Detecting and approximating decision boundaries in low dimensional spaces

TL;DR

This paper introduces a guaranteed-accuracy method for detecting and approximating decision boundaries in low-dimensional spaces, using scattered points and classification algorithms, with applications in fault detection, decision aid, and inverse scattering.

Contribution

The paper presents a novel approach to accurately approximate decision curves in 2D and 3D with fewer classifications, incorporating local refinement based on geometric properties.

Findings

Requires fewer pointwise classifications than previous methods

Ensures complete coverage of the decision curve

Applicable to fault detection, decision aid, and inverse scattering

Abstract

A method for detecting and approximating fault lines or surfaces, respectively, or decision curves in two and three dimensions with guaranteed accuracy is presented. Reformulated as a classification problem, our method starts from a set of scattered points along with the corresponding classification algorithm to construct a representation of a decision curve by points with prescribed maximal distance to the true decision curve. Hereby, our algorithm ensures that the representing point set covers the decision curve in its entire extent and features local refinement based on the geometric properties of the decision curve. We demonstrate applications of our method to problems related to the detection of faults, to Multi-Criteria Decision Aid and, in combination with Kirsch's factorization method, to solving an inverse acoustic scattering problem. In all applications we considered in this work, our method requires significantly less pointwise classifications than previously employed algorithms.