Seminar, Miaoyan Wang, Beyond Matrices: Nonparametric Tensor Estimation and Application
Speaker: Miaoyan Wang, Associate Professor, University of Wisconsin, Madison
Title: Beyond Matrices: Nonparametric Tensor Estimation and Application
Abstract: High-order tensor datasets present a ubiquitous challenge in applications of recommendation systems, neuroimaging, and social networks. Here we introduce provably-guaranteed methods for estimating a likely high-rank signal tensor from such noisy observations. We consider a generative latent variable tensor model that incorporates both high rank and low rank models, including simple hypergraphon models, and single index models. Our analysis establishes both statistical and computational limitations for signal tensor estimation. We prove that latent variable tensors of dimensional-$d$ are well approximated by decomposable tensors of rank log-$d$. Furthermore, we propose a polynomial-time spectral algorithm that provably achieves the optimal estimation rate. Notably, we show that a statistical-computational gap only emerges for latent variable tensors of order 3 or higher. The efficacy of our approach is further showcased through numerical experiments and real-world applications, demonstrating both theoretical soundness and practical merit.