Solution manifold and its statistical applications

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