# Data-driven Optimal Filtering for Linear Systems with Unknown Noise Covariances

arXiv: https://arxiv.org/abs/2305.17836 · HTML: https://mlsanity.com/paper/2305.17836/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 May 2023.

## Built on by (most cited first, 3 of 3)

- [Learning Low-dimensional Latent Dynamics from High-dimensional Observations: Non-asymptotics and Lower Bounds](https://mlsanity.com/paper/2405.06089/) (2024)
- [Differentiable Adaptive Kalman Filtering via Optimal Transport](https://mlsanity.com/paper/2508.07037/) (2025)
- [Scalar Federated Learning for Linear Quadratic Regulator](https://mlsanity.com/paper/2604.05088/) (2026)

## Abstract (arXiv)

This paper examines learning the optimal filtering policy, known as the Kalman gain, for a linear system with unknown noise covariance matrices using noisy output data. The learning problem is formulated as a stochastic policy optimization problem, aiming to minimize the output prediction error. This formulation provides a direct bridge between data-driven optimal control and, its dual, optimal filtering. Our contributions are twofold. Firstly, we conduct a thorough convergence analysis of the stochastic gradient descent algorithm, adopted for the filtering problem, accounting for biased gradients and stability constraints. Secondly, we carefully leverage a combination of tools from linear system theory and high-dimensional statistics to derive bias-variance error bounds that scale logarithmically with problem dimension, and, in contrast to subspace methods, the length of output trajectories only affects the bias term.