A computational framework for inverse queries in statistical learning problems

Posted on:2001-04-12

Degree:Ph.D

Type:Dissertation

University:University of California, Irvine

Candidate:Mundkur, Prashanth

Full Text:PDF

GTID:1460390014952279

Subject:Engineering

Abstract/Summary:

This dissertation addresses the problem of inverting the nonlinear functions that arise in solutions to statistical learning problems. Such problems arise when inverse queries need to be answered.; To provide background for the kinds of inverse queries that can arise in learning, we first briefly present, as examples of the forward problem, the three most common learning problems arising in applications, viz. regression estimation, density estimation and pattern recognition. These learning problems are usually formulated in terms of the estimation of a nonlinear learning function f from a selected class of functions using given training data. Many approaches to this estimation problem give rise to a common general form of f. Once estimated, this f then constitutes a solution to the forward learning problem.; Inverse queries are then posed as questions asked of this f that can be formulated as an inverse problem, viz. to find the solutions

x&vbm0;fx ∈Q of the equation

f

x ∈Q . This dissertation proposes a theoretical and computational framework based on simplicial approximation for tackling the problem of computing solutions to such inverse queries. This framework is motivated by elementary results from piecewise-linear topology.; Two algorithms are developed: one derives from a straightforward application of the simplicial approximation theorem and deals with inverse problems for arbitrary continuous learning functions. This algorithm is useful, but suffers from various drawbacks due to its generality. A second algorithm is developed to address these drawbacks by specializing it to learning functions from a family that satisfies certain scaling conditions; these conditions derive from the concept of scale-space that has proven useful in computer vision research. Gaussian radial basis functions lie in this family, and are general enough to be widely used in machine learning applications. Implementation details and simulation results are presented for both algorithms.; We conclude that the framework is useful as a computational tool for finding general solutions to functional inversion in inverse learning problems.

Keywords/Search Tags:

Learningproblems, Inverse, Framework, Computational, Solutions, Functions

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