Parity Requires Unified Input Dependence and Negative Eigenvalues in SSMs

TL;DR

This paper investigates the limitations of state-space models in solving parity tasks, revealing that effective models require both input dependence and negative eigenvalues, supported by theoretical proofs and experiments.

Contribution

It demonstrates that combining input-dependent and non-negative SSMs still fails at parity, establishing the necessity of negative eigenvalues and input dependence for success.

Findings

Input-dependent transition matrices are necessary for solving parity.

Combining non-negative and input-independent SSMs does not solve parity.

Negative eigenvalues are essential for effective state-tracking in SSMs.

Abstract

Recent work has shown that LRNN models such as S4D, Mamba, and DeltaNet lack state-tracking capability due to either time-invariant transition matrices or restricted eigenvalue ranges. To address this, input-dependent transition matrices, particularly those that are complex or non-triangular, have been proposed to enhance SSM performance on such tasks. While existing theorems demonstrate that both input-independent and non-negative SSMs are incapable of solving simple state-tracking tasks, such as parity, regardless of depth, they do not explore whether combining these two types in a multilayer SSM could help. We investigate this question for efficient SSMs with diagonal transition matrices and show that such combinations still fail to solve parity. This implies that a recurrence layer must both be input-dependent and include negative eigenvalues. Our experiments support this conclusion by analyzing an SSM model that combines S4D and Mamba layers.