Low-rank tensor decompositions (TDs) provide an effective framework for multiway data analysis. Traditional TD methods rely on predefined structural assumptions, such as CP or Tucker decompositions. From a probabilistic perspective, these can be viewed as using Dirac delta distributions to model the relationships between shared factors and the low-rank tensor. However, such prior knowledge is rarely available in practical scenarios, particularly regarding the optimal rank structure and contraction rules. The optimization procedures based on fixed contraction rules are complex, and approximations made during these processes often lead to accuracy loss. To address this issue, we propose a score-based model that eliminates the need for predefined structural or distributional assumptions, enabling the learning of compatibility between tensors and shared factors. Specifically, a neural network is designed to learn the energy function, which is optimized via score matching to capture the gradient of the joint log-probability of tensor entries and shared factors. Our method allows for modeling structures and distributions beyond the Dirac delta assumption. Moreover, integrating the block coordinate descent (BCD) algorithm with the proposed smooth regularization enables the model to perform both tensor completion and denoising. Experimental results demonstrate significant performance improvements across various tensor types, including sparse and continuous-time tensors, as well as visual data.

翻译：低秩张量分解为多路数据分析提供了一个有效的框架。传统的张量分解方法依赖于预定义的结构假设，例如CP分解或Tucker分解。从概率视角看，这些方法可被视为使用狄拉克δ分布来建模共享因子与低秩张量之间的关系。然而，在实际场景中此类先验知识往往难以获得，特别是在最优秩结构和收缩规则方面。基于固定收缩规则的优化过程较为复杂，且在此过程中所作的近似常导致精度损失。为解决这一问题，我们提出一种基于分数的模型，该模型无需预定义的结构或分布假设，能够学习张量与共享因子之间的兼容性。具体而言，我们设计了一个神经网络来学习能量函数，并通过分数匹配进行优化，以捕捉张量条目与共享因子的联合对数概率梯度。我们的方法能够对超越狄拉克δ假设的结构和分布进行建模。此外，将块坐标下降算法与所提出的平滑正则化相结合，使模型能够同时执行张量补全和去噪任务。实验结果表明，该方法在多种张量类型（包括稀疏张量、连续时间张量以及视觉数据）上均取得了显著的性能提升。