AN OPTIMAL STATISTICAL AND COMPUTATIONAL FRAMEWORK FOR GENERALIZED TENSOR ESTIMATION

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 nonconvexity 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.

Automated summary

This paper presents a flexible framework for generalized low-rank tensor estimation problems. By using the projected gradient descent method, the framework can adapt to the underlying low-rank structure in nonconvex problems, while providing statistical error bounds and linear convergence rates. The paper also considers a range of generalized tensor estimation problems and proves that the proposed algorithm achieves the minimax optimal convergence rate in estimation errors. Extensive experiments on simulated and real data demonstrate the superiority of the proposed framework.