  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.
