论文标题
稀疏的非负张量分解和完成，嘈杂的观察结果
Sparse Nonnegative Tensor Factorization and Completion with Noisy Observations
论文作者
论文摘要
在本文中，我们研究了三阶张量的部分和嘈杂观测值的稀疏非负张量分解和完成问题。由于稀疏性和非负性，将潜在的张量分解为一个稀疏的非负张量和一种非负张量的张量张量产物。我们建议最大程度地减少具有非负约束的观测值的最大似然估计总和，而稀疏因子的张量$ \ ell_0 $ norm。我们表明，可以在一般噪声观察下建立所提出模型的估计器的误差界限。可以得出特定噪声分布下的详细误差界限，包括加性高斯噪声，加性拉普拉斯噪声和泊松观测值。此外，最小值的下限与已建立的上限匹配，直至基础张量的大小的对数因子。这些针对张量的理论结果比矩阵获得的理论结果更好，这说明了使用非负稀疏张量模型完成和降解的优势。提供了数值实验，以验证与基于矩阵的方法相比，基于张量的方法的优越性。
In this paper, we study the sparse nonnegative tensor factorization and completion problem from partial and noisy observations for third-order tensors. Because of sparsity and nonnegativity, the underlying tensor is decomposed into the tensor-tensor product of one sparse nonnegative tensor and one nonnegative tensor. We propose to minimize the sum of the maximum likelihood estimation for the observations with nonnegativity constraints and the tensor $\ell_0$ norm for the sparse factor. We show that the error bounds of the estimator of the proposed model can be established under general noise observations. The detailed error bounds under specific noise distributions including additive Gaussian noise, additive Laplace noise, and Poisson observations can be derived. Moreover, the minimax lower bounds are shown to be matched with the established upper bounds up to a logarithmic factor of the sizes of the underlying tensor. These theoretical results for tensors are better than those obtained for matrices, and this illustrates the advantage of the use of nonnegative sparse tensor models for completion and denoising. Numerical experiments are provided to validate the superiority of the proposed tensor-based method compared with the matrix-based approach.
