# Minimax rates in sparse, high-dimensional changepoint detection

## TL;DR

This paper derives the exact minimax detection rates for sparse high-dimensional mean change detection, revealing phase transitions and dependence on sample size, with extensions to dependent data.

## Contribution

It provides the first precise minimax rates for sparse changepoint detection in high dimensions, including phase transitions and constants in different regimes.

## Key findings

- Identifies phase transition at sparsity level ~√(p log log n).
- Derives exact minimax detection rates across regimes.
- Provides extensions to dependent data scenarios.

## Abstract

We study the detection of a sparse change in a high-dimensional mean vector as a minimax testing problem. Our first main contribution is to derive the exact minimax testing rate across all parameter regimes for $n$ independent, $p$-variate Gaussian observations. This rate exhibits a phase transition when the sparsity level is of order $\sqrt{p \log \log (8n)}$ and has a very delicate dependence on the sample size: in a certain sparsity regime it involves a triple iterated logarithmic factor in~$n$. Further, in a dense asymptotic regime, we identify the sharp leading constant, while in the corresponding sparse asymptotic regime, this constant is determined to within a factor of $\sqrt{2}$. Extensions that cover spatial and temporal dependence, primarily in the dense case, are also provided.