Gramian-balanced neural SSM compression

(1) Integrate this as an offline compression pass for a linear or nearly linear SSM layer. Expose its hidden-state transition $F\in\mathbb{R}^{n\times n}$, input matrix $B\in\mathbb{R}^{n\times m}$, and output matrix $C\in\mathbb{R}^{p\times n}$. If the layer has nonlinear gates, freeze the gates on representative calibration sequences or apply the method to the underlying linear transition. Require spectral radius $\rho(F)<1$; estimate it with 10 to 20 power iterations and scale $F$ by $\min(1-\epsilon,0.99/\max(\rho(F),10^{-6}))$ when necessary. (2) Pseudocode: initialize empty $Z_p$; for $k=1,\ldots,K$, set $Y=[FZ_p\;B]$ and $Z_p=\operatorname{compress}_{\tau}(Y)$; independently initialize empty $Z_q$ and repeat with $Y=[F^{\top}Z_q\;C^{\top}]$. Compute $M=Z_q^{\top}Z_p$, run its SVD, retain the first $r$ singular values, construct $V_r,W_r$, and replace $(F,B,C)$ with $(F_r,W_r^{\top}B,CV_r)$. (3) The Stein equations and balancing projection are taken from the paper; the engineer estimates factors matrix-free through products with $F$ and $F^{\top}$, never forming $P$ or $Q$. Monitor approximate residuals using $P=Z_pZ_p^{\top}$ and $Q=Z_qZ_q^{\top}$, and choose $K$ until residual norms stop decreasing. (4) First experiment: compress a 256- or 512-state S4- or Mamba-style layer to ranks 32, 64, and 128 on WikiText-103 or Long Range Arena, comparing against uncompressed fine-tuning and magnitude-based state pruning. Measure validation perplexity or task accuracy, activation memory, recurrent FLOPs, and decoding latency. A useful success criterion is at least 2x lower state-update cost and memory with less than 1% relative perplexity degradation after brief fine-tuning; also evaluate $\|G(e^{j\omega})-G_r(e^{j\omega})\|$ on a held-out frequency grid.