Learning Parametric Dictionaries for Signals on Graphs

Abstract

Abstract-In sparse signal representation, the choice of a dictionary often involves a tradeoff between two desirable propertiesthe ability to adapt to specific signal data and a fast implementation of the dictionary. To sparsely represent signals residing on weighted graphs, an additional design challenge is to incorporate the intrinsic geometric structure of the irregular data domain into the atoms of the dictionary. In this work, we propose a parametric dictionary learning algorithm to design data-adapted, structured dictionaries that sparsely represent graph signals. In particular, we model graph signals as combinations of overlapping local patterns. We impose the constraint that each dictionary is a concatenation of subdictionaries, with each subdictionary being a polynomial of the graph Laplacian matrix, representing a single pattern translated to different areas of the graph. The learning algorithm adapts the patterns to a training set of graph signals. Experimental results on both synthetic and real datasets demonstrate that the dictionaries learned by the proposed algorithm are competitive with and often better than unstructured dictionaries learned by state-of-the-art numerical learning algorithms in terms of sparse approximation of graph signals. In contrast to the unstructured dictionaries, however, the dictionaries learned by the proposed algorithm feature localized atoms and can be implemented in a computationally efficient manner in signal processing tasks such as compression, denoising, and classification.

Multi-graph learning of spectral graph dictionaries

“…3, we illustrate the sparse approximation performance of our dictionary representation by studying the reconstruction performance in SNR on the set of testing signals for different sparsity levels. The performance is compared to that obtained by learning separately a dictionary on each graph [7], and the one obtained by the sparse decomposition in the graph wavelet dictionary [4]. We observe that multi-graph learning improves significantly the performance in comparison to SGWT.…”

“…In particular, if the generating kernel is smooth, g(L) consists of a set of N columns, each representing a localized atom, generated by the same kernel, and positioned on different nodes on the graph [4], [1]. One can thus design graph operators consisting of localized atoms in the vertex domain by taking the kernel g(·) in (1) to be a smooth polynomial function of degree K [4], [7]:…”

“…For efficient representations in many applications, these coefficients should be sparse and they should capture the most important characteristics of the graph signals [7]. Sparsity however largely depends on the design of the dictionary, which in our signal model, depends on the choice of the polynomial coefficients.…”

“…We further consider two different sizes for the training data: the training set in the first one consists of M = 400 signals per graph, while in the second one it consists of M = 1000 signals per graph. We compare the approximation performance of our algorithm to that obtained with (i) the spectral graph wavelet transform (SGWT) [4] and (ii) a polynomial graph dictionary, learned in each graph separately [7]. In Fig.…”

“…Such adaptation is generally achieved by training a dictionary that can sparsely represent a particular class of signals. Unfortunately, existing algorithms for dictionary learning techniques for graph signals [7], [8] rely on the assumption that all the training signals lie on the same graph, which is pretty constraining in practice. Moreover, classical dictionary learning techniques such as K-SVD [9] cannot handle training signals with different dimensionality, thus they cannot be applied to our problem.…”