Two ridge ratio criteria for multiple change point detection in tensors

This paper proposes two novel criteria for detecting change structures in tensor data. To measure the difference between any two adjacent tensors and to handle both dense and sparse model structures of the tensors, we define a signal-screening averaged Frobenius distance for the moving sums of tensor data and a signal-screening mode-based Frobenius distance for the moving sums of slices of tensor data. The latter is particularly useful when some mode is not suitable to be included in the Frobenius distance. Based on these two sequences, we construct two signal statistics using the ratios with adaptive-to-change ridge functions respectively, to enhance the detection capacity of the criteria. The estimated number of changes and their estimated locations are consistent to the corresponding true quantities in certain senses. The results hold when the size of the tensor and the number of change points diverge at certain rates, respectively. Numerical studies are conducted to examine the finite sample performances of the proposed methods. We also analyze two real data examples for illustration.