Adaptive Sketching for Fast and Convergent Canonical Polyadic Decomposition

This work considers the canonical polyadic decomposition (CPD) of tensors using proximally regularized sketched alternating least squares algorithms. First, it establishes a sublinear rate of convergence for proximally regularized sketched CPD algorithms under two natural conditions that are known to be satisfied by many popular forms of sketching. Second, it demonstrates that the iterative nature of CPD algorithms can be exploited algorithmically to choose more performant sketching rates. This is accomplished by introducing CPD-MWU, a proximally-regularized sketched alternating least squares algorithm that adaptively selects the sketching rate at each iteration. On both synthetic and real data we observe that for noisy tensors CPD-MWU produces decompositions of comparable accuracy to the standard CPD decomposition in less time, often half the time; for ill-conditioned tensors, given the same time budget, CPD-MWU produces decompositions with an order-of-magnitude lower relative error. For a representative real-world dataset CPD-MWU produces residual errors on average 20% lower than CPRAND-MIX and 44% lower than SPALS, two recent sketched CPD algorithms.