Parity Requires Unified Input Dependence and Negative Eigenvalues 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.
Cite
@article{arxiv.2508.07395,
title = {Parity Requires Unified Input Dependence and Negative Eigenvalues in SSMs},
author = {Behnoush Khavari and Mehran Shakerinava and Jayesh Khullar and Jerry Huang and François Rivest and Siamak Ravanbakhsh and Sarath Chandar},
journal= {arXiv preprint arXiv:2508.07395},
year = {2025}
}
Comments
5 pages. Accepted at ICML 2025 Workshop on Methods and Opportunities at Small Scale