Duality Theory for Non-Markovian Linear Gaussian Models
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910103334027264 |
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| author | Kudre, Aditya Chang, Heng-Sheng Mehta, Prashant G. |
| author_facet | Kudre, Aditya Chang, Heng-Sheng Mehta, Prashant G. |
| contents | This work develops a duality theory for partially observed linear Gaussian models in discrete time. The state process evolves according to a causal but non-Markovian (or higher-order Gauss-Markov) structure, captured by a lower-triangular transition operator, which is related to transformer, with $T$ as the context length. The main contributions are: (i) a dual control system for the linear Gaussian model, formulated as a backward difference equation (B $Δ$ E); (ii) a duality principle establishing that a specific linear-quadratic optimal control problem for the B $Δ$ E is dual to the filtering problem for the partially observed model; and (iii) an explicit optimal control formula yielding a novel (transformer-like) linear predictor, referred to as the dual filter, whose computational complexity scales linearly in the time horizon $T$, in contrast to the $O(T^3)$ cost of classical smoothing and Wiener-Hopf approaches. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03909 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Duality Theory for Non-Markovian Linear Gaussian Models Kudre, Aditya Chang, Heng-Sheng Mehta, Prashant G. Systems and Control Optimization and Control This work develops a duality theory for partially observed linear Gaussian models in discrete time. The state process evolves according to a causal but non-Markovian (or higher-order Gauss-Markov) structure, captured by a lower-triangular transition operator, which is related to transformer, with $T$ as the context length. The main contributions are: (i) a dual control system for the linear Gaussian model, formulated as a backward difference equation (B $Δ$ E); (ii) a duality principle establishing that a specific linear-quadratic optimal control problem for the B $Δ$ E is dual to the filtering problem for the partially observed model; and (iii) an explicit optimal control formula yielding a novel (transformer-like) linear predictor, referred to as the dual filter, whose computational complexity scales linearly in the time horizon $T$, in contrast to the $O(T^3)$ cost of classical smoothing and Wiener-Hopf approaches. |
| title | Duality Theory for Non-Markovian Linear Gaussian Models |
| topic | Systems and Control Optimization and Control |
| url | https://arxiv.org/abs/2604.03909 |