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Title: Reconstruction algorithms for low-rank tensors and depth-3 multilinear circuits
We give new and efficient black-box reconstruction algorithms for some classes of depth-3 arithmetic circuits. As a consequence, we obtain the first efficient algorithm for computing the tensor rank and for finding the optimal tensor decomposition as a sum of rank-one tensors when then input is a constant-rank tensor. More specifically, we provide efficient learning algorithms that run in randomized polynomial time over general fields and in deterministic polynomial time over and for the following classes: 1) Set-multilinear depth-3 circuits of constant top fan-in ((k) circuits). As a consequence of our algorithm, we obtain the first polynomial time algorithm for tensor rank computation and optimal tensor decomposition of constant-rank tensors. This result holds for d dimensional tensors for any d, but is interesting even for d=3. 2) Sums of powers of constantly many linear forms ((k) circuits). As a consequence we obtain the first polynomial-time algorithm for tensor rank computation and optimal tensor decomposition of constant-rank symmetric tensors. 3) Multilinear depth-3 circuits of constant top fan-in (multilinear (k) circuits). Our algorithm works over all fields of characteristic 0 or large enough characteristic. Prior to our work the only efficient algorithms known were over polynomially-sized finite fields (see. Karnin-Shpilka 09’). Prior more » to our work, the only polynomial-time or even subexponential-time algorithms known (deterministic or randomized) for subclasses of (k) circuits that also work over large/infinite fields were for the setting when the top fan-in k is at most 2 (see Sinha 16’ and Sinha 20’). « less
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Award ID(s):
Publication Date:
Journal Name:
STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
Page Range or eLocation-ID:
809 to 822
Sponsoring Org:
National Science Foundation
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