# Riccati State Space Models: Non-iterative Parallelization for Nonlinear Sequence Modeling

arXiv: https://arxiv.org/abs/2609.35441 · HTML: https://mlsanity.com/paper/2609.35441/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 29 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 25 of 25)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [Reformer: The Efficient Transformer](https://mlsanity.com/paper/2001.04451/) (2020)
- [Modeling Long- and Short-Term Temporal Patterns with Deep Neural Networks](https://mlsanity.com/paper/1703.07015/) (2017)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Enhancing the Locality and Breaking the Memory Bottleneck of Transformer on Time Series Forecasting](https://mlsanity.com/paper/1907.00235/) (2019)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Liquid Time-constant Networks](https://mlsanity.com/paper/2006.04439/) (2020)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [Closed-form Continuous-time Neural Models](https://mlsanity.com/paper/2106.13898/) (2021)
- [Parallelizing non-linear sequential models over the sequence length](https://mlsanity.com/paper/2309.12252/) (2023)
- [Oscillatory State-Space Models](https://mlsanity.com/paper/2410.03943/) (2024)
- [Towards Scalable and Stable Parallelization of Nonlinear RNNs](https://mlsanity.com/paper/2407.19115/) (2024)
- [Log Neural Controlled Differential Equations: The Lie Brackets Make a Difference](https://mlsanity.com/paper/2402.18512/) (2024)
- [Rough Transformers: Lightweight and Continuous Time Series Modelling through Signature Patching](https://mlsanity.com/paper/2405.20799/) (2024)
- [Parallelization of Non-linear State-Space Models: Scaling Up Liquid-Resistance Liquid-Capacitance Networks for Efficient Sequence Modeling](https://mlsanity.com/paper/2505.21717/) (2025)
- [Kalman Linear Attention: Parallel Bayesian Filtering For Efficient Language Modelling and State Tracking](https://mlsanity.com/paper/2602.10743/) (2026)
- [Liquid Resistance Liquid Capacitance Networks](https://mlsanity.com/paper/2403.08791/) (2024)
- [Predictability Enables Parallelization of Nonlinear State Space Models](https://mlsanity.com/paper/2508.16817/) (2025)
- [Fixed-Point RNNs: Interpolating from Diagonal to Dense](https://mlsanity.com/paper/2503.10799/) (2025)
- [Expanding Expressivity in Transformer Models with M"obiusAttention](https://mlsanity.com/paper/2409.12175/) (2024)
- [A Unifying Framework for Parallelizing Sequential Models with Linear Dynamical Systems](https://mlsanity.com/paper/2509.21716/) (2025)

## Abstract (arXiv)

State space models (SSMs) achieve efficient sequence processing because their affine state updates are closed under composition and can therefore be evaluated with an associative parallel scan. Nonlinear recurrent models can provide richer, state-dependent dynamics, but generally lose this compositional structure: parallel evaluation then requires iterative methods that repeatedly linearize and scan the recurrence. We ask, what state-dependent nonlinear dynamics can be designed to remain exactly composable? We answer by introducing RiccatiSSM, a nonlinear SSM, in which each state dimension follows an input-conditioned Riccati differential equation. Its quadratic state dependence makes the local Jacobian explicitly state-dependent, while its exact per-step flow under piecewise-constant inputs is a M"obius transformation. Since M"obius maps are closed under composition and compose through 2× 2 matrix multiplication, the complete nonlinear state trajectory can be evaluated exactly with a single associative parallel scan, without iterative linearization. We further derive a constrained parameterization that ensures bounded, contractive dynamics, and avoids poles in the fractional-linear state update. Across long-sequence classification, regression, and forecasting tasks, RiccatiSSM achieves competitive predictive performance while reducing runtime by 22-33% compared to the nonlinear LrcSSM under matched architectures. These results demonstrate that state-dependent nonlinear dynamics can retain exact composability and be evaluated efficiently within a single parallel scan.