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Activity Number: 438 - Statistical Methods for Topological Data Analysis
Type: Topic Contributed
Date/Time: Thursday, August 6, 2020 : 10:00 AM to 11:50 AM
Sponsor: Section on Statistical Learning and Data Science
Abstract #309869
Title: Solution Manifold and Its Statistical Applications
Author(s): Yen-Chi Chen*
Companies: University of Washington
Keywords: mixture model; local optima; non convex; gradient descent; EM algorithm; bump hunting

A solution manifold is the collection of points in a d-dimensional space satisfying a system of s equations with s< d. Solution manifolds occur in several statistical problems including hypothesis testing, curved-exponential families, constrained mixture models, partial identifications, and nonparametric set estimation. We analyze solution manifolds both theoretically and algorithmically. In terms of theory, we derive five useful results: the smoothness theorem, the stability theorem (which implies the consistency of a plug-in estimator), the convergence of a gradient flow, the local center manifold theorem and the convergence of the gradient descent algorithm. To numerically approximate a solution manifold, we propose a Monte Carlo gradient descent algorithm. In the case of likelihood inference, we design a manifold constraint maximization procedure to find the maximum likelihood estimator on the manifold. We also develop a method to approximate a posterior distribution defined on a solution manifold.

Authors who are presenting talks have a * after their name.

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