Tensor decomposition is a powerful tool for extracting physically meaningful latent factors from multi-dimensional nonnegative data, and has been an increasing interest in a variety of fields such as image processing, machine learning, and computer vision. In this paper, we propose a sparse nonnegative Tucker decomposition and completion method for the recovery of underlying nonnegative data under noisy observations. Here the underlying nonnegative data tensor is decomposed into a core tensor and several factor matrices with all entries being nonnegative and the factor matrices being sparse. The loss function is derived by the maximum likelihood estimation of the noisy observations, and the $\ell_0$ norm is employed to enhance the sparsity of the factor matrices. We establish the error bound of the estimator of the proposed model under generic noise scenarios, which is then specified to the observations with additive Gaussian noise, additive Laplace noise, and Poisson observations, respectively. Our theoretical results are better than those by existing tensor-based or matrix-based methods. Moreover, the minimax lower bounds are shown to be matched with the derived upper bounds up to logarithmic factors. Numerical examples on both synthetic and real-world data sets demonstrate the superiority of the proposed method for nonnegative tensor data completion.

翻译：张量分解是从多维非负数据中提取具有物理意义的潜在因子的有力工具，在图像处理、机器学习和计算机视觉等多个领域日益受到关注。本文提出一种稀疏非负Tucker分解与补全方法，用于在噪声观测下恢复潜在的非负数据。该方法将潜在非负数据张量分解为一个核心张量和若干因子矩阵，所有元素均保持非负性且因子矩阵具有稀疏特性。损失函数通过噪声观测的最大似然估计推导得出，并采用$\ell_0$范数以增强因子矩阵的稀疏性。我们在通用噪声场景下建立了所提模型估计量的误差界，并分别针对加性高斯噪声、加性拉普拉斯噪声及泊松观测等具体情形进行细化。理论结果表明，本文方法优于现有基于张量或矩阵的分解方法。此外，研究证明极小极大下界与所得上界在对数因子范围内相匹配。在合成数据集和真实数据集上的数值实验均验证了所提方法在非负张量数据补全任务中的优越性。