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In this paper, we develop a novel procedure for low-rank tensor regression,namely \emph{\underline{I}mportance \underline{S}ketching \underline{L}ow-rank\underline{E}stimation for \underline{T}ensors} (ISLET). The central ideabehind ISLET is \emph{importance sketching}, i.e., carefully designed sketchesbased on both the responses and low-dimensional structure of the parameter ofinterest. We show that the proposed method is sharply minimax optimal in termsof the mean-squared error under low-rank Tucker assumptions and underrandomized Gaussian ensemble design. In addition, if a tensor is low-rank withgroup sparsity, our procedure also achieves minimax optimality. Further, weshow through numerical study that ISLET achieves comparable or bettermean-squared error performance to existing state-of-the-art methods whilehaving substantial storage and run-time advantages including capabilities forparallel and distributed computing. In particular, our procedure performsreliable estimation with tensors of dimension $p = O(10^8)$ and is $1$ or $2$orders of magnitude faster than baseline methods.

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