A review on minimax rates in change point detection and localisation

Localising change points in piecewise polynomials of general degrees

In this paper we are concerned with a sequence of univariate random variables with piecewise polynomial means and independent sub-Gaussian noise. The underlying polynomials are allowed to be of arbitrary but fixed degrees. All the other model parameters are allowed to vary depending on the sample size. We propose a two-step estimation procedure based on the $\ell_0$-penalisation and provide upper bounds on the localisation error. We complement these results by deriving a global information-theoretic lower bounds, which show that our two-step estimators are nearly minimax rate-optimal. We also …

Optimal partition recovery in general graphs

We consider a graph-structured change point problem in which we observe a random vector with piecewise constant but unknown mean and whose independent, sub-Gaussian coordinates correspond to the $n$ nodes of a fixed graph. We are interested in the localisation task of recovering the partition of the nodes associated to the constancy regions of the mean vector. When the partition $\mathcal{S}$ consists of only two elements, we characterise the difficulty of the localisation problem in terms of four key parameters: the maximal noise variance $σ^2$, the size $Δ$ of the smaller element of the part…

Minimax Rates in Network Analysis: Graphon Estimation, Community Detection and Hypothesis Testing

This paper surveys some recent developments in fundamental limits and optimal algorithms for network analysis. We focus on minimax optimal rates in three fundamental problems of network analysis: graphon estimation, community detection, and hypothesis testing. For each problem, we review state-of-the-art results in the literature followed by general principles behind the optimal procedures that lead to minimax estimation and testing. This allows us to connect problems in network analysis to other statistical inference problems from a general perspective.

Adversarially robust change point detection

Change point detection is becoming increasingly popular in many application areas. On one hand, most of the theoretically-justified methods are investigated in an ideal setting without model violations, or merely robust against identical heavy-tailed noise distribution across time and/or against isolate outliers; on the other hand, we are aware that there have been exponentially growing attacks from adversaries, who may pose systematic contamination on data to purposely create spurious change points or disguise true change points. In light of the timely need for a change point detection method…

Optimal network online change point localisation

We study the problem of online network change point detection. In this setting, a collection of independent Bernoulli networks is collected sequentially, and the underlying distributions change when a change point occurs. The goal is to detect the change point as quickly as possible, if it exists, subject to a constraint on the number or probability of false alarms. In this paper, on the detection delay, we establish a minimax lower bound and two upper bounds based on NP-hard algorithms and polynomial-time algorithms, i.e., \[ \mbox{detection delay} \begin{cases} \gtrsim \log(1/α) \frac{\max\{…

基于不变性的随机过程移动和分段

Moving sum data segmentation for stochastics processes based on invariance

本文基于强不变性原理，利用移动和统计量对一类状态切换多元过程进行数据分割，证明了变点估计量的一致性并推导了极小极大最优定位速率，还给出了局部变化下的极限分布。