Mamba

State Space Update

\[\mathbf{h}_t = \mathbf{A}_t \mathbf{h}_{t-1} + \mathbf{B}_t \mathbf{x}_t\] \[\mathbf{y}_t = \mathbf{C}_t \mathbf{h}_t + \mathbf{D}\mathbf{x}_t\]

Input-Dependent Dynamics

\[\mathbf{A}_t = \exp(\Delta_t \mathbf{A})\] \[\mathbf{B}_t = (\Delta_t \mathbf{A})^{-1} \left(\exp(\Delta_t \mathbf{A}) - \mathbf{I}\right)\Delta_t \mathbf{B}\]