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 consideration sparse tensors, we propose sketching each subproblem using leverage scores to select subset the rows, with probabilistic guarantees on solution accuracy. We randomly sample rows proportional score upper bounds that efficiently special Khatri--Rao structure inherent decomposition. Crucially, for $(d+1)$-way tensor, number sketched system $O(r^d/\epsilon)$ rank $r$ $\epsilon$-accuracy solve, independent both size nonzeros tensor. Along way, provide practical generic matrix problem sampling overabundance high-leverage-score proposing include such deterministically combine repeated samples system; conjecture this lead improved theoretical bounds. Numerical results real-world large-scale tensors show method significantly faster than deterministic methods at nearly same level