2025年度 統計数理セミナー
Statistical Mathematics Seminar 2025

English

日本語

日本語

日本語

日本語

日本語

日本語

日本語

日本語

日本語

English

Optimal transport (OT) provides a set of powerful toolkits to compare measures. However, OT has a high computational complexity, i.e., super cubic w.r.t. the number of input supports. Several variants of Sliced Wasserstein (SW) have been developed in the literature to overcome this challenge. These approaches exploit the closed-form expression of the univariate OT by projecting input measures onto one-dimensional lines. However, projecting measures onto low-dimensional spaces can lead to a loss of topological information. To mitigate this issue, we propose to replace one-dimensional lines with a more advanced structure, called tree systems. This structure is metrizable by a tree metric, which yields a closed-form expression for OT on tree systems. We derive an extensive theoretical analysis to formally define tree systems, introduce the concept of splitting maps, propose novel variants of Radon transform for tree systems, and verify their infectivity. Empirically, we illustrate that the proposed approaches perform favorably compared to SW and its variants on applications with dynamic-support measures such as generative models, and diffusion models.

日本語

日本語

日本語

日本語

日本語

以下の２つの内容を紹介する。１）結晶構造には固溶体と呼ばれる相が存在する。固溶体は一見単純な構造を持ちながらも、多様な元素の組み合わせが可能であり、その構造は実験的にも、また様々なレベルの理論計算によっても調べられており、ある程度の数の結果が存在する。これらの元素組み合わせ情報「証拠理論」を用いて統一的に扱った推薦システムとその成果について紹介する。２）近年では、大規模言語モデルを活用して、科学的知識や実験プロセス、思考の過程を記述・説明する取り組みが進んでいる。しかし、こうした情報の律速となるのは人間の理解であろうと思われる。そこで、人間が最低限の知識を効率よく伝達するための手法として、「機能分解木」とその活用事例を紹介する。

日本語

国語学は人文科学の範疇であり，計量分析といえばコーパスを利用したテキスト分析はまずは想起されるが，社会科学分野でよく研究手段とされるような量的社会調査も様々なテーマでよく実施されている。本発表では講演者が過去に参加した，または現に参加している共同研究等のプロジェクトから３件の研究を紹介し，統計科学分野からの貢献の役割を考える。1件目は「鶴岡市における言語調査」，2件目は「方言に関する話者類型研究」，3件目は「新型コロナ関連用語の社会的許容度に関する研究」である。

English

In this work, we address a class of nonconvex nonsmooth optimization problems where the objective function is the sum of two smooth functions (one of which is proximable) and two nonsmooth functions (one proper, closed and proximable, and the other continuous and weakly concave). We introduce a new splitting algorithm that extends the Davis-Yin splitting (DYS) algorithm to handle such four-term nonconvex nonsmooth problems. We prove that with appropriately chosen stepsizes, our algorithm exhibits global subsequential convergence to stationary points with a stationarity measure converging at a global rate of 1/T, where T is the number of iterations. When specialized to the setting of the DYS algorithm, our results allow for larger stepsizes compared to existing bounds in the literature. Experimental results demonstrate the practical applicability and effectiveness of our proposed algorithm. This is joint work with Jan Harold Alcantara and Akiko Takeda.

日本語

日本語

日本語

Strict proper losses are fundamental loss functions inducing classifiers capable of estimating class probabilities. While practitioners have devised many loss functions, their properness is often unverified. In this work, we identify several losses as improper, calling into question the validity of class probability estimates derived from their simplex-projected outputs. Nevertheless, we show that these losses are strictly proper composite with appropriate link functions, allowing predictions to be mapped into true class probabilities. We invent the calmness condition, which we prove suffices to identify that a loss has a strictly proper composite representation, and provide the general form of the inverse link.

日本語

日本語

点集合は、犯罪や地震などのイベントの起きた場所という空間的な場合だけでなく、単語の特定の文脈での意味を表す埋め込みベクトルの集合のように、自然言語処理においても重要なデータである。その時間遷移の統計モデルは重要な研究対象であるが、分布は一般に複雑であり、単純な方法では分布の様子の遷移を捉えることができない。本研究ではこの問題に対し、機械学習の分野で提案されたガウス過程密度サンプラーと乱択化フーリエ特徴を組み合わせることにより、複雑な点分布の変化を周波数空間の主成分として捉える方法を提案し、実験的に検証した。

日本語

English

We study a class of symmetric quantum walks on Hamming graphs, where the distance between vertices specifies the transition probability. The simple quantum walk on the hypercube, which has been discussed in literature, is a special case. The zeros of certain self-reciprocal polynomials determine the dynamics of the quantum walks. We obtain a spectral representation of the wave function, where working with the coin space isomorphic to the state space and the commutative association scheme enables our systematic treatment. The limit distributions of several quantum walks are obtained. They are represented as the mixtures of a distribution that appears in random walks and another unknown distribution. This is joint work with Robert Griffiths.

English

This talk will discuss a random walk on the infinite hypercube, Xt+1 = Xt + Zt mod 2. The increments (Zt) are i.i.d. with entries that form an infinite exchangeable {0, 1} sequence, a de Finetti sequence. There is geometric killing in the random walk. A Gaussian free field (gx)x∈{0,1}∞ is associated with the random walk by taking the covariance function to be proportional to the Green function of the random walk. The Green function and a strong representation for (gx) are characterized by a negative binomial point process which involves the de Finetti measure of the increments of the random walk.

日本語

日本語