Robust Bichromatic Classification using Two Lines

TL;DR

It is found that a region bounded by two lines, so either a halfplane, strip, wedge, or double wedge, containing (most of) the blue points $B$ and few red points is found.

Abstract

Given two sets $R$ and $B$ of $n$ points in the plane, we present efficient algorithms to find a two-line linear classifier that best separates the "red" points in $R$ from the "blue" points in $B$ and is robust to outliers. More precisely, we find a region $\mathcal{W}_B$ bounded by two lines, so either a halfplane, strip, wedge, or double wedge, containing (most of) the blue points $B$, and few red points. Our running times vary between optimal $O(n\log n)$ and around $O(n^3)$, depending on the type of region $\mathcal{W}_B$ and whether we wish to minimize only red outliers, only blue outliers, or both.

Figures (9)

• Figure 1: We consider separating $R$ and $B$ by at most two lines. This gives rise to four types of regions $\mathcal{W}\xspace_B$: halfplanes, strips, wedges, and two types of double wedges: hourglasses and bowties.
• Figure 2: Perfectly separating $R$ and $B$ may require more than one line. When considering outliers, we may allow '(and minimize) only red outliers, only blue outliers, or both.
• Figure 3: (a)/(b): shrinking/extending segment $\overline{{\ell_1^*} \ell_2^*}$ until it reaches a bicolored face. (c): points $p^*$ and $q^*$ in antipodal outer faces $F$ and $F'$. Segment $\overline{p^* \ell_2^*}$ intersects exactly those lines that $\overline{q^* \ell_2^*}$ does not intersect.
• Figure 4: (left) Line $\ell$ misclassifies red points $a$ and $b$. (right) In the dual space that means $\ell$ lies in the forbidden regions of both $a$ and $b$.
• Figure 5: Four types of red lines for strip separation, with restrictions on their parameter space.

Theorems & Definitions (13)

• Lemma 1
• Lemma 2
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Lemma 10
• Lemma 11
• Lemma 12