Graph Vertex Sampling with Arbitrary Graph Signal Hilbert Spaces

Girault, Benjamin; Ortega, Antonio; Narayayan, Shrikanth S.

doi:10.1109/icassp40776.2020.9054723

Cited by 12 publications

(11 citation statements)

References 16 publications

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“…Theorem 4 generalizes Proposition 1, demonstrating that the spectral folding property is not unique to the (L, I)-GFT of bipartite graphs, and in fact, it is always satisfied if the inner product matrix Q is chosen as in (28). Note also that while other recent work [28], [35]- [37] demonstrates the benefits of using alternative inner products for signal processing on irregular domains, our work is the first to show that nondiagonal choices of Q are also useful.…”

Section: B Spectral Folding On Arbitrary Graphsmentioning

confidence: 61%

Two Channel Filter Banks on Arbitrary Graphs with Positive Semi Definite Variation Operators

Pavéz¹,

Girault²,

Ortega³

et al. 2022

Preprint

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We study the design of filter banks for signals defined on the nodes of graphs. We propose novel two channel filter banks, that can be applied to arbitrary graphs, given a positive semi definite variation operator, while using downsampling operators on arbitrary vertex partitions. The proposed filter banks also satisfy several desirable properties, including perfect reconstruction, and critical sampling, while having efficient implementations. Our results generalize previous approaches only valid for the normalized Laplacian of bipartite graphs. We consider graph Fourier transforms (GFTs) given by the generalized eigenvectors of the variation operator. This GFT basis is orthogonal in an alternative inner product space, which depends on the choices of downsampling sets and variation operators. We show that the spectral folding property of the normalized Laplacian of bipartite graphs, at the core of bipartite filter bank theory, can be generalized for the proposed GFT if the inner product matrix is chosen properly. We give a probabilistic interpretation to the proposed filter banks using Gaussian graphical models. We also study orthogonality properties of tree structured filter banks, and propose a vertex partition algorithms for downsampling. We show that the proposed filter banks can be implemented efficiently on 3D point clouds, with hundreds of thousands of points (nodes), while also improving the color signal representation quality over competing state of the art approaches.

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Section: B Spectral Folding On Arbitrary Graphsmentioning

confidence: 61%

Two Channel Filter Banks on Arbitrary Graphs with Positive Semi Definite Variation Operators

Pavéz¹,

Girault²,

Ortega³

et al. 2022

Preprint

Self Cite

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“…1. Improving the sampling to generate our graphs (so that they concentrate on the most important pairs of samples of the training set), as proposed in [3]. 2.…”

Section: Discussionmentioning

confidence: 99%

Ranking Deep Learning Generalization using Label Variation in Latent Geometry Graphs

Lassance,

Béthune,

Bontonou

et al. 2020

Preprint

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Measuring the generalization performance of a Deep Neural Network (DNN) without relying on a validation set is a difficult task. In this work, we propose exploiting Latent Geometry Graphs (LGGs) to represent the latent spaces of trained DNN architectures. Such graphs are obtained by connecting samples that yield similar latent representations at a given layer of the considered DNN. We then obtain a generalization score by looking at how strongly connected are samples of distinct classes in LGGs. This score allowed us to rank 3rd on the NeurIPS 2020 Predicting Generalization in Deep Learning (PGDL) competition.

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“…We consider the (M, Q) graph Fourier transform ((M, Q)-GFT), a generalization of the GFT to arbitrary finite dimensional Hilbert spaces [15,19] with inner product x, y Q = y Qx and variation operator M. BFB theory is built upon a spectral folding property of the eigenvectors and eigenvalues of the normalized Laplacian of bipartite graphs [20]. We follow a similar strategy and prove a new spectral folding property for the (M, Q)-GFT.…”

Section: Introductionmentioning

confidence: 99%

“…Our theory follows a similar strategy, by proving a new spectral folding property for the (M, Q) graph Fourier transform ((M, Q)-GFT). This result builds upon a generalization of the graph Fourier transform to arbitrary finite dimensional Hilbert spaces [15,19] with inner product x, y Q = y Qx and variation operator M. In our framework, the down-sampling and variation operators M completely determine the choice of inner product Q. Interestingly, our spectral domain conditions on the filters match those of [5,6] for the normalized Laplacian of bipartite graphs, and therefore we can re-use any of their filter designs, or any of the more recent improvements [8,9]. When the graph is bipartite, and M is the normalized or combinatorial Laplacian, we recover the nonZe-roDC and ZeroDC filter-banks, respectively [5,6].…”

Section: Introductionmentioning

confidence: 99%

Spectral Folding And Two-Channel Filter-Banks On Arbitrary Graphs

Pavéz

Girault

Ortega

et al. 2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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In the past decade, several multi-resolution representation theories for graph signals have been proposed. Bipartite filter-banks stand out as the most natural extension of time domain filter-banks, in part because perfect reconstruction, orthogonality and bi-orthogonality conditions in the graph spectral domain resemble those for traditional filter-banks. Therefore, many of the well known orthogonal and bi-orthogonal designs can be easily adapted for graph signals.A major limitation is that this framework can only be applied to the normalized Laplacian of bipartite graphs. In this paper we extend this theory to arbitrary graphs and positive semi-definite variation operators. Our approach is based on a different definition of the graph Fourier transform (GFT), where orthogonality is defined with respect to the Q inner product. We construct GFTs satisfying a spectral folding property, which allows us to easily construct orthogonal and bi-orthogonal perfect reconstruction filter-banks. We illustrate signal representation and computational efficiency of our filter-banks on 3D point clouds with hundreds of thousands of points.

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Graph Vertex Sampling with Arbitrary Graph Signal Hilbert Spaces

Cited by 12 publications

References 16 publications

Two Channel Filter Banks on Arbitrary Graphs with Positive Semi Definite Variation Operators

Two Channel Filter Banks on Arbitrary Graphs with Positive Semi Definite Variation Operators

Ranking Deep Learning Generalization using Label Variation in Latent Geometry Graphs

Spectral Folding And Two-Channel Filter-Banks On Arbitrary Graphs

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