Low-Rank Tensor Recovery via Variational Schatten-p Quasi-Norm and Jacobian Regularization

TL;DR

This work addresses low-rank tensor recovery by uniting CP-based tensor decompositions with implicit neural representations. It introduces a CP-pruned, function-parametric tensor model that uses a variational Schatten-$p$ quasi-norm to automatically prune redundant rank-1 components and a Jacobian-based, Hutchinson-estimated smoothness regularizer to promote spatial continuity without costly SVDs. Theoretical results establish an excess risk bound for the CP-based tensor function and show the variational norm upper-bounds unfoldings, enabling reliable, beyond-grid recovery. Empirically, CP-Pruner demonstrates superior performance on image inpainting, multispectral denoising, and point-cloud upsampling, with robust behavior across $p$, $R$, and smoothing parameters, and runs in a practical, SVD-free regime.

Abstract

Higher-order tensors are well-suited for representing multi-dimensional data, such as images and videos, which typically characterize low-rank structures. Low-rank tensor decomposition has become essential in machine learning and computer vision, but existing methods like Tucker decomposition offer flexibility at the expense of interpretability. The CANDECOMP/PARAFAC (CP) decomposition provides a natural and interpretable structure, while obtaining a sparse solutions remains challenging. Leveraging the rich properties of CP decomposition, we propose a CP-based low-rank tensor function parameterized by neural networks (NN) for implicit neural representation. This approach can model the tensor both on-grid and beyond grid, fully utilizing the non-linearity of NN with theoretical guarantees on excess risk bounds. To achieve sparser CP decomposition, we introduce a variational Schatten-p quasi-norm to prune redundant rank-1 components and prove that it serves as a common upper bound for the Schatten-p quasi-norms of arbitrary unfolding matrices. For smoothness, we propose a regularization term based on the spectral norm of the Jacobian and Hutchinson's trace estimator. The proposed smoothness regularization is SVD-free and avoids explicit chain rule derivations. It can serve as an alternative to Total Variation (TV) regularization in image denoising tasks and is naturally applicable to implicit neural representation. Extensive experiments on multi-dimensional data recovery tasks, including image inpainting, denoising, and point cloud upsampling, demonstrate the superiority and versatility of our method compared to state-of-the-art approaches. The code is available at https://github.com/CZY-Code/CP-Pruner.

Figures (8)

• Figure 1: The proposed CP-Pruner represents tensor data on or beyond meshgrids. The low-rank regularization automatically prune redundant components for a sparser CP representation. Smoothness regularization is meshgrid-independent and avoids extra TV loss for denoising task.
• Figure 2: The overview of the proposed CP-Pruner. For simplicity, we focus on the three-dimensional case, although our approach can be readily generalized to higher dimensions. (a) For holistic modeling, the variational Schatten-p quasi-norm automatically prune redundant rank-1 components for low-rankness, yielding a sparser CP decomposition. For detailed modeling, (b) implicit neural networks map spatial coordinates to sub-vectors, where Einstein summation is used to compute the tensor entry at each location; (c) The regularization based on the spectral norm of the Jacobian matrix ensures spatial smoothness.
• Figure 3: Results of multi-dimensional image inpainting by different methods on color images Sailboat, multispectral image Flowers and video Carphone with SR=0.1.
• Figure 4: Results of multi-dimensional image denoising by different methods on HSIs WDC mall (Case 1), PaviaU (Case2), Beads (Case 3), Balloons (Case4) and Fruits (Case5).
• Figure 5: Results of point cloud upsampling by different methods on Table, Airplane, Chair, and Lamp in the ShapeNet datasetchang2015shapenet. The number of observed points is 20% of the original points, with fewer than 500 points in total.

Theorems & Definitions (5)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Theorem 2