DeepAI AI Chat
Log In Sign Up

Nonconvex Low-Rank Symmetric Tensor Completion from Noisy Data

11/11/2019
by   Changxiao Cai, et al.
0

We study a noisy symmetric tensor completion problem of broad practical interest, namely, the reconstruction of a low-rank symmetric tensor from highly incomplete and randomly corrupted observations of its entries. While a variety of prior work has been dedicated to this problem, prior algorithms either are computationally too expensive for large-scale applications, or come with sub-optimal statistical guarantees. Focusing on "incoherent" and well-conditioned tensors of a constant CP rank, we propose a two-stage nonconvex algorithm — (vanilla) gradient descent following a rough initialization — that achieves the best of both worlds. Specifically, the proposed nonconvex algorithm faithfully completes the tensor and retrieves all individual tensor factors within nearly linear time, while at the same time enjoying near-optimal statistical guarantees (i.e. minimal sample complexity and optimal estimation accuracy). The estimation errors are evenly spread out across all entries, thus achieving optimal ℓ_∞ statistical accuracy. The insight conveyed through our analysis of nonconvex optimization might have implications for other tensor estimation problems.

READ FULL TEXT

page 1

page 2

page 3

page 4

04/29/2021

Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurements

Tensors, which provide a powerful and flexible model for representing mu...
11/14/2017

Near-optimal sample complexity for convex tensor completion

We analyze low rank tensor completion (TC) using noisy measurements of a...
11/14/2017

Statistically Optimal and Computationally Efficient Low Rank Tensor Completion from Noisy Entries

In this article, we develop methods for estimating a low rank tensor fro...
05/06/2022

Low-rank Tensor Learning with Nonconvex Overlapped Nuclear Norm Regularization

Nonconvex regularization has been popularly used in low-rank matrix lear...
02/20/2019

Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimization

This paper studies noisy low-rank matrix completion: given partial and c...
07/01/2020

Tensor Estimation with Nearly Linear Samples

There is a conjectured computational-statistical gap in terms of the num...