Learning the intensity of time events with change-points

TL;DR

This paper develops a new convex segmentation method with data-driven weights for learning the intensity of inhomogeneous counting processes, providing theoretical guarantees and efficient algorithms.

Contribution

It introduces a weighted total-variation penalization with data-driven weights, offering the first theoretical guarantees for segmentation with convex proxies beyond i.i.d. signals.

Findings

Proves oracle inequalities with fast convergence rates.

Demonstrates consistency in change-point detection.

Shows effectiveness on simulated and genomics data.

Abstract

We consider the problem of learning the inhomogeneous intensity of a counting process, under a sparse segmentation assumption. We introduce a weighted total-variation penalization, using data-driven weights that correctly scale the penalization along the observation interval. We prove that this leads to a sharp tuning of the convex relaxation of the segmentation prior, by stating oracle inequalities with fast rates of convergence, and consistency for change-points detection. This provides first theoretical guarantees for segmentation with a convex proxy beyond the standard i.i.d signal + white noise setting. We introduce a fast algorithm to solve this convex problem. Numerical experiments illustrate our approach on simulated and on a high-frequency genomics dataset.