  This paper studies the computational and statistical aspects of quantile and
pseudo-Huber tensor decomposition. The integrated investigation of
computational and statistical issues of robust tensor decomposition poses
challenges due to the non-smooth loss functions. We propose a projected
sub-gradient descent algorithm for tensor decomposition, equipped with either
the pseudo-Huber loss or the quantile loss. In the presence of both
heavy-tailed noise and Huber's contamination error, we demonstrate that our
algorithm exhibits a so-called phenomenon of two-phase convergence with a
carefully chosen step size schedule. The algorithm converges linearly and
delivers an estimator that is statistically optimal with respect to both the
heavy-tailed noise and arbitrary corruptions. Interestingly, our results
achieve the first minimax optimal rates under Huber's contamination model for
noisy tensor decomposition. Compared with existing literature, quantile tensor
decomposition removes the requirement of specifying a sparsity level in
advance, making it more flexible for practical use. We also demonstrate the
effectiveness of our algorithms in the presence of missing values. Our methods
are subsequently applied to the food balance dataset and the international
trade flow dataset, both of which yield intriguing findings.
