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.

翻译：高阶张量非常适合表示多维数据（如图像和视频），这类数据通常具有低秩结构。低秩张量分解已成为机器学习和计算机视觉领域的关键技术，但现有方法（如Tucker分解）在提供灵活性的同时牺牲了可解释性。CANDECOMP/PARAFAC（CP）分解提供了一种自然且可解释的结构，然而获得稀疏解仍具挑战性。利用CP分解的丰富特性，我们提出一种基于CP分解的低秩张量函数，该函数通过神经网络（NN）参数化以实现隐式神经表示。该方法能够对网格内及超网格张量进行建模，充分利用神经网络的非线性特性，并在超额风险界上提供理论保证。为实现更稀疏的CP分解，我们引入变分Schatten-p拟范数以剪枝冗余的秩-1分量，并证明该范数是任意展开矩阵Schatten-p拟范数的共同上界。为提升平滑性，我们提出基于雅可比矩阵谱范数与Hutchinson迹估计器的正则项。所提出的平滑正则方法无需奇异值分解（SVD），避免了显式的链式求导过程。该正则项可作为图像去噪任务中全变分（TV）正则的替代方案，并天然适用于隐式神经表示。在多维数据恢复任务（包括图像修复、去噪和点云上采样）上的大量实验表明，相较于现有先进方法，本方法具有优越性与普适性。代码公开于https://github.com/CZY-Code/CP-Pruner。