Characterization and inference of weighted graph topologies from observations of diffused signals

Transcription

1 Characterization and inference of weighted graph topologies from observations of diffused signals Bastien Pasdeloup*, Vincent Gripon*, Grégoire Mercier*, Dominique Pastor*, Michael G. Rabbat** * name.surname@telecom-bretagne.eu ** name.surname@mcgill.ca Preprint version of is work available on arxiv at Graph Signal Processing Workshop, Philadelphia May 26, 2016

2 Motivations GSP tools Filtering Clustering Data analysis... 1

3 Motivations GSP tools Filtering Clustering Data analysis... 1

4 Motivations Graph inference GSP tools Filtering Clustering Data analysis... 1

5 Outline Problem formulation A brief overview of related work Characterization of e space of admissible topologies Selecting a point in is space: two example approaches A few results Conclusions & Future work Bastien Pasdeloup May 26, 2016 Philadelphia

6 Outline Problem formulation A brief overview of related work Characterization of e space of admissible topologies Selecting a point in is space: two example approaches A few results Conclusions & Future work Bastien Pasdeloup May 26, 2016 Philadelphia

7 General idea of our work General idea: Signals observed on a graph have common properties. These properties should be explainable by e graph topology. Translation: Given a matrix X of signals, ere should exist independent signals Y such at X is issued from e evolution of Y on e graph. 2

8 General idea of our work General idea: Signals observed on a graph have common properties. These properties should be explainable by e graph topology. Translation: Given a matrix X of signals, ere should exist independent signals Y such at X is issued from e evolution of Y on e graph. Algorim 2

9 Context of our work General idea: Signals observed on a graph have common properties. These properties should be explainable by e graph topology. Translation: Given a matrix X of signals, ere should exist independent signals Y such at X is issued from e evolution of Y on e graph. Algorim Considering heat propagation, 2

10 Outline Problem formulation A brief overview of related work Characterization of e space of admissible topologies Selecting a point in is space: two example approaches A few results Conclusions & Future work Bastien Pasdeloup May 26, 2016 Philadelphia

11 Covariance resholding General idea: Covariance matrix is a good tool to model relations among nodes. Some of its entries are spurious, let us remove em using a reshold. Drawbacks: Need for a proper reshold. Does not eliminate all indirect covariances. 3

12 Graphical lasso based meods General idea: Sparse inverse of e covariance matrix removes e partial correlations. Drawbacks: Existence of negative entries in e result matrix. More a functional approach an a structural one. Sample covariance matrix Regularization parameter for sparsity Friedman, Hastie, Tibshirani, Sparse inverse covariance estimation wi e graphical lasso, Biostatistics, Dempster, Covariance Selection, Biometrics,

13 Smooness based meods General idea: The observed signals should be smoo on e graph. Drawbacks: Seems legit, alough e objective is different from ours. Work from Dong et. al will serve as a baseline for comparisons. Dong, Thanou, Frossard, Vandergheynst, Learning Laplacian matrix in smoo graph signal representations, arxiv, Kalofolias, How to learn a graph from smoo signals, AISTATS,

14 Diffusion based meods General idea: First version of our work, where. Drawbacks: Restrictive case: K is supposed to be constant and known. Study was only made for an infinite number of signals. Pasdeloup, Rabbat, Gripon, Pastor, Mercier, Graph reconstruction from e observation of diffused signals, Allerton,

15 Diffusion based meods General idea: First version of our work, where. Drawbacks: Restrictive case: K is supposed to be constant and known. Study was only made for an infinite number of signals. Pasdeloup, Rabbat, Gripon, Pastor, Mercier, Graph reconstruction from e observation of diffused signals, Allerton,

16 Diffusion based meods General idea: First version of our work, where. Drawbacks: Restrictive case: K is supposed to be constant and known. Study was only made for an infinite number of signals. Pasdeloup, Rabbat, Gripon, Pastor, Mercier, Graph reconstruction from e observation of diffused signals, Allerton,

17 Diffusion based meods General idea: First version of our work, where. Drawbacks: Restrictive case: K is supposed to be constant and known. Study was only made for an infinite number of signals. = Pasdeloup, Rabbat, Gripon, Pastor, Mercier, Graph reconstruction from e observation of diffused signals, Allerton,

18 Diffusion based meods General idea: First version of our work, where. Drawbacks: Restrictive case: K is supposed to be constant and known. Study was only made for an infinite number of signals. wi S diagonal matrix of signs (found by solving an optimization problem) Pasdeloup, Rabbat, Gripon, Pastor, Mercier, Graph reconstruction from e observation of diffused signals, Allerton,

19 Outline Problem formulation A brief overview of related work Characterization of e space of admissible topologies Selecting a point in is space: two example approaches A few results Conclusions & Future work Bastien Pasdeloup May 26, 2016 Philadelphia

20 What can we do now? Problem: Previous meod cannot be applied anymore, now at are unknown. Algorim 7

21 What can we do now? Problem: Previous meod cannot be applied anymore, now at are unknown. Algorim Important part of e slide :) 7

22 Assumption on e matrix to recover Current situation: We have e eigenvectors Assumption:, let us find e eigenvalues. should be positive to correspond to a valid graph Laplacian. 8

23 Example for a 3x3 random matrix Constraints enforcing positivity Admissible set of eigenvalues Actual eigenvalues of e matrix used Remark: We have set (eigenvalue 0 of e Laplacian) 9

24 Outline Problem formulation A brief overview of related work Characterization of e space of admissible topologies Selecting a point in is space: two example approaches A few results Conclusions & Future work Bastien Pasdeloup May 26, 2016 Philadelphia

25 Which point to choose in is admissible space? Enforcing simplicity: Recovers a matrix wi a null diagonal. Maximises e exchanges between nodes (no conservation). Enforcing sparsity: Common desired property for a graph. Oers: Binary entries, distance to a reference graph, smooness 10

26 Recovering a simple graph 11

27 Recovering a simple graph Simplex meod 11

28 Recovering a sparse graph Find e point at reaches most constraints Equivalent to minimizing e norm 12

29 Recovering a sparse graph Find e point at reaches most constraints Equivalent to minimizing e norm Classical proxy: Minimize e norm 12

30 Outline Problem formulation A brief overview of related work Characterization of e space of admissible topologies Selecting a point in is space: two example approaches A few results Conclusions & Future work Bastien Pasdeloup May 26, 2016 Philadelphia

31 Experimental assessment of e meods Erdős-Rényi random model Random geometric model Algorim Erdős, Rényi, On random graphs I, Publ Ma. Debrecen,

32 Error measurements Mean Error Per Reconstructed Entry: Reconstruction Error of e Retreived Eigenvalues: Reconstruction Error of e Powered Retreived Eigenvalues: 14

33 Error measurements Mean Error Per Reconstructed Entry: Reconstruction Error of e Retreived Eigenvalues: Reconstruction Error of e Powered Retreived Eigenvalues: 14

34 Results when using e groundtru eigenvectors 15

35 Results when using e groundtru eigenvectors 15

36 Results when e number of signals is limited Impact on e polytope: Perfect retrieval of e eigenvectors is not possible. PCA provides a good estimator for em. Pearson, On lines and planes of closest fit to system of points in space, Philosophical Magazine,

37 Results when e number of signals is limited Impact on e polytope: Perfect retrieval of e eigenvectors is not possible. PCA provides a good estimator for em. M = 10 Pearson, On lines and planes of closest fit to system of points in space, Philosophical Magazine,

38 Results when e number of signals is limited Impact on e polytope: Perfect retrieval of e eigenvectors is not possible. PCA provides a good estimator for em. M = 100 Pearson, On lines and planes of closest fit to system of points in space, Philosophical Magazine,

39 Results when e number of signals is limited Impact on e polytope: Perfect retrieval of e eigenvectors is not possible. PCA provides a good estimator for em. M = 1000 Pearson, On lines and planes of closest fit to system of points in space, Philosophical Magazine,

40 Convergence towards e actual eigenvectors 17

41 Application to audio signals 44,100 Hz N = 50 18

42 Outline Problem formulation A brief overview of related work Characterization of e space of admissible topologies Selecting a point in is space: two example approaches A few results Conclusions & Future work Bastien Pasdeloup May 26, 2016 Philadelphia

43 Summary of e talk 19

44 Perspectives Extensions of is work: Smooness criterion for e selection of a point? Binarity citerion? Reduce e number of constraints (currently Ongoing and close future work: )! Graph reconstruction for enhanced separability of classes of signals. 20

45 Characterization and inference of weighted graph topologies from observations of diffused signals Bastien Pasdeloup*, Vincent Gripon*, Grégoire Mercier*, Dominique Pastor*, Michael G. Rabbat** * name.surname@telecom-bretagne.eu ** name.surname@mcgill.ca Graph Signal Processing Workshop, Philadelphia May 26, 2016