# Non-Commutative State Tracking with Input-Dependent Low-Rank Updates in Mamba-3

arXiv: https://arxiv.org/abs/2609.28273 · HTML: https://mlsanity.com/paper/2609.28273/

## 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 21 of its 26 references that are in the map.
- First posted on arXiv 23 Sep 2026.

## 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, 21 of 21)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Learning Fine-Grained Bimanual Manipulation with Low-Cost Hardware](https://mlsanity.com/paper/2304.13705/) (2023)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [What Matters in Learning from Offline Human Demonstrations for Robot Manipulation](https://mlsanity.com/paper/2108.03298/) (2021)
- [HiPPO: Recurrent Memory with Optimal Polynomial Projections](https://mlsanity.com/paper/2008.07669/) (2020)
- [Gated Delta Networks: Improving Mamba2 with Delta Rule](https://mlsanity.com/paper/2412.06464/) (2024)
- [On the Parameterization and Initialization of Diagonal State Space Models](https://mlsanity.com/paper/2206.11893/) (2022)
- [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)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [The Illusion of State in State-Space Models](https://mlsanity.com/paper/2404.08819/) (2024)
- [RWKV-7 "Goose" with Expressive Dynamic State Evolution](https://mlsanity.com/paper/2503.14456/) (2025)
- [Unlocking State-Tracking in Linear RNNs Through Negative Eigenvalues](https://mlsanity.com/paper/2411.12537/) (2024)
- [Mamba-3: Improved Sequence Modeling using State Space Principles](https://mlsanity.com/paper/2603.15569/) (2026)
- [DeltaProduct: Improving State-Tracking in Linear RNNs via Householder Products](https://mlsanity.com/paper/2502.10297/) (2025)
- [Theoretical Foundations of Deep Selective State-Space Models](https://mlsanity.com/paper/2402.19047/) (2024)
- [(How) Do Language Models Track State?](https://mlsanity.com/paper/2503.02854/) (2025)
- [MTIL: Encoding Full History with Mamba for Temporal Imitation Learning](https://mlsanity.com/paper/2505.12410/) (2025)
- [RoboSSM: Scalable In-context Imitation Learning via State-Space Models](https://mlsanity.com/paper/2509.19658/) (2025)
- [Dissecting Linear Recurrent Models: How Different Gating Strategies Drive Selectivity and Generalization](https://mlsanity.com/paper/2601.12598/) (2026)

## Abstract (arXiv)

State tracking from sequential observations can require both retaining information and updating it by composing observed operations. We extend Mamba-3's diagonal transition with an input-dependent low-rank reflection term to support noncommutative state tracking, in which the order of operations matters. The rank-one update couples state coordinates along an input-dependent direction, enabling non-diagonal state transitions within a single Mamba-3 block. The extension preserves Mamba-3's exponential-trapezoidal discretization, rotary embeddings (RoPE), and readout. For training, we adapt chunkwise computation to parallelize the proposed recurrence within each chunk. Experiments cover group word problems with discrete inputs and a shell game with continuous observations, in which a policy is trained by behavioral cloning. Among the models selected for their strong performance under fixed timing, the proposed model maintains higher tracking success on longer swap sequences in the shell game with continuous observations and timing jitter. These experiments show that the proposed method achieves high accuracy on the evaluated non-commutative tracking tasks, improving on standard Mamba-3. The extension thus offers a Mamba-3-based approach to non-commutative state tracking.