Minimax rates in sparse, high-dimensional change point detection

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