An Exact Instrument for State Usage in Selective State-Space Models, and the Input-Driven Migration It Reveals
Authors: Raktim Bhattacharya
Organizations: Aerospace Engineering, Texas A&M University
Abstract
Selective state-space models such as Mamba route information through a bank of first-order modes whose input coupling is set by a learned selection mechanism. We give an exact instrument for measuring how a trained model uses these modes. Because the state matrix is diagonal, each channel's output decomposes exactly into per-mode contributions, and a per-(layer, channel, window) Gram tensor yields the exact output error of dropping any subset of modes, offline, at any budget. Validated against the reference implementation to a relative error of 2.3×10−7 on the Mamba-1 family where it is exact, the instrument predicts a layer's deployed pruning error to a median relative deviation of 5×10−7 over 4,464 configurations, its floor set by the reconstruction. Applying the instrument across the Mamba-1 family (130M--2.8B), the deployed 7B Falcon-Mamba, and Mamba-2, we find that trained models re-allocate their state space with the input: which modes carry the signal migrates across contexts, and at the most affected layers a per-input oracle roughly halves the output error of a fixed mode set. Frozen-signal counterfactuals attribute the migration primarily to the input-dependent write map Bt; the timestep usually identified with selectivity carries almost none of it. Input-scheduled mode pruning on this measurement outperforms static, Hankel-based, and layer-adaptive rankings at every scale from 130M to the deployed 7B Falcon-Mamba, and at half the state budget it matches the unpruned model. Because the scheduler reads each window's mode usage from a first pass, this demonstrates realizable headroom; we claim no deployed compute or memory saving.
Hidden states change substantially across the layers of modern language models, but most layer-wise analyses focus on one aspect of that change. We propose Layer-wise Representation Dynamics (LRD), a framework with three layer-wise measurement families: Frenet (Grassmann speed and curvature) for global subspace motion, Neighborhood Retention Score (NRS) for local nearest-neighbor retention, and Graph Filtration Mutual Information (GFMI) for alignment with the final layer. Applying LRD to 31 models (encoder-based and decoder-based embedders, plus base LLMs) on 30 MTEB tasks reveals architectural and task-level differences that are not apparent from final-layer representations alone. We then use LRD for two applications: label-free model selection and inference-time layer pruning. For selection, all three model-level scores correlate positively with downstream MTEB performance, with end-to-end subspace displacement (d_{0,L}) the strongest, and the same direction holds on a smaller base-LLM MMLU panel. For pruning, GFMI is the only measurement-guided rule that beats Random at the 15% and 20% budgets and has the best median change at every budget. Frenet is effective only at the lightest budget, while NRS does not transfer from model selection to pruning. These results show that layer-wise structure provides signal for both interpretation and deployment decisions.
Muon is a recent optimizer that orthogonalizes the update to each weight matrix with a Newton-Schulz iteration, which performs steepest descent under the spectral norm. Almost all the evidence for it comes from Transformer models, and its behavior on state-space models is largely unreported. We compare Muon with AdamW on Mamba-2 130M under a controlled protocol that varies only which weight groups are trained with Muon. The benefit is localized. Muon on the output projection alone beats Muon on the input projection or on both. The advantage is mainly one of token efficiency. It holds on two corpora and two token budgets, and persists when training continues well past the compute-optimal point. Conditioning does not explain the gain. Muon lowers the condition number of whichever projection it trains, but the better-conditioned input projection is not the one that helps.
Arslan Battalov, Karim Kramin, Alexander Markotenko +1
Recent work on looped language models suggests that many reasoning problems benefit from greater computational depth rather than from additional independent parameters. Existing studies, however, focus almost exclusively on Transformer backbones, leaving open whether this principle also applies to state-space language models. We investigate Looped Mamba and Looped Hybrid Mamba-Transformer architectures, which repeatedly apply a shared Mamba (or hybrid) block to introduce explicit finite-depth recurrent computation. On two controlled reasoning tasks-Mano (modular-arithmetic manipulation) and p-hop induction-Looped Mamba consistently outperforms parameter-matched non-looped baselines and, in several settings, matches or exceeds non-looped models of equal effective depth. We then extend the study to language model pre-training under matched iso-parameter and iso-FLOPs protocols, which jointly disentangle the effects of parameter sharing and effective depth: looped models remain competitive on downstream benchmarks with substantially fewer distinct parameters, although deeper non-looped models retain an advantage in validation perplexity under strict iso-FLOPs comparisons. Finally, we adapt Ouro's two-stage exit gate to Looped Mamba for threshold-controlled selection among recurrent-step outputs. Executing such exits on a state-space backbone, however, leaves the recurrent state without its deeper updates, and validation perplexity then degrades severely. We therefore introduce a cache-hole adaptation that aligns continued training with skipped-state inference. At the scales studied, the adapted model keeps perplexity close to full computation and matches or exceeds full-compute exit-state selection on downstream benchmarks while executing roughly half of the recurrent steps, which translates into measured inference speedups once the prefill is compute-bound.