確率・ 統計・ 行列ワークショップ 彦根 2018

JSPS科研費 [1], [2]の研究の一環として, 2018年10月24日 (水) から 25日 (木) まで, 滋賀大学データサイエンス学部 (彦根キャンパス) にて, 小さな研究集会を行いますのでご案内致します.

• [1] JP18H04092 基盤研究(A) 計算代数統計の方法の性能向上と実用化の推進 (研究代表者 竹村彰通)
• [2] JP16H02792 基盤研究(B) 期待オイラー標数法の深化と実用化，および関連する数理の展開 (研究代表者 栗木哲)

Schedule

2018年10月24日 (水)

13:30--14:30

Characterizations of indicator functions for fractional factorial designs
A polynomial indicator function of designs, introduced by Fontana, Pistone and Rogantin (2000), is a basic tool to characterize fractional factorial designs in the field of computational algebraic statistics. For the case of two-level designs, the structure of the indicator function is well-known. For example, the coefficients of indicator functions have clear meanings relating to the orthogonality for the two-level cases. The polynomial relation among the coefficients are also derived for the two-level cases, which can be used to classify designs with given sizes. However, for the cases of multi-level designs with rational factors, such relations are complicated and interpretations are difficult. In this work, we consider the structure of the indicator function of general designs and its applications.
15:00--16:00

The Hessian matrix of a graph.
Let us consider the Hessian matrix of the weighted generating function for spanning trees in a graph $\Gamma$ with $n+1$ vertices. We assume that $\Gamma$ is simple, undirected and connected. We say that a subgraph in $\Gamma$ is a spanning tree if it is connected and has $n$ edges. A partial derivative corresponds to enumerating spanning trees including some edges. We show that the Hessian does not vanish for some graph by a combinatorial proof.
16:30--17:30

Limit theorems for random analytic functions and their zeros
The study of random analytic functions (power series) and their level sets has a long history and, especially, there have been many works on Gaussian analytic functions, which are random analytic functions that are also Gaussian processes. In this talk, after we survey some recent topics around Gaussian analytic functions and related point processes, we discuss limit theorems for random analytic functions and their zeros.
(本講演は滋賀大DSセミナーと共催)

2018年10月25日 (木)

10:00--11:00

Missing data problem and Cholesky decomposition
We consider a missing data problem for a multivariate normal distribution under the conditional independence assigned by a undirected graph. If there exists a perfect DAG structure on the graph compatible missing data pattern, we give an explicit solution of the likelihood equation, where the Cholesky decomposition plays a crucial role. Furthermore, in order to generalize the formula, we introduce an algebraic structure of a vector space of real symmetric matrices that admits a nice Cholesky decomposition.
11:30--12:30

Doubly autoparallel submanifolds in the space of the probability simplex - Their characterization and classification -
We consider information geometry on the probability simplex to study doubly autoparallel submanifolds. Information geometry defines on the simplex two affine connections, respectively called the exponential and the mixture connections. The geometry naturally introduces submanifolds simultaneously autoparallel with respect to the both connections, which implies that such submanifolds are simultaneously exponential and mixture families of discrete probability distributions.We show their characterization and classification. This is a joint work with Prof. H. Ishi (Nagoya Univ.).
Lunch
14:00--15:00

Selective inference for the problem of regions via multiscale bootstrap resampling with applications to hierarchical clustering and lasso
Selective inference procedures are considered for computing approximately unbiased p-values for arbitrary shaped hypotheses which are selected after looking at the data. Our idea is to estimate the geometric quantities, namely, signed distance and mean curvature, by the multiscale bootstrap in which we change the sample size of bootstrap replicates. Our method is second-order accurate in the large sample theory of smooth boundary surfaces of the hypothesis regions, and it is also justified for regions with nonsmooth surfaces such as cones. This is joint work with Yoshikazu Terada (Osaka University / RIKEN AIP).
(本講演は滋賀大DSセミナーと共催)
15:30--16:30
ブノワ　コリンズ氏 (京都大学)/ Benoît COLLINS(Kyoto Univ.)
Norm convergence for random permutations
Consider a finite sequence of independent random permutations on n points. We will explain why, in probability, as n goes to infinity, these permutations viewed as operators on the (n-1) dimensional vector space orthogonal to the vector with all coordinates equal to 1, are asymptotically strongly free. Our proof relies on the development of a matrix version of the non-backtracking operator theory and a refined trace method. As a byproduct, we show that the non-trivial eigenvalues of random n-lifts of a fixed based graphs approximately achieve the Alon-Boppana bound with high probability in the large n limit. This result generalizes a theorem by Friedman. Time allowing, we will discuss more recent developments. This is joint work with Charles Bordenave.