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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By solving a linear inverse problem under a sparsity constraint, one can successfully recover the coefficients, if there exists such a sparse approximation for the proposed class of signals. In this framework the dictionary can be adapted to a given set of signals using dictionary learning methods. The learned dictionary often does not have useful structures for a fast implementation, i.e. fast matrix-vector multiplication. This prevents such a dictionary being used for the real applications or large scale problems. The structure can be induced on the dictionary throughout the learning progress. Examples of such structures are shift-invariance and being multi-scale. These dictionaries can be efficiently implemented using a filter bank. In this paper a well-known structure, called compressibility, is adapted to be used in the dictionary learning problem. As a result, the complexity of the implementation of a compressible dictionary can be reduced by wisely choosing a generative model. By some simulations, it has been shown that the learned dictionary provides sparser approximations, while it does not increase the computational complexity of the algorithms, with respect to the pre-designed fast structured dictionaries. 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. We consider the dictionary learning problem for the analysis model based sparse representation. A novel algorithm is proposed by adapting the synthesis model based simultaneous codeword optimisation (SimCO) algorithm to the analysis model. This algorithm assumes that the analysis dictionary contains unit Ł 2 -norm atoms and trains the dictionary by the optimisation on manifolds. This framework allows one to update multiple dictionary atoms in each iteration, leading to a computationally efficient optimisation process. We demonstrate the competitive performance of the proposed algorithm using experiments on both synthetic and real data, as compared with three baseline algorithms, Analysis K-SVD, analysis operator learning (AOL) and learning overcomplete sparsifying transforms (LOST), respectively. Many recent efforts have shown the effectiveness of dictionary learning methods in solving several computer vision problems. However, when designing dictionaries, training and testing domains may be different, due to different view points and illumination conditions. In this paper, we present a function learning framework for the task of transforming a dictionary learned from one visual domain to the other, while maintaining a domain-invariant sparse representation of a signal. Domain dictionaries are modeled by a linear or non-linear parametric function. The dictionary function parameters and domain-invariant sparse codes are then jointly learned by solving an optimization problem. Experiments on real datasets demonstrate the effectiveness of our approach for applications such as face recognition, pose alignment and pose estimation.