Witnessed k-Distance

TL;DR

This paper analyzes an efficient approximation scheme for the distance to a measure, which is robust to noise and outliers, enabling scalable geometric inference with theoretical guarantees.

Contribution

It introduces a linear-complexity approximation method for the distance to a measure, preserving inference quality close to the exact but computationally expensive version.

Findings

The approximation maintains theoretical guarantees on inference quality.

The scheme reduces computational complexity from combinatorial to linear.

It enables scalable geometric analysis in noisy data environments.

Abstract

Distance function to a compact set plays a central role in several areas of computational geometry. Methods that rely on it are robust to the perturbations of the data by the Hausdorff noise, but fail in the presence of outliers. The recently introduced distance to a measure offers a solution by extending the distance function framework to reasoning about the geometry of probability measures, while maintaining theoretical guarantees about the quality of the inferred information. A combinatorial explosion hinders working with distance to a measure as an ordinary (power) distance function. In this paper, we analyze an approximation scheme that keeps the representation linear in the size of the input, while maintaining the guarantees on the inference quality close to those for the exact (but costly) representation.