Noisy tensor completion for tensors with a sparse canonical polyadic factor

Abstract

In this paper we study the problem of noisy tensor completion for tensors that admit a canonical polyadic or CANDECOMP/PARAFAC (CP) decomposition with one of the factors being sparse. We present general theoretical error bounds for an estimate obtained by using a complexity-regularized maximum likelihood principle and then instantiate these bounds for the case of additive white Gaussian noise. We also provide an ADMM-type algorithm for solving the complexity-regularized maximum likelihood problem and validate the theoretical finding via experiments on synthetic data set. Index TermsTensor decomposition, noisy tensor completion, complexity-regularized maximum likelihood estimation, sparse CP decomposition, sparse factor models.

Sparse Nonnegative Tensor Factorization and Completion With Noisy Observations

“…Besides, the model in [54] is only effective for third-order tensors, while the model in (6) can be utilized to any order tensors and can extract more physically meaningful latent components by the sparse and nonnegative factors [56]. Remark 3.4 Jain et al [24] proposed a noisy tensor completion method based on CP decomposition, where one factor is sparse. The model ( 6) can reduce to sparse nonnegative CP decomposition and completion when the core tensor is diagonal.…”

“…These conditions can be satisfied easily in real-world applications since n is generally large. Remark 4.5 Jain et al [24] proposed a noisy tensor completion model based on CP decomposition with a special sparse factor for an n 1 × n 2 × n 3 tensor, where the third factor matrix is sparse in CP decomposition. In particular, when the constraints are nonnegative, the model in [24] reduces to sparse nonnegative CP decomposition and completion.…”

“…Moreover, Hong et al [20] proposed a generalized CP low-rank tensor decomposition that allows other loss functions besides squared error, and presented a gradient descent type algorithm to solve the resulting model, which can handle missing data by a similar approach in [1]. Besides, Jain et al [24] proposed a CP decomposition and completion method with one sparse factor under a general class of noise models for third-order tensors, and established the error bound of the estimator of their proposed model, which was further derived to the error bound for the observations with additive Gaussian noise. The method in [24] can reduce to nonnegative CP decomposition with one sparse factor under nonnegative constraint of factor matrices.…”

“…Besides, Jain et al [24] proposed a CP decomposition and completion method with one sparse factor under a general class of noise models for third-order tensors, and established the error bound of the estimator of their proposed model, which was further derived to the error bound for the observations with additive Gaussian noise. The method in [24] can reduce to nonnegative CP decomposition with one sparse factor under nonnegative constraint of factor matrices. However, the previous methods need to give the CP rank of a tensor in advance, which is NP-hard in general [18].…”

Tensor Completion Algorithms in Big Data Analytics

“…Goldfarb and Qin [65] study the problem of robust low-rank tensor completion in a convex ADMM optimization framework and provide both theoretical and practical analysis of convex and non-convex models. Recent work of Jain [84] also adopts ADMM framework to complete noisy tensors with one of the CP factors being sparse. Besides ADMM framework, Zhao et al [223] consider the probabilistic framework with a fully Bayesian treatment optimized by variational Bayesian inference approach.…”