ISLET: Fast and Optimal Low-rank Tensor Regression via Importance Sketching

In this paper, we develop a novel procedure for low-rank tensor regression, namely emphunderlineImportance underlineSketching underlineLow-rank underlineEstimation for underlineTensors (ISLET). The central idea behind ISLET is importance sketching, i.e., carefully designed sketches based on both the responses and low-dimensional structure of the parameter of interest. We show that the proposed method is sharply minimax optimal in terms of the mean-squared error under low-rank Tucker assumptions and under randomized Gaussian ensemble design. In addition, if a tensor is low-rank with group sparsity, our procedure also achieves minimax optimality. Further, we show through numerical study that ISLET achieves comparable or better mean-squared error performance to existing state-of-the-art methods while having substantial storage and run-time advantages including capabilities for parallel and distributed computing. In particular, our procedure performs reliable estimation with tensors of dimension p = O(108) and is 1 or 2 orders of magnitude faster than baseline methods.