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...

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Vydáno v:	IEEE transactions on signal processing Ročník 64; číslo 23; s. 6160 - 6173
Hlavní autoři:	Xiaowen Dong, Thanou, Dorina, Frossard, Pascal, Vandergheynst, Pierre
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	IEEE 01.12.2016
Témata:	Analytical models factor analysis Gaussian prior graph signal processing Kernel Laplace equations Laplacian matrix learning representation theory Signal processing Signal processing algorithms Signal representation Topology
ISSN:	1053-587X, 1941-0476
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Abstract	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.
AbstractList	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.
Author	Thanou, Dorina Xiaowen Dong Vandergheynst, Pierre Frossard, Pascal
Author_xml	– sequence: 1 surname: Xiaowen Dong fullname: Xiaowen Dong email: xdong@mit.edu organization: MIT Media Lab., Cambridge, MA, USA – sequence: 2 givenname: Dorina surname: Thanou fullname: Thanou, Dorina email: dorina.thanou@epfl.ch organization: Signal Process. Labs., Ecole Polytech. Fed. de Lausanne, Lausanne, Switzerland – sequence: 3 givenname: Pascal surname: Frossard fullname: Frossard, Pascal email: pascal.frossard@epfl.ch organization: Signal Process. Labs., Ecole Polytech. Fed. de Lausanne, Lausanne, Switzerland – sequence: 4 givenname: Pierre surname: Vandergheynst fullname: Vandergheynst, Pierre email: pierre.vandergheynst@epfl.ch organization: Signal Process. Labs., Ecole Polytech. Fed. de Lausanne, Lausanne, Switzerland
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Snippet	The construction of a meaningful graph plays a crucial role in the success of many graph-based representations and algorithms for handling structured data,...
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SubjectTerms	Analytical models factor analysis Gaussian prior graph signal processing Kernel Laplace equations Laplacian matrix learning representation theory Signal processing Signal processing algorithms Signal representation Topology
Title	Learning Laplacian Matrix in Smooth Graph Signal Representations
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Volume	64
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