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arXiv 2002.112552020-02math.STcs.LGstat.MEstat.MLstat.TH

An Optimal Statistical and Computational Framework for Generalized Tensor Estimation

University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of Chicago（芝加哥大学）

Rungang Han, Rebecca Willett, Anru R. Zhang

英文摘要：

This paper describes a flexible framework for generalized low-rank tensor estimation problems that includes many important instances arising from applications in computational imaging, genomics, and network analysis. The proposed estimator consists of finding a low-rank tensor fit to the data under generalized parametric models. To overcome the difficulty of non-convexity in these problems, we introduce a unified approach of projected gradient descent that adapts to the underlying low-rank structure. Under mild conditions on the loss function, we establish both an upper bound on statistical error and the linear rate of computational convergence through a general deterministic analysis. Then we further consider a suite of generalized tensor estimation problems, including sub-Gaussian tensor PCA, tensor regression, and Poisson and binomial tensor PCA. We prove that the proposed algorithm achieves the minimax optimal rate of convergence in estimation error. Finally, we demonstrate the superiority of the proposed framework via extensive experiments on both simulated and real data.

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