Solution manifold and Its Statistical Applications

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.