# Fr'echet Change Point Detection

arXiv: https://arxiv.org/abs/1911.11864 · HTML: https://mlsanity.com/paper/1911.11864/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Nov 2019.

## Built on by (most cited first, 5 of 5)

- [Optimal Transport Barycenter via Nonconvex-Concave Minimax Optimization](https://mlsanity.com/paper/2501.14635/) (2025)
- [Estimation of the Sample Frechet Mean: A Convolutional Neural Network Approach](https://mlsanity.com/paper/2210.07401/) (2022)
- [Detecting Structural Shifts in Multivariate Hawkes Processes with Fr'echet Statistics](https://mlsanity.com/paper/2308.06769/) (2023)
- [RIO-CPD: A Riemannian Geometric Method for Correlation-aware Online Change Point Detection](https://mlsanity.com/paper/2407.09698/) (2024)
- [Riemannian Change Point Detection on Manifolds with Robust Centroid Estimation](https://mlsanity.com/paper/2508.18045/) (2025)

## Abstract (arXiv)

We propose a method to infer the presence and location of change-points in the distribution of a sequence of independent data taking values in a general metric space, where change-points are viewed as locations at which the distribution of the data sequence changes abruptly in terms of either its Fr'echet mean or Fr'echet variance or both. The proposed method is based on comparisons of Fr'echet variances before and after putative change-point locations and does not require a tuning parameter except for the specification of cut-off intervals near the endpoints where change-points are assumed not to occur. Our results include theoretical guarantees for consistency of the test under contiguous alternatives when a change-point exists and also for consistency of the estimated location of the change-point if it exists, where under the null hypothesis of no change-point the limit distribution of the proposed scan function is the square of a standardized Brownian Bridge. These consistency results are applicable for a broad class of metric spaces under mild entropy conditions. Examples include the space of univariate probability distributions and the space of graph Laplacians for networks. Simulation studies demonstrate the effectiveness of the proposed methods, both for inferring the presence of a change-point and estimating its location. We also develop theory that justifies bootstrap-based inference and illustrate the new approach with sequences of maternal fertility distributions and communication networks.