Diagonal Equivalence in State Space Models: Characterization, Retrieval Rank, and Selective Extensions

Dennis Rochau

We classify real and complex state-space models (SSMs) with linear transitions, distinguishing fixed-basis diagonal equivalence from input-output equivalence and tracking coefficient-conditioning windows under coordinate changes. We characterize diagonal equivalence through commuting transition families whose members are diagonalizable over the chosen field. With time-shared token-local matrix maps, we identify this class's state-mediated kernels with unnormalized scalar-mode linear attention with multiplicative masks. Mamba 3's multiple-input, multiple-output (MIMO) construction lifts one value per local interface; SSD-MIMO routes independent values for local retrieval rank above one. HybridMamba combines Mamba 1's per-coordinate decay, Mamba 2-style heads and Mamba 3's complex rotations and two-endpoint writes, mixing real and rotating state blocks with different decay laws. They recover Mamba 1-3 single-input, single-output (SISO) configurations as exact special cases. On five large H100 bfloat16 workloads, our fused GPU kernels improve geometric-mean training time over official Mamba 1/3 in matched SISO recoveries, with Mamba 2 near parity at SSM-core scope. Memory tradeoffs…