Invited 30

## Functional Estimation, Testing and Clustering under Sparsity (Organizer: Jiashun Jin)

Conference
11:30 AM — 12:00 PM KST
Local
Jul 18 Sun, 10:30 PM — 11:00 PM EDT

### Optimal Network Testing by the Signed-Polygon Statistics

Tracy Ke (Harvard University)

7
Given a symmetric social network, we are interested in testing whether it has only one community or multiple communities. The desired tests should (a) accommodate severe degree heterogeneity, (b) accommodate mixed-memberships, (c) have a tractable null distribution, and (d) adapt automatically to different levels of sparsity and achieve the optimal phase transition. How to find such a test is a challenging problem. Many existing tests do not allow for heterogeneity or mixed-memberships and cannot (a)-(d). We propose the Signed Polygon as a class of new tests. Fixing m ≥ 3, for each m-gon in the network, define a score using the centered adjacency matrix. The sum of such scores is then the m-th order Signed Polygon statistic. The Signed Quadrilateral (SgnQ) is a special example of the Signed Polygon with m=4. We show that the SgnQ test satisfies (a)-(d), and especially, they work well for both very sparse and less sparse networks. We derive the asymptotic null distribution and the power of the SgnQ test. For the matching lower bound, we use a phase transition framework, which is more informative than the standard minimax argument. The SgnQ test is applied to a coauthorship network constructed from research papers in 36 statistics journals in a 41-year time span. We demonstrate using the SgnQ test to (a) measure the coauthorship diversity and (b) build a multi-layer community tree.
This is a collaborated work with Jiashun Jin and Shengming Luo.

### Statistical inference for linear mediation models with high-dimensional mediators

Runze Li (Penn State University)

8
Mediation analysis draws increasing attention in many scientific areas such as genomics, epidemiology and finance. In this paper, we propose new statistical inference procedures for high dimensional mediation models, in which both the outcome model and the mediator model are linear with high dimensional mediators. Traditional procedures for mediation analysis cannot be used to make statistical inference for high dimensional linear mediation models due to high-dimensionality of the mediators. We propose an estimation procedure for the indirect effects of the models via a partial penalized least squares method, and further establish its theoretical properties. We further develop a partial penalized Wald test on the indirect effects, and prove that the proposed test has a $\chi^2$ limiting null distribution. We also propose an $F$-type test for direct effects and show that the proposed test asymptotically follows a $\chi^2$-distribution under null hypothesis and a noncentral $\chi^2$-distribution under local alternatives. Monte Carlo simulations are conducted to examine the finite sample performance of the proposed tests and compare their performance with existing ones. A real data example is used to illustrate the proposed methodology.

### Perturbation Bounds for Tensors and Their Applications in High Dimensional Data Analysis

Ming Yuan (Columbia University)

7
We develop deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors, in a spirit similar to classical results for matrices. Our bounds exhibit intriguing differences between matrices and higher-order tensors. Most notably, they indicate that for higher-order tensors perturbation affects each singular value/vector in isolation. In particular, its effect on a singular vector does not depend on the multiplicity of its corresponding singular value or its distance from other singular values. Our results can be readily applied and provide a unified treatment to many different problems involving higher-order orthogonally decomposable tensors. In particular, we illustrate the implications of our bounds in several high dimensional data analysis problems.

### Q&A for Invited Session 30

0
This talk does not have an abstract.

###### Session Chair

Jiashun Jin (Carnegie Mellon University)

Invited 39

## KSS Invited Session: Interactive Particle Systems and Urn Models (Organizer: Woncheol Jang)

Conference
11:30 AM — 12:00 PM KST
Local
Jul 18 Sun, 10:30 PM — 11:00 PM EDT

### Convergence of randomized urn models with irreducible and reducible replacement

Li-Xin Zhang (Zhejiang University)

7
Generalized Friedman urn is a popular model in probability theory. Since Athreya and Ney (1972) showed the almost sure convergence of urn proportions in a randomized urn model with irreducible replacement matrix under the $L\log L$ moment assumption, this assumption has been regarded as the weakest moment assumption, but its necessariness has never been shown. In this talk, we will consider the strong and weak convergence of generalized Friedman urns. It is proved that, when the random replacement matrix is irreducible in probability, the sufficient and necessary moment assumption for the almost sure convergence of the urn proportions is that the expectation of the replacement matrix is finite, which is less stringent than the $L\log L$ moment assumption, and that when the replacement is reducible, the $L\log L$ moment assumption is the weakest sufficient condition. The rate of convergence and the strong and weak convergence of non-homogeneous generalized Friedman urns are also derived.

### Condensation phenomenon and metastability in interacting particle systems

Insuk Seo (Seoul National University)

16
In this talk, we discuss recent development in the study of the condensation phenomena appeared in interacting particle systems such as the super-critical or critical zero-range process and the inclusion process. In particular, we will focus on the metastable behavior, i.e., the evolution of the condensate after it is formed.

This talk is based on joint works with S. Kim, C, Landim, and D. Marcondes.

### Time correlation exponents in planar last passage percolation

Riddhipratim Basu (International Centre for Theoretical Sciences-TIFR)

8
Planar last passage percolation (LPP) models are canonical examples of stochastic growth in the Kardar-Parisi-Zhang universality class, where one considers oriented paths (moving forward in the "time" direction) between points in a random environment accruing the integral of the noise along itself as its weight. The maximal weight of a path joining two points is called the last passage time between the points. Although these models are expected to exhibit universal features under mild conditions on the underlying i.i.d. noise, rigorous progress has mostly been limited to a handful of exactly solvable models. One of the questions in this class of models that has drawn a lot of recent attention is that of the two time correlation, i.e., two understand the correlation decay of the last passage times to a sequence of points varying along the time direction (i.e., the diagonal direction) starting from different initial data. I shall describe some results obtaining the exponents governing the short and long range correlations in the context of the exactly solvable model of planar exponential last passage percolation starting from flat and step initial data.

Based on joint works with Shirshendu Ganguly and Lingfu Zhang.

### Q&A for Invited Session 39

0
This talk does not have an abstract.

###### Session Chair

Panki Kim (Seoul National University)