Minimax rates in sparse, high-dimensional changepoint detection
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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 √(p (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. We also identify the leading constants in the rate to within a factor of 2 in both sparse and dense asymptotic regimes. Extensions to cases of spatial and temporal dependence are provided.
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