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arXiv · 2512.23068

Breaking the Memory Wall: Exact Analytical Differentiation via Tiled Operator-Space Evolution

Shuhuan Wang·Yuzhen Xie·Jiayi Li·Yinliang Diao
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Abstract

Selective State Space Models (SSMs) achieve linear-time inference, yet their gradient-based sensitivity analysis remains bottlenecked by O(L) memory scaling during backpropagation. This memory constraint precludes genomic-scale modeling (L > 10^5) on consumer-grade hardware. We introduce Phase Gradient Flow (PGF), a framework that computes exact analytical derivatives by operating directly in the state-space manifold, bypassing the need to materialize the intermediate computational graph. By reframing SSM dynamics as Tiled Operator-Space Evolution (TOSE), our method delivers O(1) memory complexity relative to sequence length, yielding a 94% reduction in peak VRAM and a 23x increase in throughput compared to standard Autograd. Unlike parallel prefix scans that exhibit numerical divergence in stiff ODE regimes, PGF ensures stability through invariant error scaling, maintaining near-machine precision across extreme sequences. We demonstrate the utility of PGF on an impulse-response benchmark with 128,000-step sequences - a scale where conventional Autograd encounters prohibitive memory overhead, often leading to out-of-memory (OOM) failures in multi-layered models. Our work enables chromosome-scale sensitivity analysis on a single GPU, bridging the gap between theoretical infinite-context models and practical hardware limitations.

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Shuhuan Wang, Yuzhen Xie, Jiayi Li, Yinliang Diao. 2025-12-28. Breaking the Memory Wall: Exact Analytical Differentiation via Tiled Operator-Space Evolution. https://arxiv.org/abs/2512.23068

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