近年のデータ量の増加に伴い，データの統計的性質を推定する機械学習はますます重要になってきている．機械学習ではデータの時系列が独立かつ同一の分布に従うモデルを仮定して研究を行うことが多いが，より現実的な，データ間の相関を考慮したモデルの研究に近年注目が集まっている．本研究では，機械学習や通信システムの性能解析の道具として，近年注目を集めている情報幾何に関する新しい成果をえた．具体的には，マルコフ遷移行列全体を多様体とみなした際に，可逆なマルコフ遷移行列全体が指数型遷移行列族かつ混合型遷移行列族になっていることを証明した．可逆なマルコフ過程は様々な分野で重要な遷移行列のクラスと考えられている．例えば，マルコフ連鎖モンテカルロ法で現れるのは可逆マルコフ過程である．可逆マルコフ過程は混合時間がスペクトルギャップで特徴付されるなど，これまでにもよい性質を持つことが知られていた．本研究の成果によって，可逆マルコフ遷移行列は幾何学的にもよい性質を持つことが明らかになった．
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また，本研究では，マルコフ過程のランピングの逆操作として，より大きな状態空間を持つマルコフ過程への埋め込みに関する成果も得た．より具体的には，確率分布族のマルコフ埋込の拡張概念として，遷移行列のマルコフ埋込を新たに定義し，そのマルコフ埋込がある種の整合性条件を満たす唯一の埋込法であることを証明した．また，ランピング可能な遷移行列の全体が，指数型遷移行列族と混合型遷移行列族から成る葉層構造に分解できることを解明した．

The convergence rate of a Markov chain towardsits stationary distribution is typically assessed using the concept of worst-casetotal variation *mixing time*. However, this quantity is pessimistic and is challengingto infer from a single stream of data. The goal of our research program, in collaborationwith Pierre Alquier (ESSEC Business School, Singapore) is to advocate for theuse of the average-mixing time as a more optimistic and demonstrablyeasier-to-estimate alternative.We had already demonstrated that estimatingthe average mixing time was both possible and beneficial in some settings andour plan was to improve our estimation procedures, both theoretically and in practice.Towards our goal, we have obtained thefollowing results in FY24:1/ We improved minimax ProbablyApproximately Correct (PAC) rates for estimating individual β-mixingcoefficients of discrete time, time homogeneous Markov chains over countablestate spaces by obtaining finite sample bounds in the *general ergodic* setting.Until now, rates in the general ergodic setting were only known for the MeanAbsolute Deviation (MAD), and PAC rates were only available in the *uniformlyergodic* setting, i.e. in the restrictive setting where the (worst-case) mixingtime is finite. Our new bound depends on the *average* mixing time instead, andthus remains valid even when the mixing time is infinite.2/ Building upon 1/ we obtained a PAC boundfor estimating the average mixing time from a single countable Markovian trajectoryin a *general ergodic setting*. Until now, only rates in the uniformly ergodicsetting were available.3/ Finally, we obtained estimation ratesfor Kernel Mean Embeddings (KME) with time-dependent data. Specifically, we extendedour previous estimation results from an iid setting to mixing processes byobtaining confidence intervals involving a covariance parameter in the ReproducingKernel Hilbert Space (RKHS) and mixing coefficients for both φ-mixing and β-mixing processes. In particular, we demonstrated that whenthe underlying process is a countable Markov chain, our framework based on theaverage mixing time is readily applicable.We presented our research results on the averagemixing time at the following venues:[1] IEEE East Asian School of InformationTheory 2024 (EASIT'24)Shonan, Kanagawa, July 30-August 2, 2024[2] Bernoulli-IMS 11th World Congress inProbability and Statistics (Bernoulli-IMS'24)&nbsp;Bochum, Germany, August 12-16, 2024[3] The 27th Information-Based InductionSciences Workshop (IBIS'24)Omiya, Saitama, November 4-7, 2024&nbsp;[4] The 46th Symposium on Information Theoryand its Applications (SITA'24)Awara, Fukui, December 10-13, 2024