Learning Low-dimensional Latent Dynamics from High-dimensional Observations: Non-asymptotics and Lower Bounds

TL;DR

This paper develops an algorithm for learning low-dimensional latent dynamics from high-dimensional observations, providing optimal sample complexity bounds and extending to meta-learning scenarios with multiple systems.

Contribution

It introduces a novel algorithm with provable guarantees for recovering low-dimensional models from high-dimensional data, and explores meta-learning extensions that surpass traditional bounds.

Findings

Sample complexity of order $ ilde{O}(n/\, ext{ extbackslash epsilon}^2)$ is optimal up to logarithmic factors.

The linear dependence on observation dimension $n$ is due to high-dimensional noise in the observer's column space.

Meta-learning from multiple datasets can break the standard sample complexity lower bounds.

Abstract

In this paper, we focus on learning a linear time-invariant (LTI) model with low-dimensional latent variables but high-dimensional observations. We provide an algorithm that recovers the high-dimensional features, i.e. column space of the observer, embeds the data into low dimensions and learns the low-dimensional model parameters. Our algorithm enjoys a sample complexity guarantee of order $\tilde{\mathcal{O}}(n/\epsilon^2)$, where $n$ is the observation dimension. We further establish a fundamental lower bound indicating this complexity bound is optimal up to logarithmic factors and dimension-independent constants. We show that this inevitable linear factor of $n$ is due to the learning error of the observer's column space in the presence of high-dimensional noises. Extending our results, we consider a meta-learning problem inspired by various real-world applications, where the observer column space can be collectively learned from datasets of multiple LTI systems. An end-to-end algorithm is then proposed, facilitating learning LTI systems from a meta-dataset which breaks the sample complexity lower bound in certain scenarios.