| 研究課題/領域番号 |
23K13024
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| 研究種目 |
若手研究
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| 配分区分 | 基金 |
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| 審査区分 |
小区分12040:応用数学および統計数学関連
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| 研究機関 | 国立研究開発法人理化学研究所 |
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研究代表者 |
Wolfer Geoffrey 国立研究開発法人理化学研究所, 革新知能統合研究センター, 基礎科学特別研究員 (10965784)
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| 研究期間 (年度) |
2023-04-01 – 2026-03-31
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| 研究課題ステータス |
交付 (2023年度)
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| 配分額 *注記 |
4,420千円 (直接経費: 3,400千円、間接経費: 1,020千円)
2025年度: 1,430千円 (直接経費: 1,100千円、間接経費: 330千円)
2024年度: 1,170千円 (直接経費: 900千円、間接経費: 270千円)
2023年度: 1,820千円 (直接経費: 1,400千円、間接経費: 420千円)
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| キーワード | Information geometry / Markov chains / Identity testing / Markov Chain Monte Carlo / Data science |
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| 研究開始時の研究の概要 |
This project has three objectives: (1) to deepen our theoretical understanding of the information geometry of Markov models (2) to develop new geometrical tools and methods for the data-science community (3) to demonstrate their applicability by addressing modern inference problems in Markov chains.
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| 研究実績の概要 |
The primary focus of my research in FY23 was demonstrating how geometric techniques can be applied to modern problems in Markov chains within the field of data science.
In the theory we developed, allowable mappings between Markov chains all possess an operational (data-processing) interpretation, and as a result, the applicability of our theory follows by design. This year, we showed how our Markov embeddings aid in developing reduction techniques for modern Markov chain inference problems. Specifically, we devised a method to reduce identity testing of reversible Markov chains to that of symmetric ones. Our method recovers state-of-the-art sample complexity in most regimes and applies beyond the immediate problem, including tolerant testing settings.
In parallel work, we observed that one could recover many of the established “reversiblizations” of (non-reversible) Markov chains by regarding them as a geometric projection of the chain onto the reversible set. In particular, although it was previously understood that the Metropolis-Hastings algorithm is a form of reversiblization corresponding to a specific divergence, we demonstrated how exploring different notions of divergences can give rise to other widely used sampling algorithms. Notably, we recover the Markov Chain Monte Carlo (MCMC) algorithm based on Barker dynamics, which has recently gained popularity. This is important because it can help practioners develop new MCMC algorithms using a similar approach.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
1: 当初の計画以上に進展している
理由
The progress on the applicative side of the project is already considerable.
In terms of output, we successfully secured immediate acceptance of both our works at our top choices of venues: one oral presentation at the international conference on Geometric Science of Information (GSI'2023) and one publication in the IEEE Transactions on Information Theory journal, the reference in the field of information theory. These achievements serve as significant milestones, demonstrating the applicability of our framework to modern problems in probability and statistics.
Furthermore, our research has ignited interest within the statistical physics community, as we were invited to write an academic survey on the topic of information geometry of Markov chains.
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| 今後の研究の推進方策 |
Given the substantial progress in the more applicative aspects of the project in FY23, I will recenter the emphasis on the theoretical facets of my project in FY24.
Specifically, I will make progress on geometric classifications of families of transition matrices, at least for spaces of size up to 3. This will be a significant stepping stone towards a Cencov-type theorem in the context of Markov chains.
In parallel, I will continue to explore connection between information geometry of Markov chains and core questions in statistics such as mixing properties of Markov chains, Markov Chain Monte Carlo algorithms or hypothesis testing in order to further demonstrate the value of my approach.
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