A Robust Spectral Algorithm for Overcomplete Tensor Decomposition
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We give a spectral algorithm for decomposing overcomplete order-4 tensors, so long as their components satisfy an algebraic non-degeneracy condition that holds for nearly all (all but an algebraic set of measure 0) tensors over (ℝ^d)^⊗ 4 with rank n ≤ d^2. Our algorithm is robust to adversarial perturbations of bounded spectral norm. Our algorithm is inspired by one which uses the sum-of-squares semidefinite programming hierarchy (Ma, Shi, and Steurer STOC'16, arXiv:1610.01980), and we achieve comparable robustness and overcompleteness guarantees under similar algebraic assumptions. However, our algorithm avoids semidefinite programming and may be implemented as a series of basic linear-algebraic operations. We consequently obtain a much faster running time than semidefinite programming methods: our algorithm runs in time Õ(n^2d^3) ≤Õ(d^7), which is subquadratic in the input size d^4 (where we have suppressed factors related to the condition number of the input tensor).
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