High-Dimensional Change-Point Estimation: Combining Filtering with Convex Optimization

TL;DR

This paper introduces a new high-dimensional change-point estimation method that combines filtering with convex optimization, effectively exploiting low-dimensional structures in noisy, high-dimensional data for reliable detection.

Contribution

It proposes a novel algorithm integrating filtered derivative and atomic norm regularization, suitable for online high-dimensional change-point detection with proven statistical guarantees.

Findings

Method reliably detects change-points when the product of change magnitude and spacing exceeds a Gaussian width threshold.

Algorithm is computationally scalable and effective in online high-dimensional settings.

Theoretical analysis provides conditions for successful change-point estimation in structured signals.

Abstract

We consider change-point estimation in a sequence of high-dimensional signals given noisy observations. Classical approaches to this problem such as the filtered derivative method are useful for sequences of scalar-valued signals, but they have undesirable scaling behavior in the high-dimensional setting. However, many high-dimensional signals encountered in practice frequently possess latent low-dimensional structure. Motivated by this observation, we propose a technique for high-dimensional change-point estimation that combines the filtered derivative approach from previous work with convex optimization methods based on atomic norm regularization, which are useful for exploiting structure in high-dimensional data. Our algorithm is applicable in online settings as it operates on small portions of the sequence of observations at a time, and it is well-suited to the high-dimensional setting both in terms of computational scalability and of statistical efficiency. The main result of this paper shows that our method performs change-point estimation reliably as long as the product of the smallest-sized change (the Euclidean-norm-squared of the difference between signals at a change-point) and the smallest distance between change-points (number of time instances) is larger than a Gaussian width parameter that characterizes the low-dimensional complexity of the underlying signal sequence.