Learning Laplacian Matrix in Smooth Graph Signal Representations

The construction of a meaningful graph plays a crucial role in the success of many graph-based representations and algorithms for handling structured data, especially in the emerging field of graph signal processing. However, a meaningful graph is not always readily available from the data, nor easy to define depending on the application domain. In particular, it is often desirable in graph signal processing applications that a graph is chosen such that the data admit certain regularity or smoothness on the graph. In this paper, we address the problem of learning graph Laplacians, which is equivalent to learning graph topologies, such that the input data form graph signals with smooth variations on the resulting topology. To this end, we adopt a factor analysis model for the graph signals and impose a Gaussian probabilistic prior on the latent variables that control these signals. We show that the Gaussian prior leads to an efficient representation that favors the smoothness property of the graph signals. We then propose an algorithm for learning graphs that enforces such property and is based on minimizing the variations of the signals on the learned graph. Experiments on both synthetic and real world data demonstrate that the proposed graph learning framework can efficiently infer meaningful graph topologies from signal observations under the smoothness prior

Graph Signal Processing: Overview, Challenges and Applications

Research in Graph Signal Processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing. We then summarize recent developments in developing basic GSP tools, including methods for sampling, filtering or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning. We finish by providing a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE

Graph learning under sparsity priors

Graph signals offer a very generic and natural representation for data that lives on networks or irregular structures. The actual data structure is however often unknown a priori but can sometimes be estimated from the knowledge of the application domain. If this is not possible, the data structure has to be inferred from the mere signal observations. This is exactly the problem that we address in this paper, under the assumption that the graph signals can be represented as a sparse linear combination of a few atoms of a structured graph dictionary. The dictionary is constructed on polynomials of the graph Laplacian, which can sparsely represent a general class of graph signals composed of localized patterns on the graph. We formulate a graph learning problem, whose solution provides an ideal fit between the signal observations and the sparse graph signal model. As the problem is non-convex, we propose to solve it by alternating between a signal sparse coding and a graph update step. We provide experimental results that outline the good graph recovery performance of our method, which generally compares favourably to other recent network inference algorithms

Learning parametric dictionaries for graph signals

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

Geometric deep learning: going beyond Euclidean data

Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging, regulatory networks in genetics, and meshed surfaces in computer graphics. In many applications, such geometric data are large and complex (in the case of social networks, on the scale of billions), and are natural targets for machine learning techniques. In particular, we would like to use deep neural networks, which have recently proven to be powerful tools for a broad range of problems from computer vision, natural language processing, and audio analysis. However, these tools have been most successful on data with an underlying Euclidean or grid-like structure, and in cases where the invariances of these structures are built into networks used to model them. Geometric deep learning is an umbrella term for emerging techniques attempting to generalize (structured) deep neural models to non-Euclidean domains such as graphs and manifolds. The purpose of this paper is to overview different examples of geometric deep learning problems and present available solutions, key difficulties, applications, and future research directions in this nascent field