Tensor reconstruction beyond constant rank

We give reconstruction algorithms for subclasses of depth-3 arithmetic circuits. In particular, we obtain the first efficient algorithm for finding tensor rank, and an optimal tensor decomposition as a sum of rank-one tensors, when given black-box access to a tensor of super-constant rank. Specifically, we obtain the following results: 1. A deterministic algorithm that reconstructs polynomials computed by σ[k]V[d] σ circuits in time poly(n, d, c)·poly(k)^kk10, 2. A randomized algorithm that reconstructs polynomials computed by multilinear σ[k]Q[d] σ circuits in time poly(n, d, c)·k^kkkO(k), 3. A randomized algorithm that reconstructs polynomials computed by set-multilinear σ[k]Q[d] σ circuits in time poly(n, d, c)·k^kkkO(k), where c = log q if F = Fq is a finite field, and c equals the maximum bit complexity of any coefficient of f if F is infinite. Prior to our work, polynomial time algorithms for the case when the rank, k, is constant, were given by Bhargava, Saraf and Volkovich [5]. Another contribution of this work is correcting an error from a paper of Karnin and Shpilka [20] (with some loss in parameters) that also affected Theorem 1.6 of [5]. Consequently, the results of [20, 5] continue to hold, with a slightly worse setting of parameters. For fixing the error we systematically study the relation between syntactic and semantic notions of rank of σΦσ circuits, and the corresponding partitions of such circuits. We obtain our improved running time by introducing a technique for learning rank preserving coordinate-subspaces. Both [20] and [5] tried all choices of finding the "correct" coordinates, which, due to the size of the set, led to having a fast growing function of k at the exponent of n. We manage to find these spaces in time that is still growing fast with k, yet it is only a fixed polynomial in n.