We present a natural generalization of the recent low rank + sparse matrixdecomposition and consider the decomposition of matrices into components ofmultiple scales. Such decomposition is well motivated in practice as datamatrices often exhibit local correlations in multiple scales. Concretely, wepropose a multi-scale low rank modeling that represents a data matrix as a sumof block-wise low rank matrices with increasing scales of block sizes. We thenconsider the inverse problem of decomposing the data matrix into itsmulti-scale low rank components and approach the problem via a convexformulation. Theoretically, we show that under various incoherence conditions,the convex program recovers the multi-scale low rank components \revised{eitherexactly or approximately}. Practically, we provide guidance on selecting theregularization parameters and incorporate cycle spinning to reduce blockingartifacts. Experimentally, we show that the multi-scale low rank decompositionprovides a more intuitive decomposition than conventional low rank methods anddemonstrate its effectiveness in four applications, including illuminationnormalization for face images, motion separation for surveillance videos,multi-scale modeling of the dynamic contrast enhanced magnetic resonanceimaging and collaborative filtering exploiting age information.
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