Sparse Representation on Graphs by Tight Wavelet Frames and Applications

In this paper, we introduce a unified theory of tight wavelet frames on non-flat domains in both continuum setting, i.e. on manifolds, and discrete setting, i.e. on graphs; discuss how fast tight wavelet frame transforms can be computed and how they can be effectively used to process graph data. We start from defining multiresolution analysis (MRA) generated by a single generator on manifolds, and discuss the conditions needed for a generator to form an MRA. With a given MRA, we show how MRA-based tight frames can be constructed and the conditions needed for them to generate a tight wavelet frame on manifolds. In particular, we show that under suitable conditions, framelets on R constructed from the unitary extension principle [1] can also generate tight frame systems on manifolds. We also discuss how the transition from the continuum to the discrete setting can be naturally defined, which leads to multi-level discrete tight wavelet frame transforms (decomposition and reconstruction) on graphs. In order for the proposed discrete tight wavelet frame transforms to be useful in applications, we show how the transforms can be computed efficiently and accurately. More importantly, numerical simulations show that the proposed discrete tight wavelet frame transform maps piecewise smooth data to a set of sparse coefficients. This indicates that the proposed tight wavelet frames indeed provide sparse representation on graphs. Finally, we consider two specific applications: graph data denoising and semi-supervised clustering. Utilizing the proposed sparse representation, we introduce l1-norm based optimization models for denoising and semi-supervised clustering, which are inspired by the models used in image restoration and image segmentation.