# Spectral Filtering for Complex Linear Dynamical Systems

arXiv: https://arxiv.org/abs/2601.22400 · HTML: https://mlsanity.com/paper/2601.22400/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 21 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Builds on (most cited first, 17 of 17)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Hungry Hungry Hippos: Towards Language Modeling with State Space Models](https://mlsanity.com/paper/2212.14052/) (2022)
- [Diagonal State Spaces are as Effective as Structured State Spaces](https://mlsanity.com/paper/2203.14343/) (2022)
- [Finite Sample Analysis of Stochastic System Identification](https://mlsanity.com/paper/1903.09122/) (2019)
- [Relative Loss Bounds for On-line Density Estimation with the Exponential Family of Distributions](https://mlsanity.com/paper/1301.6677/) (2013)
- [Learning Linear Dynamical Systems with Semi-Parametric Least Squares](https://mlsanity.com/paper/1902.00768/) (2019)
- [Learning Linear Dynamical Systems via Spectral Filtering](https://mlsanity.com/paper/1711.00946/) (2017)
- [Universality of Linear Recurrences Followed by Non-linear Projections: Finite-Width Guarantees and Benefits of Complex Eigenvalues](https://mlsanity.com/paper/2307.11888/) (2023)
- [A New Approach to Learning Linear Dynamical Systems](https://mlsanity.com/paper/2301.09519/) (2023)
- [Introduction to Online Control](https://mlsanity.com/paper/2211.09619/) (2022)
- [No-Regret Prediction in Marginally Stable Systems](https://mlsanity.com/paper/2002.02064/) (2020)
- [Improved rates for prediction and identification of partially observed linear dynamical systems](https://mlsanity.com/paper/2011.10006/) (2020)
- [Provable Benefits of Complex Parameterizations for Structured State Space Models](https://mlsanity.com/paper/2410.14067/) (2024)
- [FutureFill: Fast Generation from Convolutional Sequence Models](https://mlsanity.com/paper/2410.03766/) (2024)
- [Universal Sequence Preconditioning](https://mlsanity.com/paper/2502.06545/) (2025)
- [An Uncertainty Principle for Linear Recurrent Neural Networks](https://mlsanity.com/paper/2502.09287/) (2025)

## Built on by (most cited first, 2 of 2)

- [The Power of Second Order Methods for Sequence Preconditioning](https://mlsanity.com/paper/2605.08390/) (2026)
- [A Memory Efficient Unified Algorithm for Online Learning of Linear Dynamical Systems](https://mlsanity.com/paper/2607.02050/) (2026)

## Abstract (arXiv)

We study the problem of learning complex-valued linear dynamical systems (CLDS) with sector-bounded spectrum. This class captures oscillatory and long-memory dynamics arising in signal processing, structured state space models, and quantum systems. We introduce a spectral filtering method based on the Slepian basis and show that learnability is governed by an effective dimension independent of the ambient state dimension. As a consequence, we obtain dimension-free regret bounds for sequence prediction in CLDS with spectrum contained in a sector of the unit disk.