2023 Fiscal Year Research-status Report
Improving Information Geometry of Markov Chains for Data-Science
| Project/Area Number |
23K13024
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| Research Institution | Institute of Physical and Chemical Research |
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Principal Investigator |
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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| Project Period (FY) |
2023-04-01 – 2026-03-31
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| Keywords | Information geometry / Markov chains / Identity testing / Markov Chain Monte Carlo |
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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
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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| Strategy for Future Research Activity |
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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| Causes of Carryover |
The reason for incurring an amount to be used in the next fiscal year is that I have decided to postpone the purchase of new equipment (laptop and monitor).
In the next fiscal year, I will use this amount to either purchase equipment, or to complement the funds for attending international conferences and publications fees. This is required, given the current economic condition with the weaker yen.
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