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

arXiv: https://arxiv.org/abs/2512.23068 · HTML: https://mlsanity.com/paper/2512.23068/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Core SSM architectures for language, one of 10 directions in State-space sequence models (S4, Mamba).
- Builds on 19 of its 29 references that are in the map.
- First posted on arXiv 28 Dec 2025.

## Placed in

- State-space sequence models (S4, Mamba) / [Core SSM architectures for language](https://mlsanity.com/ssm/core-ssm-architectures-for-language/)

## Builds on (most cited first, 19 of 19)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [FlashAttention-2: Faster Attention with Better Parallelism and Work Partitioning](https://mlsanity.com/paper/2307.08691/) (2023)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Training Deep Nets with Sublinear Memory Cost](https://mlsanity.com/paper/1604.06174/) (2016)
- [RWKV: Reinventing RNNs for the Transformer Era](https://mlsanity.com/paper/2305.13048/) (2023)
- [Retentive Network: A Successor to Transformer for Large Language Models](https://mlsanity.com/paper/2307.08621/) (2023)
- [Hungry Hungry Hippos: Towards Language Modeling with State Space Models](https://mlsanity.com/paper/2212.14052/) (2022)
- [On the Parameterization and Initialization of Diagonal State Space Models](https://mlsanity.com/paper/2206.11893/) (2022)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Hyena Hierarchy: Towards Larger Convolutional Language Models](https://mlsanity.com/paper/2302.10866/) (2023)
- [xLSTM: Extended Long Short-Term Memory](https://mlsanity.com/paper/2405.04517/) (2024)
- [The Reversible Residual Network: Backpropagation Without Storing Activations](https://mlsanity.com/paper/1707.04585/) (2017)
- [HyenaDNA: Long-Range Genomic Sequence Modeling at Single Nucleotide Resolution](https://mlsanity.com/paper/2306.15794/) (2023)
- [Parallelizing Linear Recurrent Neural Nets Over Sequence Length](https://mlsanity.com/paper/1709.04057/) (2017)

## Abstract (arXiv)

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.