確率場と機械学習
キーワード：確率場、機械学習
2020.04.
| 白井 朋之（しらい ともゆき） | データ更新日：2023.10.02 |
主な研究テーマ
ランダム複体のパーシステントホモロジー
キーワード：パーシステントホモロジー，ランダム複体
2013.04.
キーワード：パーシステントホモロジー，ランダム複体
2013.04.
行列式確率の研究
キーワード：行列式点過程
2009.09～2013.09.
キーワード：行列式点過程
2009.09～2013.09.
Random walks on graphs and spectral geometry
キーワード：ランダムウォーク、スペクトル幾何
1996.09.
キーワード：ランダムウォーク、スペクトル幾何
1996.09.
Fermion random fields and its generalization
キーワード：ランダム場，fermion, boson, alpha-determinant
1996.09.
キーワード：ランダム場，fermion, boson, alpha-determinant
1996.09.
従事しているプロジェクト研究
データ記述科学創出に向けた数学的基盤構築
2022.10～2027.03, 代表者：白井朋之, 九州大学, 九州大学
データ記述科学創出に向けた数学的基盤構築.
2022.10～2027.03, 代表者：白井朋之, 九州大学, 九州大学
データ記述科学創出に向けた数学的基盤構築.
Disordered complex networks
2018.06～2022.03, 代表者：Subhroshekhar Ghosh and Tomoyuki Shirai, National University of Singapore, Singapore Japan.
2018.06～2022.03, 代表者：Subhroshekhar Ghosh and Tomoyuki Shirai, National University of Singapore, Singapore Japan.
CREST
2015.10～2021.03, 代表者：平岡裕章, 東北大学, JST.
2015.10～2021.03, 代表者：平岡裕章, 東北大学, JST.
研究業績
主要著書
| 1. | 白井 朋之, Finite Markov Chains and Markov Decision Processes, Springer Verlag, 5, 189--206, 2014.07, [URL]. |
主要原著論文
| 1. | Makoto Katori and Tomoyuki Shirai, Zeros of the i.i.d. Gaussian Laurent Series on an Annulus: Weighted Szegő Kernels and Permanental-Determinantal Point Processes, Communications in Mathematical Physics, https://doi.org/10.1007/s00220-022-04365-2, 392, 1099-1151, 2022.06, On an annulus 𝔸𝑞:={𝑧∈ℂ:𝑞0 is identified with the weighted Szegő kernel of 𝔸𝑞 with the weight parameter r studied by McCullough and Shen. The GAF and the zero point process are rotationally invariant and have a symmetry associated with the q-inversion of coordinate 𝑧↔𝑞/𝑧 and the parameter change 𝑟↔𝑞2/𝑟. When 𝑟=𝑞 they are invariant under conformal transformations which preserve 𝔸𝑞. Conditioning the GAF by adding zeros, new GAFs are induced such that the covariance kernels are also given by the weighted Szegő kernel of McCullough and Shen but the weight parameter r is changed depending on the added zeros. We also prove that the zero point process of the GAF provides a permanental-determinantal point process (PDPP) in which each correlation function is expressed by a permanent multiplied by a determinant. Dependence on r of the unfolded 2-correlation function of the PDPP is studied. If we take the limit 𝑞→0, a simpler but still non-trivial PDPP is obtained on the unit disk 𝔻. We observe that the limit PDPP indexed by 𝑟∈(0,∞) can be regarded as an interpolation between the determinantal point process (DPP) on 𝔻 studied by Peres and Virág (𝑟→0) and that DPP of Peres and Virág with a deterministic zero added at the origin (𝑟→∞).. |
| 2. | Subhroshekhar Ghosh, Naoto Miyoshi and Tomoyuki Shirai, Disordered complex networks: energy optimal lattices and persistent homology, IEEE Transactions on Information Theory, doi: 10.1109/TIT.2022.3163604, 2022.04, Disordered complex networks are of fundamental interest in statistical physics, and they have attracted recent interest as stochastic models for information transmission over wireless networks. While mathematically tractable, a network based on the regulation Poisson point process model offers challenges vis-a-vis network efficiency. Strongly correlated alternatives, such as networks based on random matrix spectra (the Ginibre network), on the other hand offer formidable challenges in terms of tractability and robustness issues. In this work, we demonstrate that network models based on random perturbations of Euclidean lattices interpolate between Poisson and rigidly structured networks, and allow us to achieve the best of both worlds: significantly improve upon the Poisson model in terms of network efficacy measured by the Signal to Interference plus Noise Ratio (abbrv. SINR) and the related concept of coverage probabilities, at the same time retaining a considerable measure of mathematical and computational simplicity and robustness to erasure and noise. We investigate the optimal choice of the base lattice in this model, connecting it to the celebrated problem optimality of Euclidean lattices with respect to the Epstein Zeta function, which is in turn related to notions of lattice energy. This leads us to the choice of the triangular lattice in 2D and face centered cubic lattice in 3D, whose Gaussian perturbations we consider. We provide theoretical analysis and empirical investigations to demonstrate that the coverage probability decreases with increasing strength of perturbation, eventually converging to that of the Poisson network. In the regime of low disorder, our studies suggest an approximate statistical behaviour of the coverage function near a base station as a log-normal distribution with parameters depending on the Epstein Zeta function of the lattice, and related approximate dependencies for a power-law constant that governs the network coverage probability at large thresholds. In 2D, we determine the disorder strength at which the perturbed triangular lattice (abbrv. PTL) and the Ginibre networks are the closest measured by comparing their network topologies via a comparison of their Persistence Diagrams in the total variation as well as the symmetrized nearest neighbour distances. We demonstrate that, at this very same disorder, the PTL and the Ginibre networks exhibit very similar coverage probability distributions, with the PTL performing at least as well as the Ginibre. Thus, the PTL network at this disorder strength can be taken to be an effective substitute for the Ginibre network model, while at the same time offering the advantages of greater tractability both from theoretical and empirical perspectives.. |
| 3. | Makoto Katori and Tomoyuki Shirai, Partial Isometry, Duality, and Determinantal Point Processes, Random Matrices: Theory and Applications, https://dx.doi.org/10.1142/S2010326322500253, 2250025 (70 pages), 2021.10. |
| 4. | Yasuaki Hiraoka, Tomoyuki Shirai, Khanh Duy Trinh, Limit theorems for persistence diagrams, Annals of Applied Probability, 10.1214/17-AAP1371, 28, 5, 2740-2780, 2018.10, [URL], The persistent homology of a stationary point process on RN is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.. |
| 5. | Yasuaki Hiraoka, Tomoyuki Shirai, Minimum spanning acycle and lifetime of persistent homology in the Linial–Meshulam process, Random Structures and Algorithms, 10.1002/rsa.20718, 51, 2, 315-340, 2017.09, [URL], This paper studies a higher dimensional generalization of Frieze's ζ(3) -limit theorem on the d-Linial–Meshulam process. First, we define spanning acycles as a higher dimensional analogue of spanning trees, and connect its minimum weight to persistent homology. Then, our main result shows that the expected weight of the minimum spanning acycle behaves in Θ (n^{d-1}).. |
| 6. | Alexander Igorevich Bufetov, Tomoyuki Shirai, Quasi-symmetries and rigidity for determinantal point processes associated with de Branges spaces, Proceedings of the Japan Academy Series A: Mathematical Sciences, 10.3792/pjaa.93.1, 93, 1, 1-5, 2017.01, [URL], In this note, we show that determinantal point processes on the real line corresponding to de Branges spaces of entire functions are rigid in the sense of Ghosh-Peres and, under certain additional assumptions, quasi-invariant under the group of diffeomorphisms of the line with compact support.. |
| 7. | Tomoyuki Shirai, Trinh Khanh Duy, The mean spectral measures of random Jacobi matrices related Gaussian beta ensembles, Electoric Communications of Probability, 10.1214/ECP.v20-4252, 20, 68, 1-13, 2015.10. |
| 8. | Tomoyuki Shirai, Hirofumi Osada, Absolute continuity and singularity of Palm measures of the Ginibre point process, Probability Theory and Related Fields, 10.1007/s00440-015-0644-6, 20, 68, 725-770, 2015.07. |
| 9. | Tomoyuki Shirai, Ginibre-type point processes and their asymptotic behavior, JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, 10.2969/jmsj/06720763, 67, 2, 763-787, 2015.04. |
| 10. | Tomoyuki Shirai, Limit theorem for random analytic functions and their zeros, RIMS Kôkyûroku Bessatsu, to appear, 2012.07. |
| 11. | Takuya Ohwa, Yusuke Higuchi and Tomoyuki Shirai, Exact computation for the cover times of certain classes of trees, Journal of Math-for-Industry, 2, A, 93-98, 2010.04. |
| 12. | Tomoyuki Shirai, Yoichiro Takahashi, Random point fields associated with certain Fredholm determinants I Fermion, poisson and boson point processes, Journal of Functional Analysis, 10.1016/S0022-1236(03)00171-X, 205, 2, 414-463, 2003.12, [URL], We introduce certain classes of random point fields, including fermion and boson point processes, which are associated with Fredholm determinants of certain integral operators and study some of their basic properties: limit theorems, correlation functions, Palm measures etc. Also we propose a conjecture on an α-analogue of the determinant and permanent.. |
| 13. | Tomoyuki Shirai, Yoichiro Takahashi, Random point fields associated with certain Fredholm determinants II Fermion shifts and their ergodic and Gibbs properties, Annals of Probability, 10.1214/aop/1055425789, 31, 3, 1533-1564, 2003.07, [URL], We construct and study a family of probability measures on the configuration space over countable discrete space associated with nonnegative definite symmetric operators via determinants. Under a mild condition they turn out unique Gibbs measures. Also some ergodic properties, including the entropy positivity, are discussed in the lattice case.. |
主要総説, 論評, 解説, 書評, 報告書等
主要学会発表等
| 1. | Tomoyuki Shirai, Zeros of the i.i.d. Gaussian Laurent series on an annulus, The Statistical Physics of Continuum Particle Systems with Strong Interactions, 2022.08. |
| 2. | Tomoyuki Shirai, Determinantal point processes associated with extended kernels and spanning trees on series-parallel graphs, Function theory and dynamics of point processes, 2017.06. |
| 3. | 白井 朋之, Probabilistic apsects of persistent homology, La Trobe-Kyushu Joint Seminar on Mathematics for Industry, 2016.06. |
| 4. | 白井 朋之, Persistent homology and minimum spanning acycle for certain random complexes, Workshop on "High-Dimensional Expanders 2016", 2016.06. |
| 5. | 白井 朋之, Lifetime Sum of Persistent Homology and Minimum Spanning Acycles in Random Simplicial Complexes, Topological Data Analysis on Materials Science, 2015.02. |
| 6. | 白井 朋之, Persistent homology of certain random simplicial complexes, 13thSALSIS The 13th workshop on "Stochastic Analysis on Large Scale Interacting Systems", 2014.11. |
| 7. | 白井 朋之, Absolute continuity and singularity for the Ginibre point process and its Palm measures, UK-Japan Stochastic Analysis School , 2014.09. |
| 8. | 白井 朋之, ガウス型べき級数の実零点過程の相関関数とパフィアン, 日本数学会, 2012.09. |
| 9. | 白井 朋之, 確率論とフェルミオン構造, 研究会「物質科学の数学的手法と数理物理, 2012.06. |
| 10. | Tomoyuki Shirai, Zeros of random analytic functions, 平成23年度確率論シンポジウム, 2011.12. |
| 11. | Tomoyuki Shirai, Ginibre point process and its Palm measures: absolute continuity and singularity, 10th Workshop on Stochastic Analysis on Large Scale Interacting Systems , 2011.12. |
| 12. | Tomoyuki Shirai, Determinantal point processes and the zeros of analytic functions, Functions in number theory and their probabilistic aspects, 2010.12, [URL]. |
学会活動
学協会役員等への就任
2016.04～2018.03, 日本数学会, 運営委員.
2018.04～2020.03, 日本数学会, 運営委員.
2006.04～2008.03, 日本数学会, 運営委員.
学会大会・会議・シンポジウム等における役割
2019.05.27～2019.05.31, Workshop on Probabilistic Methods in Statistical Mechanics of Random Media and Random Fields, 主催.
2019.09.02～2019.09.06, Japanese-German Open Conference on Stochastic Analysis 2019, Scientific Committe.
2019.07.31～2019.08.09, The 12th Mathematical Society of Japan, Seasonal Institute (MSJ-SI) Stochastic Analysis, Random Fields and Integrable Probability, 主催.
2008.03.01～2008.03.01, 日本数学会, 座長.
学会誌・雑誌・著書の編集への参加状況
2017.06～2023.06, Journal of Mathematical Society of Japan, 国際, 編集委員.
2012.09, International Journal of Math-for-Industry, 国際, 編集委員.
学術論文等の審査
| 年度 | 外国語雑誌査読論文数 | 日本語雑誌査読論文数 | 国際会議録査読論文数 | 国内会議録査読論文数 | 合計 |
|---|---|---|---|---|---|
| 2019年度 | 2 | 2 | |||
| 2018年度 | 2 | 2 | |||
| 2017年度 | 9 | 9 | |||
| 2009年度 | 5 | 5 | |||
| 2008年度 | 5 | 5 | |||
| 2007年度 | 4 | 4 | |||
| 2006年度 | 5 | 5 |
その他の研究活動
海外渡航状況, 海外での教育研究歴
National University of Singapore, Singapore, 2023.02～2023.03.
La Trobe University, Australia, 2022.11～2022.11.
National University of Singapore, Singapore, 2022.08～2022.09.
KIAS, Korea, 2019.05～2019.05.
王立スウェーデン工科大学（KTH), Sweden, 2019.09～2019.09.
Leiden大学, Netherlands, 2019.05～2019.06.
Magodor Kasba， Marrakech, Morocco, 2019.04～2019.04.
National University of Singapore, Singapore, 2019.03～2019.03.
Leiden大学, Netherlands, 2019.03～2019.03.
復旦大学, China, 2018.11～2018.11.
Leiden大学, Netherlands, 2018.08～2018.08.
National University of Singapore, Singapore, 2018.06～2018.06.
TU Kaiserslautern, Germany, 2017.09～2017.09.
Euler International Mathematical Institute, St. Petersburg, Russia, Russia, 2017.06～2017.06.
Leiden大学, Netherlands, 2017.03～2017.03.
CIRM, France, 2017.02～2017.03.
Ｈｏｔｅｌ Ｓｏｕｒｃｅ ｉｎ Ｌｅｓ Ｄｉａｂｌｅｒｅｔｓ, Switzerland, 2016.06～2016.06.
Technische Universität München, Germany, 2013.03～2013.03.
Institute for Mathematics Sciences, National University of Singapore, Singapore, 2012.09～2012.09.
University of Warwick , Glasgow University, UnitedKingdom, 2012.03～2012.04.
Hawaii University, UnitedStatesofAmerica, 2011.10～2011.10.
University of Washington, Microsoft Redmond, IBM Watson , Microsoft New England, UnitedStatesofAmerica, 2009.11～2009.11.
Chunbug University, Hankyong Univerisity, Korea, 2009.01～2009.01.
University of Washington, UnitedStatesofAmerica, 2006.10～2007.09.
受賞
日本数学会 2016年JMSJ論文賞, 日本数学会, 2016.03.
研究資金
科学研究費補助金の採択状況(文部科学省、日本学術振興会)
2023年度～2027年度, 基盤研究(B), 代表, 行列式点過程の普遍性とランダム現象の解析.
2022年度～2026年度, 学術変革領域研究(A), 代表, データ記述科学創出に向けた数学的基盤構築.
2020年度～2022年度, 挑戦的研究（萌芽）, 代表, 行列式確率場と機械学習.
2017年度～2019年度, 挑戦的研究（萌芽）, 代表, ホモロジー論的視点からのパーコレーションの高次元化の研究 .
2018年度～2022年度, 基盤研究(B), 代表, 行列式点過程の視点によるランダム現象の解析とその応用.
2012年度～2016年度, 基盤研究(A), 分担, ２次元クーロンポテンシャルによって相互作用する無限粒子系の確率幾何と確率力学 .
2014年度～2016年度, 挑戦的萌芽研究, 代表, ランダムグラフとパーシステントホモロジー.
2014年度～2017年度, 基盤研究(B), 代表, 行列式過程とその一般化に関する研究.
2010年度～2013年度, 基盤研究(B), 代表, 行列式構造をもつ確率過程の研究.
2006年度～2009年度, 基盤研究(C), 代表, ランダムな解析関数の零点過程の研究.
2003年度～2005年度, 若手研究(B), 代表, フェルミオン・ボソンランダム場の一般化に関する研究.
2000年度～2002年度, 奨励研究(A), 代表, 確率論のスペクトル論的研究.
1998年度～1999年度, 奨励研究(A), 代表, 確率論とスペクトル理論の融合的研究（ランダム行列とランダムウォークへの応用）.
日本学術振興会への採択状況(科学研究費補助金以外)
2021年度～2021年度, 二国間交流, 代表, ランダム媒質と確率場の統計力学における確率論的方法2.
2019年度～2019年度, 二国間交流, 代表, ランダム媒質と確率場の統計力学における確率論的方法.
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