mlsanityData to 14 Oct 2026

Dual Filter: A Transformer-like Inference Architecture for Hidden Markov Models

arXiv 2505.00818 ↗First posted 1 May 2025cs.LG cs.SY eess.SYcited by 0

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Abstract (arXiv)

This paper presents a mathematical framework for causal nonlinear prediction in settings where observations are generated from an underlying hidden Markov model (HMM). Both the problem formulation and the proposed solution are motivated by the decoder-only transformer architecture, in which a finite sequence of observations (tokens) is mapped to the conditional probability of the next token. Our objective is not to construct a mathematical model of a transformer. Rather, our interest lies in deriving, from first principles, transformer-like architectures that solve the prediction problem for which the transformer is designed. The proposed framework is based on an original optimal control approach, where the prediction objective (MMSE) is reformulated as an optimal control problem. An analysis of the optimal control problem is presented leading to a fixed-point equation on the space of probability measures. To solve the fixed-point equation, we introduce the dual filter, an iterative algorithm that closely parallels the architecture of decoder-only transformers. These parallels are discussed in detail along with the relationship to prior work on mathematical modeling of transformers as transport on the space of probability measures. Numerical experiments are provided to illustrate the performance of the algorithm using parameter values typical of research-scale transformer models.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 43 of its 82 references that are in the map. First posted on arXiv 1 May 2025.

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References · 43

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  12. 2020Approximate information state for approximate planning and reinforcement learning in partially observed systemscited by 70
31 more
  1. 2021Combiner: Full Attention Transformer with Sparse Computation Costcited by 64
  2. 2024Aligning Large Language Models with Representation Editing: A Control Perspectivecited by 64
  3. 2020A Mathematical Theory of Attentioncited by 46
  4. 2023What's the Magic Word? A Control Theory of LLM Promptingcited by 41
  5. 2024From Self-Attention to Markov Models: Unveiling the Dynamics of Generative Transformerscited by 34
  6. 2025A Unified Perspective on the Dynamics of Deep Transformerscited by 32
  7. 2023Causal Interpretation of Self-Attention in Pre-Trained Transformerscited by 30
  8. 2024Measure-to-measure interpolation using Transformerscited by 29
  9. 2023Hidden Markov Transformer for Simultaneous Machine Translationcited by 25
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  11. 2024ControlAgent: Automating Control System Design via Novel Integration of LLM Agents and Domain Expertisecited by 21
  12. 2023Can a Transformer Represent a Kalman Filter?cited by 20
  13. 2023Prompt Engineering Through the Lens of Optimal Controlcited by 16
  14. 2024The Asymptotic Behavior of Attention in Transformerscited by 16
  15. 2024A Law of Next-Token Prediction in Large Language Modelscited by 15
  16. 2025Survey on Algorithms for multi-index modelscited by 15
  17. 2015A variational approach to path estimation and parameter inference of hidden diffusion processescited by 10
  18. 2021Common Information based Approximate State Representations in Multi-Agent Reinforcement Learningcited by 8
  19. 2021Introducing the Hidden Neural Markov Chain frameworkcited by 6
  20. 2024Decision Transformer as a Foundation Model for Partially Observable Continuous Controlcited by 5
  21. 2024A variational approach to sampling in diffusion processescited by 4
  22. 2024Arrow of Time in Estimation and Control: Duality Theory Beyond the Linear Gaussian Modelcited by 3
  23. 2019What is the Lagrangian for Nonlinear Filtering?cited by 2
  24. 2019An Optimal Control Derivation of Nonlinear Smoothing Equationscited by 2
  25. 2022Duality for Nonlinear Filtering II: Optimal Controlcited by 2
  26. 2024State space models, emergence, and ergodicity: How many parameters are needed for stable predictions?cited by 2
  27. 2018An Approach to Duality in Nonlinear Filteringcited by 1
  28. 2022Duality for Nonlinear Filtering I: Observabilitycited by 1
  29. 2023Another Look at Partially Observed Optimal Stochastic Control: Existence, Ergodicity, and Approximations without Belief-Reductioncited by 1
  30. 2023Variance Decay Property for Filter Stabilitycited by 1
  31. 2024Backward Map for Filter Stability Analysiscited by 1

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