Parity Requires Unified Input Dependence and Negative Eigenvalues in SSMs

Behnoush Khavari, Jayesh Khullar, Mehran Shakerinava, Jerry Huang, Siamak Ravanbakhsh, Sarath Chandar

Keywords: SSMs, Linear Recurrent Neural Networks, State-tracking, expressivity

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 like 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 be both input-dependent and include negative eigenvalues. Our experiments support this conclusion by analyzing an SSM model that combines S4D and Mamba layers.

Submission Number: 92