We present a natural generalization of the recent low rank + sparse matrix
decomposition and consider the decomposition of matrices into components of
multiple scales. Such decomposition is well motivated in practice as data
matrices often exhibit local correlations in multiple scales. Concretely, we
propose a multi-scale low rank modeling that represents a data matrix as a sum
of block-wise low rank matrices with increasing scales of block sizes. We then
consider the inverse problem of decomposing the data matrix into its
multi-scale low rank components and approach the problem via a convex
formulation. Theoretically, we show that under various incoherence conditions,
the convex program recovers the multi-scale low rank components \revised{either
exactly or approximately}. Practically, we provide guidance on selecting the
regularization parameters and incorporate cycle spinning to reduce blocking
artifacts. Experimentally, we show that the multi-scale low rank decomposition
provides a more intuitive decomposition than conventional low rank methods and
demonstrate its effectiveness in four applications, including illumination
normalization for face images, motion separation for surveillance videos,
multi-scale modeling of the dynamic contrast enhanced magnetic resonance
imaging and collaborative filtering exploiting age information
