Abstract
Higher-order tensors are well-suited for representing multi-dimensional data, such as color images and videos. Low-rank tensor representation has become essential in machine learning and computer vision, but existing methods like Tucker decomposition offer flexibility at the expense of interpretability. In contrast, while the CANDECOMP/PARAFAC (CP) decomposition provides a more natural and interpretable tensor structure, obtaining sparse solutions remains challenging. Leveraging the rich properties of CP decomposition, we propose a CP-based low-rank tensor function parameterized by neural networks for implicit neural representation (CP-INR). This approach enables continuous data representation beyond structured grids, fully exploiting the non-linearity of tensor data with theoretical guarantees on excess risk bounds. To achieve a sparse CP decomposition, we introduce a variational form of the Schatten-p quasi-norm and prove its relationship to multilinear rank minimization. For smoothness, we propose a regularization term based on the spectral norm of the Jacobian and Hutchinson's trace estimator. Our 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 continuous data. 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.
Abstract (translated)
高阶张量非常适合表示多维数据，如彩色图像和视频。低秩张量表示在机器学习和计算机视觉中变得至关重要，但现有的方法如Tucker分解以牺牲可解释性为代价提供了灵活性。相比之下，尽管CANDECOMP/PARAFAC（CP）分解提供了一种更加自然且易于解释的张量结构，但在获得稀疏解方面仍存在挑战。利用CP分解丰富的特性，我们提出了一种基于CP分解并由神经网络参数化的低秩张量函数，用于隐式神经表示（CP-INR）。这种方法使连续数据表示超越了有结构的网格，并充分利用张量数据的非线性，同时在理论上保证了超额风险界限。为了实现稀疏CP分解，我们引入了Schatten-p准范数的一种变分形式，并证明它与多线性秩最小化之间的关系。对于平滑度，我们提出了一种基于雅可比矩阵谱范数和Hutchinson迹估计器的正则化项。我们的提出的平滑正则化方法无需SVD计算且避免了显式的链式法则导数推导，可以作为图像去噪任务中总变差（TV）正则化的替代方案，并自然适用于连续数据。在多维数据恢复任务上的广泛实验，包括图像修复、去噪和点云上采样，展示了我们方法相对于最先进的方法的优越性和灵活性。
URL
https://arxiv.org/abs/2506.22134
