Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery

TL;DR

This paper introduces Low-Rank Tensor Function Representation (LRTFR), a novel method for continuous, high-resolution data representation beyond traditional meshgrid limitations, applicable to various multi-dimensional data recovery tasks.

Contribution

The paper proposes LRTFR, a continuous tensor function model with low-rank factorization, unifying low-rank and smooth regularizations for improved multi-dimensional data representation.

Findings

Outperforms state-of-the-art methods in image inpainting and denoising.

Effective in hyperparameter optimization beyond meshgrid resolution.

Achieves superior point cloud upsampling results.

Abstract

Since higher-order tensors are naturally suitable for representing multi-dimensional data in real-world, e.g., color images and videos, low-rank tensor representation has become one of the emerging areas in machine learning and computer vision. However, classical low-rank tensor representations can only represent data on finite meshgrid due to their intrinsical discrete nature, which hinders their potential applicability in many scenarios beyond meshgrid. To break this barrier, we propose a low-rank tensor function representation (LRTFR), which can continuously represent data beyond meshgrid with infinite resolution. Specifically, the suggested tensor function, which maps an arbitrary coordinate to the corresponding value, can continuously represent data in an infinite real space. Parallel to discrete tensors, we develop two fundamental concepts for tensor functions, i.e., the tensor function rank and low-rank tensor function factorization. We theoretically justify that both low-rank and smooth regularizations are harmoniously unified in the LRTFR, which leads to high effectiveness and efficiency for data continuous representation. Extensive multi-dimensional data recovery applications arising from image processing (image inpainting and denoising), machine learning (hyperparameter optimization), and computer graphics (point cloud upsampling) substantiate the superiority and versatility of our method as compared with state-of-the-art methods. Especially, the experiments beyond the original meshgrid resolution (hyperparameter optimization) or even beyond meshgrid (point cloud upsampling) validate the favorable performances of our method for continuous representation.