Bilinear Input Modulation for Mamba: Koopman Bilinear Forms for Memory Retention and Multiplicative Computation

Selective State Space Models (SSMs), notably Mamba, employ diagonal state transitions that limit both memory retention and bilinear computational capacity. We propose a factorized bilinear input modulation that augments the SSM with a state-input product, interpretable as a finite-dimensional Koopman bilinear form. After introducing a shared state across channels (Coupled SSM), the modulation admits three implementations. Coupled Bilinear Input Modulation (seq-BIM) retains the full bilinear product on the input side at the cost of sequential computation, Coupled Gated Modulation (GM) linearizes it into a gate modulation that is compatible with the parallel scan, and Parallel Bilinear Input Modulation (p-BIM) places the same bilinear product on the state transition while remaining parallel-scannable. Experiments on a multiple input-delay pendulum (memory retention) and NARMA-10 (bilinear computation) reveal a clear dissociation. GM substantially improves memory retention but not bilinear computation, while both seq-BIM and p-BIM improve both. A pathway ablation confirms that the two downstream routes of the bilinear signal serve complementary roles. The improvement is statistically robust, with the bilinear variants consistently outperforming the other variants on bilinear computation. Furthermore, only the bilinear variants benefit from increasing the SSM state dimension, while coupling or gate modulation alone show no improvement, establishing the bilinear mechanism as uniquely capable of exploiting larger state spaces.