26 research outputs found
Quantile and pseudo-Huber Tensor Decomposition
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
Exact Clustering in Tensor Block Model: Statistical Optimality and Computational Limit
High-order clustering aims to identify heterogeneous substructures in
multiway datasets that arise commonly in neuroimaging, genomics, social network
studies, etc. The non-convex and discontinuous nature of this problem pose
significant challenges in both statistics and computation. In this paper, we
propose a tensor block model and the computationally efficient methods,
\emph{high-order Lloyd algorithm} (HLloyd), and \emph{high-order spectral
clustering} (HSC), for high-order clustering. The convergence guarantees and
statistical optimality are established for the proposed procedure under a mild
sub-Gaussian noise assumption. Under the Gaussian tensor block model, we
completely characterize the statistical-computational trade-off for achieving
high-order exact clustering based on three different signal-to-noise ratio
regimes. The analysis relies on new techniques of high-order spectral
perturbation analysis and a "singular-value-gap-free" error bound in tensor
estimation, which are substantially different from the matrix spectral analyses
in the literature. Finally, we show the merits of the proposed procedures via
extensive experiments on both synthetic and real datasets.Comment: 65 page