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1. Reconstruction algorithms for low-rank tensors and depth-3 multilinear circuits

   https://doi.org/10.1145/3406325.3451096  

   Bhargava, Vishwas; Saraf, Shubhangi; Volkovich, Ilya (June 2021, STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing)

   null (Ed.)

   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 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’). 

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2. Nonconvex Low-Rank Symmetric Tensor Completion from Noisy Data

   Cai, Changxiao; Li, Gen; Poor, H. Vincent; Chen, Yuxin (December 2019, Advances in neural information processing systems)

   We study a completion problem of broad practical interest: the reconstruction of a low-rank symmetric tensor from highly incomplete and randomly corrupted observations of its entries. While a variety of prior work has been dedicated to this problem, prior algorithms either are computationally too expensive for large-scale applications, or come with sub-optimal statistical guarantees. Focusing on incoherent'' and well-conditioned tensors of a constant CP rank, we propose a two-stage nonconvex algorithm --- (vanilla) gradient descent following a rough initialization --- that achieves the best of both worlds. Specifically, the proposed nonconvex algorithm faithfully completes the tensor and retrieves all low-rank tensor factors within nearly linear time, while at the same time enjoying near-optimal statistical guarantees (i.e. minimal sample complexity and optimal statistical accuracy). The insights conveyed through our analysis of nonconvex optimization might have implications for other tensor estimation problems. 

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3. Nonconvex Low-Rank Symmetric Tensor Completion from Noisy Data

   Cai, Changxiao; Li, Gen; Poor, H. Vincent; Chen, Yuxin (December 2019, Advances in neural information processing systems)

   We study a completion problem of broad practical interest: the reconstruction of a low-rank symmetric tensor from highly incomplete and randomly corrupted observations of its entries. While a variety of prior work has been dedicated to this problem, prior algorithms either are computationally too expensive for large-scale applications, or come with sub-optimal statistical guarantees. Focusing on incoherent'' and well-conditioned tensors of a constant CP rank, we propose a two-stage nonconvex algorithm --- (vanilla) gradient descent following a rough initialization --- that achieves the best of both worlds. Specifically, the proposed nonconvex algorithm faithfully completes the tensor and retrieves all low-rank tensor factors within nearly linear time, while at the same time enjoying near-optimal statistical guarantees (i.e. minimal sample complexity and optimal statistical accuracy). The insights conveyed through our analysis of nonconvex optimization might have implications for other tensor estimation problems. 

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4. Beyond the Signs: Nonparametric tensor completion via sign series.

   Lee, Chanwoo; Wang, Miaoyan (December 2021, 2021 Advances in Neural Information Processing Systems 34 (NeurIPS))

   We consider the problem of tensor estimation from noisy observations with possibly missing entries. A nonparametric approach to tensor completion is developed based on a new model which we coin as sign representable tensors. The model represents the signal tensor of interest using a series of structured sign tensors. Unlike earlier methods, the sign series representation effectively addresses both low- and high-rank signals, while encompassing many existing tensor models— including CP models, Tucker models, single index models, structured tensors with repeating entries—as special cases. We provably reduce the tensor estimation problem to a series of structured classification tasks, and we develop a learning reduction machinery to empower existing low-rank tensor algorithms for more challenging high-rank estimation. Excess risk bounds, estimation errors, and sample complexities are established. We demonstrate the outperformance of our approach over previous methods on two datasets, one on human brain connectivity networks and the other on topic data mining. 

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5. Nonconvex Low-Rank Tensor Completion from Noisy Data

   https://doi.org/10.1287/opre.2021.2106  

   Cai, Changxiao; Li, Gen; Poor, H. Vincent; Chen, Yuxin (March 2022, Operations Research)

   We study a noisy tensor completion problem of broad practical interest, namely, the reconstruction of a low-rank tensor from highly incomplete and randomly corrupted observations of its entries. Whereas a variety of prior work has been dedicated to this problem, prior algorithms either are computationally too expensive for large-scale applications or come with suboptimal statistical guarantees. Focusing on “incoherent” and well-conditioned tensors of a constant canonical polyadic rank, we propose a two-stage nonconvex algorithm—(vanilla) gradient descent following a rough initialization—that achieves the best of both worlds. Specifically, the proposed nonconvex algorithm faithfully completes the tensor and retrieves all individual tensor factors within nearly linear time, while at the same time enjoying near-optimal statistical guarantees (i.e., minimal sample complexity and optimal estimation accuracy). The estimation errors are evenly spread out across all entries, thus achieving optimal [Formula: see text] statistical accuracy. We also discuss how to extend our approach to accommodate asymmetric tensors. The insight conveyed through our analysis of nonconvex optimization might have implications for other tensor estimation problems. 

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