Role of eigenvalue placement in LTV SSMs versus LTI SSMs

Ascertain why moving the input-dependent eigenvalues of the time-varying state transition matrices toward marginal stability degrades training and performance in linear time-varying SSMs such as S6/Mamba and RG-LRU, whereas eigenvalues near the unit circle are beneficial for linear time-invariant SSMs such as S4, S4D, S5, and LRU; characterize the mechanism governing this contrasting behavior.

Background

Empirical results on the LRA benchmark show that LTI-based SSMs outperform certain LTV-based SSMs despite the latter’s greater expressivity. The authors attempted to improve LTV models by initializing eigenvalues closer to marginal stability—a strategy that benefits LTI models—but observed degraded learning or failure.

They explicitly state that the observed discrepancy in how eigenvalue placement affects LTI vs. LTV SSMs is not well understood, pointing to a concrete unresolved issue about spectral design and its impact on learning dynamics in input-selective, time-varying SSMs.