# Flash PD-SSM: Memory-Optimized Structured Sparse State-Space Models

arXiv: https://arxiv.org/abs/2605.19150 · HTML: https://mlsanity.com/paper/2605.19150/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 49 of its 59 references that are in the map.
- First posted on arXiv 18 May 2026.

## Builds on (most cited first, 30 of 49)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Can a Suit of Armor Conduct Electricity? A New Dataset for Open Book Question Answering](https://mlsanity.com/paper/1809.02789/) (2018)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [The LAMBADA dataset: Word prediction requiring a broad discourse context](https://mlsanity.com/paper/1606.06031/) (2016)
- [On Layer Normalization in the Transformer Architecture](https://mlsanity.com/paper/2002.04745/) (2020)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Combining Recurrent, Convolutional, and Continuous-time Models with Linear State-Space Layers](https://mlsanity.com/paper/2110.13985/) (2021)
- [HiPPO: Recurrent Memory with Optimal Polynomial Projections](https://mlsanity.com/paper/2008.07669/) (2020)
- [Linear Transformers Are Secretly Fast Weight Programmers](https://mlsanity.com/paper/2102.11174/) (2021)
- [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)
- [Parallelizing Linear Transformers with the Delta Rule over Sequence Length](https://mlsanity.com/paper/2406.06484/) (2024)
- [Diagonal State Spaces are as Effective as Structured State Spaces](https://mlsanity.com/paper/2203.14343/) (2022)
- [The UEA multivariate time series classification archive, 2018](https://mlsanity.com/paper/1811.00075/) (2018)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [Griffin: Mixing Gated Linear Recurrences with Local Attention for Efficient Language Models](https://mlsanity.com/paper/2402.19427/) (2024)
- [Neural Networks and the Chomsky Hierarchy](https://mlsanity.com/paper/2207.02098/) (2022)
- [Repeat After Me: Transformers are Better than State Space Models at Copying](https://mlsanity.com/paper/2402.01032/) (2024)
- [The Parallelism Tradeoff: Limitations of Log-Precision Transformers](https://mlsanity.com/paper/2207.00729/) (2022)
- [The Illusion of State in State-Space Models](https://mlsanity.com/paper/2404.08819/) (2024)
- [An Empirical Study of Mamba-based Language Models](https://mlsanity.com/paper/2406.07887/) (2024)
- [Mega: Moving Average Equipped Gated Attention](https://mlsanity.com/paper/2209.10655/) (2022)

## Abstract (arXiv)

State-space models (SSMs) face a fundamental trade-off between efficiency and expressivity that is mainly dictated by the structure of the model's transition matrix. Unstructured transition matrices enable maximal expressivity, as measured by their ability to model finite-state automaton (FSA) transitions, but come at a prohibitively high compute and memory cost. In contrast, most structured transition matrix forms are highly efficient both in runtime and memory consumption, but suffer from limited expressivity. Building on recent work on structured sparse SSMs, we propose Flash PD-SSM, a novel SSM that achieves comparable throughput to widely-used structured SSMs with significantly better expressivity guarantees. Flash PD-SSM maintains a trainable set of structured sparse matrices, a single one of which is discretely selected at each time-step, enabling FSA expressiveness at the level of unstructured matrices while maintaining the efficiency required for training models at scale. First, we validate Flash PD-SSM against a suite of alternative models on synthetic mechanistic and state-tracking tasks, finding that its theoretical expressivity is achieved in practice. Second, on multivariate time-series tasks involving sequences of length over 17,000, we find that Flash PD-SSM defines a new state-of-the-art (SoTA) accuracy among competing SSM methods. Finally, we demonstrate that Flash PD-SSM is an effective drop-in replacement for hybrid LLMs, yielding improvements both in natural language state-tracking and in common language modeling scenarios. The model exhibits increased throughput and decreased memory consumption compared to SSMs widely used in frontier language models.