Sharp phase transitions in high-dimensional changepoint detection

We study a hypothesis testing problem in the context of high-dimensional changepoint detection. Given a matrix $X \in \R^{p \times n}$ with independent Gaussian entries, the goal is to determine whether or not a sparse, non-null fraction of rows in $X$ exhibits a shift in mean at a common index between $1$ and $n$. We focus on three aspects of this problem: the sparsity of non-null rows, the presence of a single, common changepoint in the non-null rows, and the signal strength associated with the changepoint. Within an asymptotic regime relating the data dimensions $n$ and $p$ to the signal sparsity and strength, the information-theoretic limits of this testing problem are characterized by a formula that determines whether or not there exists a testing procedure whose sum of Type I and II errors tends to zero as $n,p \to \infty$. The formula, called the \emph{detection boundary}, partitions the parameter space into a two regions: one where it is possible to detect the presence of a single aligned changepoint (detectable region), and another where no test is able to consistently distinguish the mean matrix from one with constant rows (undetectable region).