In this paper, we introduce a new (constructive) characterization 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 with defining the quasi-affine systems on a given
manifold \cM that is formed by generalized dilations and shifts of a finite
collection of wavelet functions Ξ¨:={Οjβ:1β€jβ€r}βL2β(R).
We further require that Οjβ is generated by some refinable function Ο
with mask ajβ. We present the condition needed for the masks {ajβ:0β€jβ€r} so that the associated quasi-affine system generated by Ξ¨ is a
tight frame for L_2(\cM). Then, we discuss how the transition from the
continuum (manifolds) to the discrete setting (graphs) can be naturally done.
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 by proposing the fast tight wavelet frame transforms for graph data
(WFTG). Finally, we consider two specific applications of the proposed WFTG:
graph data denoising and semi-supervised clustering. Utilizing the sparse
representation provided by the WFTG, we propose β1β-norm based
optimization models on graphs for denoising and semi-supervised clustering. On
one hand, our numerical results show significant advantage of the WFTG over the
spectral graph wavelet transform (SGWT) by [1] for both applications. On the
other hand, numerical experiments on two real data sets show that the proposed
semi-supervised clustering model using the WFTG is overall competitive with the
state-of-the-art methods developed in the literature of high-dimensional data
classification, and is superior to some of these methods