In sparse signal representation, the choice of a dictionary often involves a tradeoff between two desirable properties – the 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.
The recent work in @cite_12 tries to bridge the gap between the graph-based transform methods and the purely numerical dictionary learning algorithms by proposing an algorithm to learn structured graph dictionaries. The learned dictionaries have a structure that is derived from the graph topology, while its parameters are learned from the data. This work is the closest to ours in a sense that both graph dictionaries consist of subdictionaries that are based on the graph Laplacian. However, it does not necessarily lead to efficient implementations as the obtained dictionary is not necessarily a smooth matrix function (see, e.g., @cite_34 for more on matrix functions) of the graph Laplacian matrix.
@cite_34 @cite_12
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Matrix functions are used in many areas of linear algebra and arise in numerous applications in science and engineering. The most common matrix function is the matrix inverse; it is not treated specifically in this chapter, but is covered in Section 1.5 and Section 51.3. This chapter is concerned with general matrix functions as well as specific cases such as matrix square roots, trigonometric functions, and the exponential and logarithmic functions. We propose a method for learning dictionaries towards sparse approximation of signals defined on vertices of arbitrary graphs. Dictionaries are expected to describe effectively the main spatial and spectral components of the signals of interest, so that their structure is dependent on the graph information and its spectral representation. We first show how operators can be defined for capturing different spectral components of signals on graphs. We then propose a dictionary learning algorithm built on a sparse approximation step and a dictionary update function, which iteratively leads to adapting the structured dictionary to the class of target signals. Experimental results on synthetic and natural signals on graphs demonstrate the efficiency of the proposed algorithm both in terms of sparse approximation and support recovery performance.