Provable Benefits of Complex Parameterizations for Structured State Space Models

TL;DR

This paper provides a theoretical analysis showing that complex parameterizations in structured state space models (SSMs) are more efficient and practical than real parameterizations, supported by experiments demonstrating improved performance.

Contribution

It establishes formal theoretical gaps between real and complex diagonal SSMs, explaining the benefits of complex parameterizations for expressiveness and learnability.

Findings

Complex SSMs can express all real SSM mappings with moderate dimension.

Real SSMs require exponentially large parameters to match complex SSM expressiveness.

Experiments support the theoretical advantages of complex parameterizations and suggest new architectural improvements.

Abstract

Structured state space models (SSMs), the core engine behind prominent neural networks such as S4 and Mamba, are linear dynamical systems adhering to a specified structure, most notably diagonal. In contrast to typical neural network modules, whose parameterizations are real, SSMs often use complex parameterizations. Theoretically explaining the benefits of complex parameterizations for SSMs is an open problem. The current paper takes a step towards its resolution, by establishing formal gaps between real and complex diagonal SSMs. Firstly, we prove that while a moderate dimension suffices in order for a complex SSM to express all mappings of a real SSM, a much higher dimension is needed for a real SSM to express mappings of a complex SSM. Secondly, we prove that even if the dimension of a real SSM is high enough to express a given mapping, typically, doing so requires the parameters of the real SSM to hold exponentially large values, which cannot be learned in practice. In contrast, a complex SSM can express any given mapping with moderate parameter values. Experiments corroborate our theory, and suggest a potential extension of the theory that accounts for selectivity, a new architectural feature yielding state of the art performance.