Generalized Low-rank plus Sparse Tensor Estimation by Fast Riemannian Optimization

TL;DR

This paper introduces a flexible and efficient framework for estimating a tensor composed of low-rank and sparse components, applicable to various models and data types, with proven convergence and error bounds.

Contribution

It proposes a novel Riemannian gradient descent algorithm with gradient pruning for fast tensor estimation, handling both linear and non-linear models with theoretical guarantees.

Findings

The algorithm converges linearly under suitable conditions.

Statistical error bounds are established and shown to be sharp in specific models.

Applications to real datasets reveal new insights in trade flow and co-authorship networks.

Abstract

We investigate a generalized framework to estimate a latent low-rank plus sparse tensor, where the low-rank tensor often captures the multi-way principal components and the sparse tensor accounts for potential model mis-specifications or heterogeneous signals that are unexplainable by the low-rank part. The framework is flexible covering both linear and non-linear models, and can easily handle continuous or categorical variables. We propose a fast algorithm by integrating the Riemannian gradient descent and a novel gradient pruning procedure. Under suitable conditions, the algorithm converges linearly and can simultaneously estimate both the low-rank and sparse tensors. The statistical error bounds of final estimates are established in terms of the gradient of loss function. The error bounds are generally sharp under specific statistical models, e.g., the robust tensor PCA and the community detection in hypergraph networks with outlier vertices. Moreover, our method achieves non-trivial error bounds for heavy-tailed tensor PCA whenever the noise has a finite $2+\varepsilon$ moment. We apply our method to analyze the international trade flow dataset and the statistician hypergraph co-authorship network, both yielding new and interesting findings.