# KOSS: Kalman-Optimal Selective State Spaces for Long-Term Sequence Modeling

arXiv: https://arxiv.org/abs/2512.16723 · HTML: https://mlsanity.com/paper/2512.16723/

## Key facts

- Cited by 1 papers in the map (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 29 of its 57 references that are in the map.
- First posted on arXiv 18 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, 29 of 29)

- [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 Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [FEDformer: Frequency Enhanced Decomposed Transformer for Long-term Series Forecasting](https://mlsanity.com/paper/2201.12740/) (2022)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [HiPPO: Recurrent Memory with Optimal Polynomial Projections](https://mlsanity.com/paper/2008.07669/) (2020)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Long Range Language Modeling via Gated State Spaces](https://mlsanity.com/paper/2206.13947/) (2022)
- [Transformer Quality in Linear Time](https://mlsanity.com/paper/2202.10447/) (2022)
- [FiLM: Frequency improved Legendre Memory Model for Long-term Time Series Forecasting](https://mlsanity.com/paper/2205.08897/) (2022)
- [KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics](https://mlsanity.com/paper/2107.10043/) (2021)
- [How to Train Your HiPPO: State Space Models with Generalized Orthogonal Basis Projections](https://mlsanity.com/paper/2206.12037/) (2022)
- [CKConv: Continuous Kernel Convolution For Sequential Data](https://mlsanity.com/paper/2102.02611/) (2021)
- [Is Mamba Effective for Time Series Forecasting?](https://mlsanity.com/paper/2403.11144/) (2024)
- [Can recurrent neural networks warp time?](https://mlsanity.com/paper/1804.11188/) (2018)
- [Effectively Modeling Time Series with Simple Discrete State Spaces](https://mlsanity.com/paper/2303.09489/) (2023)
- [Xformer: Hybrid X-Shaped Transformer for Image Denoising](https://mlsanity.com/paper/2303.06440/) (2023)
- [Gated Orthogonal Recurrent Units: On Learning to Forget](https://mlsanity.com/paper/1706.02761/) (2017)
- [GENERator: A Long-Context Generative Genomic Foundation Model](https://mlsanity.com/paper/2502.07272/) (2025)
- [From Markov to Laplace: How Mamba In-Context Learns Markov Chains](https://mlsanity.com/paper/2502.10178/) (2025)
- [A Distance Correlation-Based Approach to Characterize the Effectiveness of Recurrent Neural Networks for Time Series Forecasting](https://mlsanity.com/paper/2307.15830/) (2023)
- [DeFT-Mamba: Universal Multichannel Sound Separation and Polyphonic Audio Classification](https://mlsanity.com/paper/2409.12413/) (2024)
- [An Information-Theoretic Analysis of Discrete-Time Control and Filtering Limitations by the I-MMSE Relationships](https://mlsanity.com/paper/2304.09274/) (2023)

## Built on by (most cited first, 1 of 1)

- [To See Far, Look Close: Evolutionary Forecasting for Long-term Time Series](https://mlsanity.com/paper/2601.23114/) (2026)

## Abstract (arXiv)

Recent selective state space models (SSMs), such as Mamba and Mamba-2, have demonstrated strong performance in sequence modeling owing to input-dependent selection mechanisms. However, these mechanisms lack theoretical grounding and cannot support context-aware selection from latent state dynamics. To address these limitations, we propose KOSS, a Kalman-optimal Selective State Space model that formulates selection as latent state uncertainty minimization. Derived from estimation theory, KOSS adopts a continuous-time latent update driven by a Kalman gain that dynamically modulates information propagation based on content and context, enabling a closed-loop, context-aware selectivity mechanism. To ensure stable computation and near-linear scalability, KOSS employs global spectral differentiation for frequency-domain derivative estimation, along with a segment-wise scan for hardware-efficient processing. On a selective copying task with distractors, KOSS achieves over 79% accuracy while baselines drop below 20%, demonstrating robust context-aware selection. Furthermore, across nine long-term forecasting benchmarks, KOSS reduces MSE by 2.92--36.23% and consistently outperforms state-of-the-art models in both accuracy and stability. To assess real-world applicability, a case study on secondary surveillance radar (SSR) tracking confirms KOSS's robustness under irregular intervals and noisy conditions and demonstrates its effectiveness in real-world applications. Finally, supplementary experiments verify Kalman gain convergence and the frequency response of spectral differentiation, providing theoretical support for the proposed closed-loop design.