Consistent change-point detection with kernels

TL;DR

This paper analyzes the kernel change-point algorithm (KCP), demonstrating its ability to accurately detect multiple distribution change-points in complex, structured data using a penalized kernel criterion with high probability.

Contribution

It provides a non-asymptotic theoretical guarantee for KCP's accuracy in estimating the number and locations of change-points, applicable to complex data structures.

Findings

KCP retrieves the correct number of change-points with high probability.

KCP estimates change-point locations at the optimal rate.

Detects all types of distribution changes using characteristic kernels.

Abstract

In this paper we study the kernel change-point algorithm (KCP) proposed by Arlot, Celisse and Harchaoui (2012), which aims at locating an unknown number of change-points in the distribution of a sequence of independent data taking values in an arbitrary set. The change-points are selected by model selection with a penalized kernel empirical criterion. We provide a non-asymptotic result showing that, with high probability, the KCP procedure retrieves the correct number of change-points, provided that the constant in the penalty is well-chosen; in addition, KCP estimates the change-points location at the optimal rate. As a consequence, when using a characteristic kernel, KCP detects all kinds of change in the distribution (not only changes in the mean or the variance), and it is able to do so for complex structured data (not necessarily in $\mathbb{R}^d$). Most of the analysis is conducted assuming that the kernel is bounded; part of the results can be extended when we only assume a finite second-order moment.