Robust Classification of Dynamic Bichromatic point Sets in R2

TL;DR

This paper introduces efficient algorithms and data structures for robustly classifying dynamic red and blue point sets in the plane, allowing for at most k outliers and minimizing the maximum misclassification distance.

Contribution

It presents the first semi-online dynamic data structure and both exact and approximation algorithms for robust linear separation with outlier tolerance.

Findings

Exact algorithm runs in O(nk + n log n) time.

Approximation algorithm achieves (1+ε)-approximation in O(ε^{-1/2}((n + k^2) log n)) time.

New semi-online data structures enable efficient maintenance of such separators.

Abstract

Let $R \cup B$ be a set of $n$ points in $\mathbb{R}^2$, and let $k \in 1..n$. Our goal is to compute a line that "best" separates the "red" points $R$ from the "blue" points $B$ with at most $k$ outliers. We present an efficient semi-online dynamic data structure that can maintain whether such a separator exists. Furthermore, we present efficient exact and approximation algorithms that compute a linear separator that is guaranteed to misclassify at most $k$, points and minimizes the distance to the farthest outlier. Our exact algorithm runs in $O(nk + n \log n)$ time, and our $(1+\varepsilon)$-approximation algorithm runs in $O(\varepsilon^{-1/2}((n + k^2) \log n))$ time. Based on our $(1+\varepsilon)$-approximation algorithm we then also obtain a semi-online data structure to maintain such a separator efficiently.