Robust bilinear matrix recovery by Tensor Low-Rank Representation

For low-rank recovery and error correction, Low-Rank Representation (LRR) row-reconstructs given data matrix X by seeking a low-rank representation, while Inductive Robust Principal Component Analysis (IRPCA) aims to calculate a low-rank projection to column-reconstruct X. But either column or row information of Xis lost by LRR and IRPCA. In addition, the matrix X itself is chosen as the dictionary by LRR, but (grossly) corrupted entries may greatly depress its performance. To solve these issues, we propose a simultaneous low-rank representation and dictionary learning framework termed Tensor LRR (TLRR) for robust bilinear recovery. TLRR reconstructs given matrix X along both row and column directions by computing a pair of low-rank matrices alternately from a nuclear norm minimization problem for constructing a low-rank tensor subspace. As a result, TLRR in the optimizations can be regarded as enhanced IRPCA with noises removed by low-rank representation, and can also be considered as enhanced LRR with a clean informative dictionary using a low-rank projection. The comparison with other criteria shows that TLRR exhibits certain advantages, for instance strong generalization power and robustness enhancement to the missing values. Simulations verified the validity of TLRR for recovery.