HiPPO-Prophecy: State-Space Models can Provably Learn Dynamical Systems in Context

TL;DR

This paper introduces a theoretical framework showing how State Space Models can learn dynamical systems in context without fine-tuning, by extending the HiPPO method to approximate derivatives and predict future states.

Contribution

It provides the first theoretical explanation and explicit weight construction for SSMs to learn dynamical systems in context, including error bounds and empirical validation.

Findings

Explicit weight construction enables SSMs to predict next states.

Continuous SSMs can approximate derivatives of input signals.

Discretized SSMs effectively predict future states in experiments.

Abstract

This work explores the in-context learning capabilities of State Space Models (SSMs) and presents, to the best of our knowledge, the first theoretical explanation of a possible underlying mechanism. We introduce a novel weight construction for SSMs, enabling them to predict the next state of any dynamical system after observing previous states without parameter fine-tuning. This is accomplished by extending the HiPPO framework to demonstrate that continuous SSMs can approximate the derivative of any input signal. Specifically, we find an explicit weight construction for continuous SSMs and provide an asymptotic error bound on the derivative approximation. The discretization of this continuous SSM subsequently yields a discrete SSM that predicts the next state. Finally, we demonstrate the effectiveness of our parameterization empirically. This work should be an initial step toward understanding how sequence models based on SSMs learn in context.