2014/05/07 by Ming Yuan, Cun‐Hui Zhang, Yuan, Ming +1 · 8 citations
Computer Science · Engineering · Mathematics · #Algebra over a field #Computer science #Eigenvalues and eigenvectors #FOS: Computer and information sciences #FOS: Mathematics #Image and Signal Denoising Methods #Information Theory (cs.IT) #Machine Learning (stat.ML) #Mathematical optimization #Mathematics #Matrix norm #Norm (philosophy) #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Physics #Probability (math.PR) #Pure mathematics #Sparse and Compressive Sensing Techniques #Tensor (intrinsic definition) #Tensor contraction #Tensor decomposition and applications #Tensor product
paper · pdf · doi:10.48550/arxiv.1405.1773
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2014/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Many problems can be formulated as recovering a low-rank tensor. Although an increasingly common task, tensor recovery remains a challenging problem because of the delicacy associated with the decomposition of higher order tensors. To overcome these difficulties, existing approaches often proceed by unfolding tensors into matrices and then apply techniques for matrix completion. We show here that such matricization fails to exploit the tensor structure and may lead to suboptimal procedure. More specifically, we investigate a convex optimization approach to tensor completion by directly minimizing a tensor nuclear norm and prove that this leads to an improved sample size requirement. To establish our results, we develop a series of algebraic and probabilistic techniques such as characterization of subdifferetial for tensor nuclear norm and concentration inequalities for tensor martingales, which may be of independent interests and could be useful in other tensor related problems.