Rank-One Matrix Completion With Automatic Rank Estimation via L1-Norm Regularization

Abstract

Completing a matrix from a small subset of its entries, i.e., matrix completion is a challenging problem arising from many real-world applications, such as machine learning and computer vision. One popular approach to solve the matrix completion problem is based on low-rank decomposition/factorization. Low-rank matrix decomposition-based methods often require a prespecified rank, which is difficult to determine in practice. In this paper, we propose a novel low-rank decomposition-based matrix completion method with automatic rank estimation. Our method is based on rank-one approximation, where a matrix is represented as a weighted summation of a set of rank-one matrices. To automatically determine the rank of an incomplete matrix, we impose L1-norm regularization on the weight vector and simultaneously minimize the reconstruction error. After obtaining the rank, we further remove the L1-norm regularizer and refine recovery results. With a correctly estimated rank, we can obtain the optimal solution under certain conditions. Experimental results on both synthetic and real-world data demonstrate that the proposed method not only has good performance in rank estimation, but also achieves better recovery accuracy than competing methods.

An L 1-and-L 2-Norm-Oriented Latent Factor Model for Recommender Systems

“…On the other hand, matrix completion [36][37][38][39] or feature representation [40,41] models adopt both L 1 and L 2 norms to construct their Penalty, thereby achieving model sparsity [45] or generality [38]. Nonetheless, as mentioned before, Penalty and Loss are two different and critical components of an LF model's learning objective.…”

Improvement of the Classification Accuracy of Steady-State Visual Evoked Potential-Based Brain-Computer Interfaces by Combining L1-MCCA with SVM

“…Based on tensor analysis [29] and sparse regularization, Zhang Yu et al [30] have proposed the L1-regularized multiway canonical correlation analysis (L1-MCCA). L1 regularization [31][32][33] was used to further strengthen MCCA's test channel array optimization so that L1-MCCA has the best projection vector effect when learning reference signal optimization. During the data training process, the L1-MCCA has the stronger ability to learn projection vectors and prevent signal overfitting.…”

Feature Extraction for Incomplete Data Via Low-Rank Tensor Decomposition With Feature Regularization

“…One natural solution is to fill in the missing values and then view the recovered tensors as the extracted features. Many tensor completion techniques have been extended from matrix completion cases [23], [24], which are widely used for predicting missing data given partially observed entries and have drawn much attention in many applications such as image/video recovery [25], [26]. For example, Liu et al [25] defined the Tuckerbased tensor nuclear norm by combining nuclear norms of all matrices unfolded along each mode and proposed a high accuracy low-rank tensor completion algorithm (HaLRTC) for estimating missing values in tensor visual data.…”