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Practical Leverage-Based Sampling for Low-Rank Tensor Decomposition
The low-rank canonical polyadic tensor decomposition is useful in data analysis and can be computed by solving a sequence of overdetermined least squares subproblems. Motivated by consideration of sparse tensors, we propose sketching each subproblem using leverage scores to select a subset of the rows, with probabilistic guarantees on the solution accuracy. We randomly sample rows proportional to leverage score upper bounds that can be efficiently computed using the special Khatri--Rao subproblem structure inherent in tensor decomposition. Crucially, for a (d+1)-way tensor, the number of rows in the sketched system is O(rd/ϵ) for a decomposition of rank r and ϵ-accuracy in the least squares solve, independent of both the size and the number of nonzeros in the tensor. Along the way, we provide a practical solution to the generic matrix sketching problem of sampling overabundance for high-leverage-score rows, proposing to include such rows deterministically and combine repeated samples in the sketched system; we conjecture that this can lead to improved theoretical bounds. Numerical results on real-world large-scale tensors show the method is significantly faster than deterministic methods at nearly the same level of accuracy.
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