Simultaneous Tensor Completion and Denoising by Noise Inequality Constrained Convex Optimization

TL;DR

This paper introduces a novel convex optimization approach for tensor completion and denoising that uses noise inequality constraints, improving robustness and ease of parameter tuning in noisy data scenarios.

Contribution

It proposes new tensor completion models with noise inequality constraints, derives efficient proximal mappings, and develops an accelerated primal-dual optimization algorithm.

Findings

Effective tensor denoising demonstrated on visual data

Outperforms existing methods in tensor completion tasks

Robust to different noise distributions like Gaussian and Laplace

Abstract

Tensor completion is a technique of filling missing elements of the incomplete data tensors. It being actively studied based on the convex optimization scheme such as nuclear-norm minimization. When given data tensors include some noises, the nuclear-norm minimization problem is usually converted to the nuclear-norm `regularization' problem which simultaneously minimize penalty and error terms with some trade-off parameter. However, the good value of trade-off is not easily determined because of the difference of two units and the data dependence. In the sense of trade-off tuning, the noisy tensor completion problem with the `noise inequality constraint' is better choice than the `regularization' because the good noise threshold can be easily bounded with noise standard deviation. In this study, we tackle to solve the convex tensor completion problems with two types of noise inequality constraints: Gaussian and Laplace distributions. The contributions of this study are follows: (1) New tensor completion and denoising models using tensor total variation and nuclear-norm are proposed which can be characterized as a generalization/extension of many past matrix and tensor completion models, (2) proximal mappings for noise inequalities are derived which are analytically computable with low computational complexity, (3) convex optimization algorithm is proposed based on primal-dual splitting framework, (4) new step-size adaptation method is proposed to accelerate the optimization, and (5) extensive experiments demonstrated the advantages of the proposed method for visual data retrieval such as for color images, movies, and 3D-volumetric data.