Sparse non-negative tensor factorization using columnwise coordinate descent

Abstract

Many applications in computer vision, biomedical informatics, and graphics deal with data in the matrix or tensor form. Non-negative matrix and tensor factorization, which extract data-dependent non-negative basis functions, have been commonly applied for the analysis of such data for data compression, visualization, and detection of hidden information (factors). In this paper, we present a fast and flexible algorithm for sparse non-negative tensor factorization (SNTF) based on columnwise coordinate descent (CCD). Different from the traditional coordinate descent which updates one element at a time, CCD updates one column vector simultaneously. Our empirical results on higher-mode images, such as brain MRI images, gene expression images, and hyperspectral images show that the proposed algorithm is 1-2 orders of magnitude faster than several state-of-the-art algorithms.

Low Multilinear Rank Approximation of Tensors and Application in Missing Traffic Data

“…To estimate the missing entries in a low rank tensor, tensor completion methods which try to minimize the tensor rank have been proposed [17][18][19][20]. Liu et al proposed tensor completion based on trace norm, optimizing the objective function by block coordinate descent algorithm (BCD) [19] and alternating direction method of multipliers (ADMM) [17,18].…”

“…Liu et al proposed tensor completion based on trace norm, optimizing the objective function by block coordinate descent algorithm (BCD) [19] and alternating direction method of multipliers (ADMM) [17,18]. Gandy et al [20] built the objective function based on multi-rank instead of trace norm, optimizing the objective function by ADMM.…”

“…However, we focus on the most general tensor decomposition (i.e., Tucker) and optimize the problem on Grassmann manifold for ensuring the uniqueness of the final solution. Our method is also different from the method based on Tucker decomposition presented in [17,18], where the presented method is used for comparing with low rank tensor completion (LRTC), and their objective function does not consider the existed noise and the uniqueness of the optimal solution and is optimized by block coordinate descent method (BCD).…”

Abnormal event detection in crowded scenes using sparse representation

“…To optimize b we employ the idea ''columnwise coordinate descent'' in [21]: each column of b can be optimized simultaneously while fixing other columns:…”

Analysis of Large-Scale Traffic Dynamics in an Urban Transportation Network Using Non-Negative Tensor Factorization

“…The multi-way structure of tensor provides a natural way to encode the underlying multiple dependencies in the sequential multivariate data. For example, in video processing, tensors are widely used to represent temporal streams of multi-dimensional data, such as 2D images in video frames [34] and user product rating profiles in recommendation systems [35].…”