Low-rank tensor estimation has been frequently applied in many real-world problems. Despite successful applications, existing Schatten 1-norm minimization (SNM) methods may become very slow or even not applicable for large-scale problems. To address this difficulty, we therefore propose an efficient and scalable core tensor Schatten 1-norm minimization method for simultaneous tensor decomposition and completion, with a much lower computational complexity. We first induce the equivalence relation of Schatten 1-norm of a low-rank tensor and its core tensor. Then the Schatten 1-norm of the core tensor is used to replace that of the whole tensor, which leads to a much smaller-scale matrix SNM problem. Finally, an efficient algorithm with a rank-increasing scheme is developed to solve the proposed problem with a convergence guarantee. Extensive experimental results show that our method is usually more accurate than the state-of-the-art methods, and is orders of magnitude faster.

Low-rank tensor estimation has been frequently applied in many real-world problems. Despite successful applications, existing Schatten 1-norm minimization (SNM) methods may become very slow or even not applicable for large-scale problems. To address this difficulty, we therefore propose an efficient and scalable core tensor Schatten 1-norm minimization method for simultaneous tensor decomposition and completion, with a much lower computational complexity. We first induce the equivalence relation of Schatten 1-norm of a low-rank tensor and its core tensor. Then the Schatten 1-norm of the core tensor is used to replace that of the whole tensor, which leads to a much smaller-scale matrix SNM problem. Finally, an efficient algorithm with a rank-increasing scheme is developed to solve the proposed problem with a convergence guarantee. Extensive experimental results show that our method is usually more accurate than the state-of-the-art methods, and is orders of magnitude faster.

We analyze the statistical performance of a recently proposed convex tensor decomposition algorithm. Conventionally tensor decomposition has been formulated as non-convex optimization problems, which hindered the analysis of their performance. We show under some conditions that the mean squared error of the convex method scales linearly with the quantity we call the normalized rank of the true tensor. The current analysis naturally extends the analysis of convex low-rank matrix estimation to tensors. Furthermore, we show through numerical experiments that our theory can precisely predict the scaling behaviour in practice.

We study a new class of structured Schatten norms for tensors that includes two recently proposed norms ("overlapped" and "latent") for convex-optimizationbased tensor decomposition. We analyze the performance of "latent" approach for tensor decomposition, which was empirically found to perform better than the "overlapped" approach in some settings. We show theoretically that this is indeed the case. In particular, when the unknown true tensor is low-rank in a specific unknown mode, this approach performs as well as knowing the mode with the smallest rank. Along the way, we show a novel duality result for structured Schatten norms, which is also interesting in the general context of structured sparsity. We confirm through numerical simulations that our theory can precisely predict the scaling behaviour of the mean squared error.

Low-rank tensor estimation has been frequently applied in many real-world problems. Despite successful applications, existing Schatten 1-norm minimization (SNM) methods may become very slow or even not applicable for large-scale problems. To address this difficulty, we therefore propose an efficient and scalable core tensor Schatten 1-norm minimization method for simultaneous tensor decomposition and completion, with a much lower computational complexity. We first induce the equivalence relation of Schatten 1-norm of a low-rank tensor and its core tensor. Then the Schatten 1-norm of the core tensor is used to replace that of the whole tensor, which leads to a much smaller-scale matrix SNM problem. Finally, an efficient algorithm with a rank-increasing scheme is developed to solve the proposed problem with a convergence guarantee. Extensive experimental results show that our method is usually more accurate than the state-of-the-art methods, and is orders of magnitude faster.

Low-rank tensor estimation has been frequently applied in many real-world problems. Despite successful applications, existing Schatten 1-norm minimization (SNM) methods may become very slow or even not applicable for large-scale problems. To address this difficulty, we therefore propose an efficient and scalable core tensor Schatten 1-norm minimization method for simultaneous tensor decomposition and completion, with a much lower computational complexity. We first induce the equivalence relation of Schatten 1-norm of a low-rank tensor and its core tensor. Then the Schatten 1-norm of the core tensor is used to replace that of the whole tensor, which leads to a much smaller-scale matrix SNM problem.

Several important applications, such as streaming PCA and semidefinite programming, involve a large-scale positive-semidefinite (psd) matrix that is presented as a sequence of linear updates. Because of storage limitations, it may only be possible to retain a sketch of the psd matrix. This paper develops a new algorithm for fixed-rank psd approximation from a sketch. The approach combines the Nystrom approximation with a novel mechanism for rank truncation. Theoretical analysis establishes that the proposed method can achieve any prescribed relative error in the Schatten 1-norm and that it exploits the spectral decay of the input matrix. Computer experiments show that the proposed method dominates alternative techniques for fixed-rank psd matrix approximation across a wide range of examples.

Low-rank tensor estimation has been frequently applied in many real-world problems. Despite successful applications, existing Schatten 1-norm minimization (SNM) methods may become very slow or even not applicable for large-scale problems. To address this difficulty, we therefore propose an efficient and scalable core tensor Schatten 1-norm minimization method for simultaneous tensor decomposition and completion, with a much lower computational complexity. We first induce the equivalence relation of Schatten 1-norm of a low-rank tensor and its core tensor. Then the Schatten 1-norm of the core tensor is used to replace that of the whole tensor, which leads to a much smaller-scale matrix SNM problem. Finally, an efficient algorithm with a rank-increasing scheme is developed to solve the proposed problem with a convergence guarantee. Extensive experimental results show that our method is usually more accurate than the state-of-the-art methods, and is orders of magnitude faster.

We propose a new class of structured Schatten norms for tensors that includes two recently proposed norms (overlapped'' and "latent'') for convex-optimization-based tensor decomposition. Based on the properties of the structured Schatten norms, we mathematically analyze the performance of "latent'' approach for tensor decomposition, which was empirically found to perform better than the "overlapped'' approach in some settings. We show theoretically that this is indeed the case. In particular, when the unknown true tensor is low-rank in a specific mode, this approach performs as well as knowing the mode with the smallest rank. Along the way, we show a novel duality result for structures Schatten norms, which is also interesting in the general context of structured sparsity. We confirm through numerical simulations that our theory can precisely predict the scaling behaviour of the mean squared error. "

We discuss structured Schatten norms for tensor decomposition that includes two recently proposed norms ("overlapped" and "latent") for convex-optimization-based tensor decomposition, and connect tensor decomposition with wider literature on structured sparsity. Based on the properties of the structured Schatten norms, we mathematically analyze the performance of "latent" approach for tensor decomposition, which was empirically found to perform better than the "overlapped" approach in some settings. We show theoretically that this is indeed the case. In particular, when the unknown true tensor is low-rank in a specific mode, this approach performs as good as knowing the mode with the smallest rank. Along the way, we show a novel duality result for structures Schatten norms, establish the consistency, and discuss the identifiability of this approach. We confirm through numerical simulations that our theoretical prediction can precisely predict the scaling behaviour of the mean squared error.