.

## Built on by (most cited first, 30 of 37)

- [A New Approach to Learning Linear Dynamical Systems](https://mlsanity.com/paper/2301.09519/) (2023)
- [Spectral State Space Models](https://mlsanity.com/paper/2312.06837/) (2023)
- [Task-Optimal Exploration in Linear Dynamical Systems](https://mlsanity.com/paper/2102.05214/) (2021)
- [Unconstrained Dynamic Regret via Sparse Coding](https://mlsanity.com/paper/2301.13349/) (2023)
- [Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systems](https://mlsanity.com/paper/2307.06538/) (2023)
- [Learning Low Dimensional State Spaces with Overparameterized Recurrent Neural Nets](https://mlsanity.com/paper/2210.14064/) (2022)
- [Flash STU: Fast Spectral Transform Units](https://mlsanity.com/paper/2409.10489/) (2024)
- [FutureFill: Fast Generation from Convolutional Sequence Models](https://mlsanity.com/paper/2410.03766/) (2024)
- [Deep inference of latent dynamics with spatio-temporal super-resolution using selective backpropagation through time](https://mlsanity.com/paper/2111.00070/) (2021)
- [Learning Linear Dynamics from Bilinear Observations](https://mlsanity.com/paper/2409.16499/) (2024)
- [Universal Sequence Preconditioning](https://mlsanity.com/paper/2502.06545/) (2025)
- [Universal Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2508.11990/) (2025)
- [Global Convergence Using Policy Gradient Methods for Model-free Markovian Jump Linear Quadratic Control](https://mlsanity.com/paper/2111.15228/) (2021)
- [Time varying regression with hidden linear dynamics](https://mlsanity.com/paper/2112.14862/) (2021)
- [Fairness in Forecasting of Observations of Linear Dynamical Systems](https://mlsanity.com/paper/2209.05274/) (2022)
- [PAC-Bayes Generalisation Bounds for Dynamical Systems Including Stable RNNs](https://mlsanity.com/paper/2312.09793/) (2023)
- [Online Control in Population Dynamics](https://mlsanity.com/paper/2406.01799/) (2024)
- [Combining Federated Learning and Control: A Survey](https://mlsanity.com/paper/2407.11069/) (2024)
- [Provable Length Generalization in Sequence Prediction via Spectral Filtering](https://mlsanity.com/paper/2411.01035/) (2024)
- [Exact Recovery for System Identification with More Corrupt Data than Clean Data](https://mlsanity.com/paper/2305.10506/) (2023)
- [A New Approach to Controlling Linear Dynamical Systems](https://mlsanity.com/paper/2504.03952/) (2025)
- [Elastic Spectral State Space Models for Train-Once Budgeted Inference](https://mlsanity.com/paper/2601.22488/) (2026)
- [PAC-Bayesian-Like Error Bound for a Class of Linear Time-Invariant Stochastic State-Space Models](https://mlsanity.com/paper/2212.14838/) (2022)
- [The Complexity of Sequential Prediction in Dynamical Systems](https://mlsanity.com/paper/2402.06614/) (2024)
- [SpectraLDS: Provable Distillation for Linear Dynamical Systems](https://mlsanity.com/paper/2505.17868/) (2025)
- [The Power of Second Order Methods for Sequence Preconditioning](https://mlsanity.com/paper/2605.08390/) (2026)
- [System Identification via Nuclear Norm Regularization](https://mlsanity.com/paper/2203.16673/) (2022)
- [Dynamic Structure Estimation from Bandit Feedback using Nonvanishing Exponential Sums](https://mlsanity.com/paper/2206.00861/) (2022)
- [Linear RNNs Provably Learn Linear Dynamic Systems](https://mlsanity.com/paper/2211.10582/) (2022)
- [Spectral learning of Bernoulli linear dynamical systems models](https://mlsanity.com/paper/2303.02060/) (2023)

## Abstract (arXiv)

We give a polynomial-time algorithm for learning latent-state linear dynamical systems without system identification, and without assumptions on the spectral radius of the system's transition matrix. The algorithm extends the recently introduced technique of spectral filtering, previously applied only to systems with a symmetric transition matrix, using a novel convex relaxation to allow for the efficient identification of phases.