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Most high-dimensional matrix recovery problems are studied under theassumption that the target matrix has certain intrinsic structures. For imagedata related matrix recovery problems, approximate low-rankness and smoothnessare the two most commonly imposed structures. For approximately low-rank matrixrecovery, the robust principal component analysis (PCA) is well-studied andproved to be effective. For smooth matrix problem, 2d fused Lasso and othertotal variation based approaches have played a fundamental role. Although bothlow-rankness and smoothness are key assumptions for image data analysis, thetwo lines of research, however, have very limited interaction. Motivated bytaking advantage of both features, we in this paper develop a framework namedprojected robust PCA (PRPCA), under which the low-rank matrices are projectedonto a space of smooth matrices. Consequently, a large class of image matricescan be decomposed as a low-rank and smooth component plus a sparse component. Akey advantage of this decomposition is that the dimension of the core low-rankcomponent can be significantly reduced. Consequently, our framework is able toaddress a problematic bottleneck of many low-rank matrix problems: singularvalue decomposition (SVD) on large matrices. Theoretically, we provide explicitstatistical recovery guarantees of PRPCA and include classical robust PCA as aspecial case.

Projected Robust PCA with Application to Smooth Image Recovery