Organized 09

## Random Matrices and Infinite Particle Systems (Organizer: Hirofumi Osada)

Conference
11:30 AM — 12:00 PM KST
Local
Jul 20 Tue, 10:30 PM — 11:00 PM EDT

### Dynamical universality for random matrices

5
We establish an invariance principle corresponding to the universality of random matrices. More precisely, we prove dynamical universality of random matrices in the sense that, if random point fields $\mu ^N$ of $N$-particle systems describing eigenvalues of random matrices or log-gases with general self-interaction potentials $V^N$ converge to some random point field $\mu$, then the associated natural $\mu ^N$-reversible diffusion processes represented by solutions of stochastic differential equations (SDE) converge to some $\mu$-reversible diffusion processes given by a solution of the infinite-dimensional stochastic differential equations (ISDE). Our results are general theorems and can be applied to various random point fields related to random matrices such as sine, Airy, Bessel, and Ginibre random point fields. The representations of finite-dimensional SDEs describing the $N$-particle systems are very complicated in general. The limit ISDE has simple and universal representations, nevertheless, according to a class of random matrices such as bulk, soft-edge, and hard-edge scaling. We thus prove ISDE such that the infinite-dimensional Dyson model and Airy, Bessel, and Ginibre interacting Brownian motions are universal dynamical objects. The key ingredients are (1) Local uniform convergence of correlation functions to that of the limit point process. (2) The uniqueness of a weak solution of the limit ISDE, which deduces the uniqueness of Dirichlet forms. Concerning (2), we use the result in [1] and [2].

[1] Hirofumi Osada, Hideki Tanemura, Infinite-dimensional stochastic differential equations and tail $\sigma$-fields, Probability Theory and Related Fields 177, 1137-1242 (2020).
[2] Yosuke Kawamoto, Hirofumi Osada, Hideki Tanemura, Uniqueness of Dirichlet forms related to infinite systems of interacting Brownian motions, (online) Potential Anal.

### Signal processing via the stochastic geometry of spectrogram level sets

Subhroshekhar Ghosh (National University of Singapore)

5
Spectrograms are fundamental tools in the detection, estimation and analysis of signals in the time-frequency analysis paradigm. The spectrogram of a signal (usually corrupted with noise) is the squared magnitude of its short time Fourier transform (STFT), which in turn is a generalised version of the classical Fourier transform, augmented with a window in the time domain. Signal analysis via spectrograms has traditionally explored their peaks, i.e. their maxima, complemented by a recent interest in their zeros or minima. In particular, recent investigations have demonstrated connections between Gabor spectrograms of Gaussian white noise and Gaussian analytic functions (abbrv. GAFs) in different geometries. However, the zero sets (or the maxima or minima) of GAFs have a complicated stochastic structure, which makes a direct theoretical analysis of usual spectrogram based techniques via GAFs a difficult proposition. These techniques, in turn, largely rely on statistical observables from the analysis of spatial data, whose distributional properties for spectrogram extrema are mostly understood only at an empirical level. In this work, we investigate spectrogram analysis via the stochastic, geometric and analytical properties of their level sets. We obtain theorems demonstrating the efficacy of a spectrogram level sets based approach to the detection and estimation of signals, framed in a concrete inferential set-up. Exploiting these ideas as theoretical underpinnings, we propose a level sets based algorithm for signal analysis that is intrinsic to given spectrogram data. We substantiate the effectiveness of the algorithm by exten sive empirical studies. Our results also have theoretical implications for spectrogram zero based approaches to signal analysis.

Based on joint work with Meixia Lin and Dongfang Sun.

### Logarithmic derivatives and local densities of point processes arising from random matrices

5
We talk about a distribution (a generalized function) theory for point processes. We show that a logarithmic derivative in the distributional sense can indicate the local density of the point process. This theory is especially effective for point processes appearing in random matrix theory. In particular, using this result, we solve infinite-dimensional stochastic differential equations associated with the point process given by de Branges spaces, so-called integrable kernels, and random matrices such as Airy, sine, and Bessel point processes. [2] Conventionally, the point process that describes an infinite particle system is described by the Dobrushin-Lanford-Ruelle (DLR) equation. The point process of an infinite particle system appearing in a random matrix has a logarithmic potential as an interaction potential. Because the logarithmic potential is not integrable at infinity, the DLR equation cannot describe the point process as it is. Logarithmic derivative for point process is a concept introduced in [1] to settle this problem. There must be a logarithmic derivative and local density of the point process to solve the infinite-dimensional stochastic differential equation. [3] With our result, the existence of a logarithmic derivative with suitable integrability is sufficient for the construction of the stochastic dynamics as a solution of infinite-dimensional stochastic differential equations.

[1] Hirofumi Osada, Infinite-dimensional stochastic differential equations related to random matrices, Probability theory and related fields, 2012, 153(3-4), 471--509.
[2] Alexander I Bufetov, Andrey V Dymov, Hirofumi Osada, The logarithmic derivative for point processes with equivalent Palm measures, J. Math. Soc. Japan, 71(2), 2019, 451--469.
[3] Hirofumi Osada, Hideki Tanemura, Infinite-dimensional stochastic differential equations and tail $\sigma$-fields, Probability Theory and Related Fields 177, 1137-1242 (2020).

### Stochastic differential equations for infinite particle systems of jump type with long range interactions

Hideki Tanemura (Keio university)

5
Infinite dimensional stochastic differential equations (ISDEs) describing systems with an infinite number of particles are considered. Each particle undergoes Levy process, and interaction between particles is given by long range interaction potential, which is not only of Ruelle's class but also logarithmic. We discuss the existence and uniqueness of strong solutions of the ISDEs.

This talk is based on a collaboration with Shota Esaki (Fukuoka University).

### Q&A for Organized Contributed Session 09

0
This talk does not have an abstract.

Organized 18

## Advanced Learning Methods for Complex Data Analysis (Organizer: Xinlei Wang)

Conference
11:30 AM — 12:00 PM KST
Local
Jul 20 Tue, 10:30 PM — 11:00 PM EDT

### Peel learning for pathway-related outcome prediction

Rui Feng (University of Pennsylvania)

6
Traditional regression models are limited in outcome prediction due to their parametric nature. Current deep learning methods allow for various effects and interactions and have shown improved performance, but they typically need to be trained on a large amount of data to obtain reliable results. Gene expression studies often have small sample sizes but high dimensional correlated predictors so that traditional deep learning methods are not readily applicable. In this talk, I present peel learning, a novel neural network that incorporates the prior relationship among genes. In each layer of learning, overall structure is peeled into multiple local substructures. Within the substructure, dependency among variables is reduced through linear projections. The overall structure is gradually simplified over layers and weight parameters are optimized through a revised backpropagation. We applied PL to a small lung transplantation study to predict recipients’ postsurgery primary graft dysfunction using donors’ gene expressions within several immunology pathways, where PL showed improved prediction accuracy compared to conventional penalized regression, classification trees, feed-forward neural network, and a neural network assuming prior network structure. Through simulation studies, we also demonstrated the advantage of adding specific structure among predictor variables in neural network, over no or uniform group structure, which is more favorable in smaller studies. The empirical evidence is consistent with our theoretical proof of improved upper bound of PL’s complexity over ordinary neural networks.

### Principal boundary for data on manifolds

Zhigang Yao (National University of Singapore)

7
We will discuss the problem of finding principal components to the multivariate datasets, that lie on an embedded nonlinear Riemannian manifold within the higher-dimensional space. Our aim is to extend the geometric interpretation of PCA, while being able to capture the non-geodesic form of variation in the data. We introduce the concept of a principal sub-manifold, a manifold passing through the center of the data, and at any point of the manifold, it moves in the direction of the highest curvature in the space spanned by the eigenvectors of the local tangent space PCA. We show the principal sub-manifold yields the usual principal components in Euclidean space. We illustrate how to find, use and interpret the principal sub-manifold, with which a classification boundary can be defined for data sets on manifolds.

### Probabilistic semi-supervised learning via sparse graph structure learning

Li Wang (University of Texas at Arlington)

5
We present a probabilistic semi-supervised learning (SSL) framework based on sparse graph structure learning. Different from existing SSL methods with either a predefined weighted graph heuristically constructed from the input data or a learned graph based on the locally linear embedding assumption, the proposed SSL model is capable of learning a sparse weighted graph from the unlabeled high-dimensional data and a small amount of labeled data, as well as dealing with the noise of the input data. Our representation of the weighted graph is indirectly derived from a unified model of density estimation and pairwise distance preservation in terms of various distance measurements, where latent embeddings are assumed to be random variables following an unknown density function to be learned and pairwise distances are then calculated as the expectations over the density for the model robustness to the data noise. Moreover, the labeled data based on the same distance representations is leveraged to guide the estimated density for better class separation and sparse graph structure learning. A simple inference approach for the embeddings of unlabeled data based on point estimation and kernel representation is presented. Extensive experiments on various data sets show the promising results in the setting of SSL compared with many existing methods, and significant improvements on small amounts of labeled data.

### Bayesian modeling for paired data in genome-wide association studies with application to breast cancer

Min Chen (University of Texas at Dallas)

4
Genome-wide association studies (GWAS) has emerged as a useful tool to identify common genetic variants that are linked to complex diseases. Conventional GWAS are based on the case-control design where the individuals in cases and controls are independent. In cancer research, matched pair designs, which compare tumor tissues with normal ones from the same subjects, are becoming increasingly popular. Such designs succeed in identifying somatic mutations in tumors while controlling both genetic and environmental factors. Somatic variation is one of the most important cancer risk factors that contribute to continuous monitoring and early detection of various cancers. However, most GWAS analysis methods, developed for unrelated samples in case-control studies, cannot be employed in the matched pair designs. A novel framework is proposed in this manuscript to accommodate for the particularity of matched-data in association studies of somatic mutation effects. In addition, we develop a Bayesian model to combine multiple markers to further improve the power of mapping genome regions to cancer risks.

### Q&A for Organized Contributed Session 18

0
This talk does not have an abstract.

###### Session Chair

Xinlei Wang (Southern Methodist University)

Organized 27

## Bayesian Inference for Complex Models (Organizer: Joungyoun Kim)

Conference
11:30 AM — 12:00 PM KST
Local
Jul 20 Tue, 10:30 PM — 11:00 PM EDT

### Nonparametric Bayesian latent factor model for multivariate functional data with covariate dependency

Yeonseung Chung (Korea Advanced Institute of Science and Technology (KAIST))

2
Nowadays, multivariate functional data are frequently encountered in many fields of science. While there exist a variety of methodologies for univariate functional clustering, the approach for multivariate functional clustering are less studied. Moreover, there is little research for the functional clustering methods incorporating additional covariate information. In this paper, we propose a Bayesian nonparametric sparse latent factor model for covariate-dependent multivariate functional clustering. Multiple functional curves are represented by basis coefficients for splines, which are reduced to latent factors. Then, the factors and covariates are jointly modeled using a Dirichlet process (DP) mixture of Gaussians to facilitate a model-based covariate dependent multivariate functional clustering. The method is further extended to dynamic multivariate functional clustering to handle sequential multivariate functional data. The proposed methods are illustrated through a simulation study and applications to Canadian weather and air pollution data.

### Bayesian model selection for ultrahigh-dimensional doubly-intractable distributions

Jaewoo Park (Yonsei University)

3
Doubly intractable distributions commonly arise in many complex statistical models in physics, epidemiology, ecology, social science, among other disciplines. With an increasing number of model parameters, they often result in ultrahigh-dimensional posterior distributions; this is a challenging problem and is crucial for developing the computationally feasible approach. A particularly important application of ultrahigh-dimensional doubly intractable models is network psychometrics, which gets attention in item response analysis. However, its parameter estimation method, maximum pseudo-likelihood estimator (MPLE) combining with lasso certainly ignores the dependent structure, so that it is inaccurate. To tackle this problem, we propose a novel Markov chain Monte Carlo methods by using Bayesian variable selection methods to identify strong interactions automatically. With our new algorithm, we address some inferential and computational challenges: (1) likelihood functions involve doubly-intractable normalizing functions, and (2) increasing number of items can lead to ultrahigh dimensionality in the model. We illustrate the application of our approaches to challenging simulated and real item response data examples for which studying local dependence is very difficult. The proposed algorithm shows significant inferential gains over existing methods in the presence of strong dependence among items.

### Post-processed posteriors for banded covariances

Kwangmin Lee (Seoul National University)

5
We consider Bayesian inference of banded covariance matrices and propose a post- processed posterior. The post-processing of the posterior consists of two steps. In the first step, posterior samples are obtained from the conjugate inverse-Wishart posterior which does not satisfy any structural restrictions. In the second step, the posterior samples are transformed to satisfy the structural restriction through a post-processing function. The conceptually straightforward procedure of the post-processed posterior makes its computation efficient and can render interval estimators of functionals of covariance matrices. We show that it has nearly optimal minimax rates for banded covariances among all possible pairs of priors and post-processing functions. Furthermore, we prove that, the expected coverage probability of the $(1-\alpha)$100% highest posterior density region of the post-processed posterior is asymptotically $1-\alpha$ with respect to a conventional posterior distribution. It implies that the highest posterior density region of the post-processed posterior is, on average, a credible set of a conventional posterior. The advantages of the post-processed posterior are demonstrated by a simulation study and a real data analysis.

### Adaptive Bayesian inference for current status data on a grid

Minwoo Chae (Pohang University of Science and Technology)

3
We study a Bayesian approach to the inference of an event time distribution in the current status model where observation times are supported on a grid of potentially unknown sparsity and multiple subjects share the same observation time. The model leads to a very simple likelihood, but statistical inferences are non-trivial due to the unknown sparsity of the grid. In particular, for an inference based on the maximum likelihood estimator, one needs to estimate the density of the event time distribution which is challenging because the event time is not directly observed. We consider Bayes procedures with a Dirichlet prior on the event time distribution. With this prior, the Bayes estimator and credible sets can be easily computed via a Gibbs sampler algorithm. Our main contribution is to provide thorough investigation of frequentist's properties of the posterior distribution. Specifically, it is shown that the posterior convergence rate is adaptive to the unknown sparsity of the grid. If the grid is sufficiently sparse, we further prove the Bernstein-von Mises theorem which guarantees frequentist's validity of Bayesian credible sets. A numerical study is also conducted for illustration.

### Q&A for Organized Contributed Session 27

0
This talk does not have an abstract.

###### Session Chair

Joungyoun Kim (Yonsei Univesity)

Organized 28

## Recent advances in Time Series Analysis (Organizer: Changryoung Baek)

Conference
11:30 AM — 12:00 PM KST
Local
Jul 20 Tue, 10:30 PM — 11:00 PM EDT

### Resampling long-range dependent time series

Shuyang Bai (University of Georgia)

3
For time series exhibiting long-range dependence, inference through resampling is of particular interest since the asymptotic distributions are often difficult to determine statistically. On the other hand, due to the strong dependence and the non-standard scaling, designing versatile resampling strategies and establishing their validity is challenging. We shall introduce some progress on this direction.

### Robust test for structural instability in dynamic factor models

Changryong Baek (Sungkyunkwan University)

3
In this paper, we consider a robust test for structural breaks in dynamic factor models. Our framework considers structural changes when the underlying high dimensional time series is contaminated by some outlying observations, which is typically observed in many real applications such as fMRI, economics and finance. We propose a test based on the robust estimation of vector autoregressive model for principal component factors using minimum density power divergence estimator. Simulations study shows excellent finite sample performance, higher powers while achieving good sizes in all cases considered. Our method is illustrated to resting state fMRI series to detect brain connectivity changes. It shows that brain connectivity indeed changes even in the resting state and this is not an artifact of outlier effects.

### On scaling in high dimensions

Gustavo Didier (Tulane University)

3
Scaling relationships have been found in a wide range of phenomena that includes coastal landscapes, hydrodynamic turbulence, the metabolic rates of animals and Internet traffic. For scale invariant systems, also called fractals, a continuum of time scales contributes to the observed dynamics, and the analyst's focus is on identifying mechanisms that relate the scales, often in the form of exponents. In this talk, we will look into the little explored topic of scale invariance in high dimensions, which is especially important in the modern era of "Big Data". We will discuss the role played by wavelets in the analysis of self-similar stochastic processes and visit recent contributions to the wavelet modeling of high- and multidimensional scaling systems.

This is joint work with P. Abry (CNRS and ENS-Lyon), B.C. Boniece (Washington University in St Louis) and H. Wendt (CNRS and Université de Toulouse).

### Thresholding and graphical local Whittle estimation

Marie Duker (Cornell University)

2
The long-run variance matrix and its inverse, the so-called precision matrix, give, respectively, information about correlations and partial correlations between dependent component series of multivariate time series around zero frequency. This talk will present non-asymptotic theory for estimation of the long-run variance and precision matrices for high-dimensional Gaussian time series under general assumptions on the dependence structure including long-range dependence. The presented results for thresholding and penalizing versions of the classical local Whittle estimator ensure consistent estimation in a possibly high-dimensional regime. The key technical result is a concentration inequality of the local Whittle estimator for the long-run variance matrix around the true model parameters. In particular, it handles simultaneously the estimation of the memory parameters which enter the underlying model.

### Cotrending: testing for common deterministic trends in varying means model

Vladas Pipiras (University of North Carolina at Chapel Hill)

3
In a varying means model, the temporary evolution of a p-vector system is determined by p deterministic nonparametric functions superimposed by error terms, possibly dependent cross sectionally. The basic interest is in linear combinations across the p dimensions that make the deterministic functions constant over time. The number of such linearly independent linear combinations is referred to as a cotrending dimension, and their spanned space as a cotrending space. This work puts forward a framework to test statistically for cotrending dimension and space. Connections to principal component analysis and cointegration are also considered. Finally, a simulation study to assess the finite-sample performance of the proposed tests, and applications to several real data sets are also provided.

### Q&A for Organized Contributed Session 28

0
This talk does not have an abstract.

###### Session Chair

Changryoung Baek (Sungkyunkwan University)