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Global and local uncertainty principles for signals on graphs

  • Nathanael Perraudin (a1), Benjamin Ricaud (a2), David I Shuman (a3) and Pierre Vandergheynst (a2)
Abstract

Uncertainty principles such as Heisenberg's provide limits on the time-frequency concentration of a signal, and constitute an important theoretical tool for designing linear signal transforms. Generalizations of such principles to the graph setting can inform dictionary design, lead to algorithms for reconstructing missing information via sparse representations, and yield new graph analysis tools. While previous work has focused on generalizing notions of spreads of graph signals in the vertex and graph spectral domains, our approach generalizes the methods of Lieb in order to develop uncertainty principles that provide limits on the concentration of the analysis coefficients of any graph signal under a dictionary transform. One challenge we highlight is that the local structure in a small region of an inhomogeneous graph can drastically affect the uncertainty bounds, limiting the information provided by global uncertainty principles. Accordingly, we suggest new notions of locality, and develop local uncertainty principles that bound the concentration of the analysis coefficients of each atom of a localized graph spectral filter frame in terms of quantities that depend on the local structure of the graph around the atom's center vertex. Finally, we demonstrate how our proposed local uncertainty measures can improve the random sampling of graph signals.

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Copyright
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Corresponding author
Corresponding author: Nathanael Perraudin Email: nathanael.perraudin@epfl.ch
References
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[1]Shuman, D.I.; Narang, S.K.; Frossard, P.; Ortega, A.; Vandergheynst, P.: The emerging field of signal processing on graphs: extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Process. Mag., 30(3) (2013), 8398.
[2]Sandryhaila, A.; Moura, J.; M.F.: Discrete signal processing on graphs: Frequency analysis. IEEE. Trans. Signal Process., 62(12) (2014), 30423054.
[3]Crovella, M.; Kolaczyk, E.: Graph wavelets for spatial traffic analysis, in Proc. IEEE INFOCOM, vol. 3, March 2003, 18481857, San Francisco, CA, USA.
[4]Maggioni, M.; Bremer, J.C.; Coifman, R.R.; Szlam, A.D.: Biorthogonal diffusion wavelets for multiscale representations on manifolds and graphs, in Proc. SPIE Wavelet XI, vol. 914, September 2005.
[5]Szlam, A.D.; Maggioni, M.; Coifman, R.R.; Bremer, J.C. Jr.: Diffusion-driven multiscale analysis on manifolds and graphs: top-down and bottom-up constructions, in Proc. SPIE Wavelets, vol. 5914, August 2005, 445455.
[6]Coifman, R.R.; Maggioni, M.: Diffusion wavelets. Appl. Comput. Harmon. Anal., 21(1) (2006), 5394.
[7]Bremer, J.C.; Coifman, R.R.; Maggioni, M.; Szlam, A.D.: Diffusion wavelet packets. Appl. Comput. Harmon. Anal., 21(1) (2006), 95112.
[8]Lafon, S.; Lee, A.B.: Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning, and data set parameterization. IEEE Trans. Pattern Anal. Mach. Intell., 28(9) (2006), 13931403.
[9]Wang, W.; Ramchandran, K.: Random multiresolution representations for arbitrary sensor network graphs, in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing vol. 4, May 2006, 161164, Toulouse, France.
[10]Narang, S.K.; Ortega, A.: Lifting based wavelet transforms on graphs, in Proc. APSIPA ASC, Sapporo, Japan, October 2009, 441444.
[11]Jansen, M.; Nason, G.P.; Silverman, B.W.: Multiscale methods for data on graphs and irregular multidimensional situations. J. R. Stat. Soc. Ser. B Stat. Methodol., 71(1) (2009), 97125.
[12]Gavish, M.; Nadler, B.; Coifman, R.R.: Multiscale wavelets on trees, graphs and high dimensional data: Theory and applications to semi supervised learning, in Proc. Int. Conf. Mach. Learn., Haifa, Israel, June 2010, 367374.
[13]Hammond, D.K.; Vandergheynst, P.; Gribonval, R.: Wavelets on graphs via spectral graph theory. Appl. Comput. Harmon. Anal., 30(2) (2011), 129150.
[14]Ram, I.; Elad, M.; Cohen, I.: Generalized tree-based wavelet transform. IEEE Trans. Signal Process., 59(9) (2011), 41994209.
[15]Narang, S.K.; Ortega, A.: Perfect reconstruction two-channel wavelet filter-banks for graph structured data. IEEE. Trans. Signal Process., 60(6) (2012), 27862799.
[16]Leonardi, N.; Van De Ville, D.: Tight wavelet frames on multislice graphs. IEEE Trans. Signal Process., 61(13) (2013), 33573367.
[17]Ekambaram, V.N.; Fanti, G.C.; Ayazifar, B.; Ramchandran, K.: Critically-sampled perfect-reconstruction spline-wavelet filter banks for graph signals, in Proc. Global Conf. Signal and Information Processing, Austin, TX, December 2013, 475478.
[18]Narang, S.K.; Ortega, A.: Compact support biorthogonal wavelet filterbanks for arbitrary undirected graphs. IEEE Trans. Signal Process., 61(19) (2013), 46734685.
[19]Liu, P.; Wang, X.; Gu, Y.: Coarsening graph signal with spectral invariance, in Proc. IEEE Int. Conf. Accoustics, Speech, and Signal Process., Florence, Italy, May 2014, 10701074.
[20]Sakiyama, A.; Tanaka, Y.: Oversampled graph Laplacian matrix for graph filter banks. IEEE Trans. Signal Process., 62(24) (2014), 64256437.
[21]Nguyen, H.Q.; Do, M.N.: Downsampling of signals on graphs via maximum spanning trees. IEEE Trans. Signal Process., 63(1) (2015), 182191.
[22]Shuman, D.I.; Wiesmeyr, C.; Holighaus, N.; Vandergheynst, P.: Spectrum-adapted tight graph wavelet and vertex-frequency frames. IEEE Trans. Signal Process., 63(16) (2015), 42234235.
[23]Shuman, D.I.; Ricaud, B.; Vandergheynst, P.: Vertex-frequency analysis on graphs. Appl. Comput. Harmon. Anal., 40(2) (2016), 260291.
[24]Shuman, D.I.; Faraji, M.; Vandergheynst, P.: A multiscale pyramid transform for graph signals. IEEE. Trans. Signal Process., vol. 64, 2016, 21192134.
[25]Matolcsi, T.; Szücs, J.: Intersections Des Mesures Spectrales Conjugées, vol. 277, CR Acad. Sci., Paris, 1973, 841843.
[26]Donoho, D.L.; Stark, P.B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math, 49(3) (1989), 906931.
[27]Donoho, D.L.; Huo, X.: Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inf. Theory, 47(7) (2001), 28452862.
[28]Elad, M.; Bruckstein, A.M.: A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Trans. Inf. Theory, 48(9) (2002), 25582567.
[29]Gribonval, R.; Nielsen, M.: Sparse representations in unions of bases. IEEE Trans. Inf. Theory, 49(12) (2003), 33203325.
[30]Candes, E.J.; Romberg, J.: Quantitative robust uncertainty principles and optimally sparse decompositions. Found. Comput. Math., 6(2) (2006), 227254.
[31]Ghobber, S.; Jaming, P.: On uncertainty principles in the finite dimensional setting. Linear Algebra Appl., 435(4) (2011), 751768.
[32]Ricaud, B.; Torrésani, B.: Refined support and entropic uncertainty inequalities. IEEE Trans. Inf. Theory, 59(7) (2013), 42724279.
[33]Ricaud, B.; Shuman, D.I.; Vandergheynst, P.: n the sparsity of wavelet coefficients for signals on graphs, in SPIE Wavelets and Sparsity, San Diego, California, August 2013.
[34]McGraw, P.N.; Menzinger, M.: Laplacian spectra as a diagnostic tool for network structure and dynamics. Phys. Rev. E, 77(3) (2008), 031102-1–031102-14.
[35]Saito, N.; Woei, E.: On the phase transition phenomenon of graph Laplacian eigenfunctions on trees. RIMS Kokyuroku, vol. 1743 (2011), 7790.
[36]Folland, G.; Sitaram, A.: The uncertainty principle: A mathematical survey. J. Fourier Anal. Appl., 3(3) (1997), 207238.
[37]Mallat, S.G.: A Wavelet Tour of Signal Processing: the sparse way, 3rd ed. Academic Press, 2008.
[38]Agaskar, A.; Lu, Y.M.: An uncertainty principle for functions defined on graphs, in Proc. SPIE, vol. 8138, San Diego, CA, August 2011, 81380T-1–81380T-11.
[39]Agaskar, A.; Lu, Y.M.: Uncertainty principles for signals defined on graphs: bounds and characterizations, in Proc. IEEE Int. Conf. Acc., Speech, and Signal Process., Kyoto, Japan, March 2012, 34933496.
[40]Agaskar, A.; Lu, Y.M.: A spectral graph uncertainty principle. IEEE Trans. Inf. Theory, 59(7) (2013), 43384356.
[41]Pasdeloup, B.; Alami, R.; Gripon, V.; Rabbat, M.: Toward an uncertainty principle for weighted graphs, in Proc. Eur. Signal Process. Conf. (EUSIPCO), August 2015, 14961500, Nice, France.
[42]Tsitsvero, M.; Barbarossa, S.; Di Lorenzo, P.: Signals on graphs: Uncertainty principle and sampling. IEEE Trans. Signal Process., 64(18) (2016), 48454860.
[43]Slepian, D.; Pollak, H.O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty. Bell System Tech. J., 40(1) (1961), 4363.
[44]Pesenson, I.Z.: Sampling solutions of schrödinger equations on combinatorial graphs, in IEEE 2015 Int. Conf. Sampling Theory and Applications (SampTA), 2015, 8285, Washington, DC, USA.
[45]Maassen, H.; Uffink, J.: Generalized entropic uncertainty relations. Phys. Rev. Lett., 60(12) (1988), 11031106.
[46]Reed, M.; Simon, B.: Methods of Modern Mathematical Physics, Vol. 2.: Fourier Analysis, Self-Adjointness, Academic Press, 1975.
[47]Grady, L.J.; Polimeni, J.R.: Discrete Calculus, in Applied Analysis on Graphs for Computational Science, Springer-Verlag London, 2010.
[48]Lieb, E.H.: Integral bounds for radar ambiguity functions and Wigner distributions. J. Math. Phys., 31(3) (1990), 594.
[49]Sandryhaila, A.; Moura, J.; M.F.: Discrete signal processing on graphs. IEEE. Trans. Signal Process., 61(7) (2013), 16441656.
[50]Chung, F.; R.K.: Spectral Graph Theory, in Vol. 92 of the CBMS Regional Conference Series in Mathematics. American Mathematical Society, 1997, pp. 212.
[51]Rényi, A.: On measures of entropy and information, in Proc. Fourth Berkeley Symp. Mathematics, Statistics and Probability, 1961, 547561, University of California Press.
[52]Ricaud, B.; Torrésani, B.: A survey of uncertainty principles and some signal processing applications. Adv. Comput. Math., 40(3) (2014), 629650.
[53]Dekel, Y.; Lee, J.R.; Linial, N.: Eigenvectors of random graphs: Nodal domains. Random Structures Algorithms, 39(1) (2011), 3958.
[54]Dumitriu, I.; Pal, S.: Sparse regular random graphs: Spectral density and eigenvectors. Ann. Probab., 40(5) (2012), 21972235.
[55]Tran, L.V.; Vu, V.H.; Wang, K.: Sparse random graphs: Eigenvalues and eigenvectors. Random Struct. Algo., 42(1) (2013), 110134.
[56]Brooks, S.; Lindenstrauss, E.: Non-localization of eigenfunctions on large regular graphs. Israel J. Math., 193(1) (2013), 114.
[57]Nakatsukasa, Y.; Saito, N.; Woei, E.: Mysteries around the graph Laplacian eigenvalue 4. Linear Algebra Appl., 438(8) (2013), 32313246.
[58]Beckner, W.: Inequalities in Fourier analysis. Ann. Math., 102(1) (1975), 159182.
[59]Gilbert, J.; Rzeszotnik, Z.: The norm of the {Fourier} transform on finite abelian groups. Ann. Inst. Fourier, 60(4) (2010), 13171346.
[60]Christensen, O.: Frames and Bases: An Introductory Course, in Applied and Numerical Harmonic Analysis, 2008, Birkhäuser Basel.
[61]Kovačević, J.; Chebira, A.: Life beyond bases: The advent of frames (part I). IEEE Signal Process. Mag., 24(4) (2007), 86104.
[62]Kovačević, J.; Chebira, A.: Life beyond bases: The advent of frames (part II). IEEE Signal Process. Mag., 24(5) (2007), 115125.
[63]Metzger, B.; Stollmann, P.: Heat kernel estimates on weighted graphs. Bull. London Math. Soc., 32(4) (2000), 477483.
[64]Leonardi, N.; Van De Ville, D.: Wavelet frames on graphs defined by FMRI functional connectivity, in Proc. IEEE Int. Symp. Biomed. Imag., Chicago, IL, March 2011, 21362139.
[65]Thanou, D.; Shuman, D.I.; Frossard, P.: Learning parametric dictionaries for signals on graphs. IEEE. Trans. Signal Process., 62(15) (2014), 38493862.
[66]Feichtinger, H.; Onchis-Moaca, D.; Ricaud, B.; Torrésani, B.; Wiesmeyr, C.: A method for optimizing the ambiguity function concentration, in Proc. Eur. Signal Processing Conf. (EUSIPCO), August 2012, 804808, Bucharest, Romania.
[67]Perraudin, N.; Vandergheynst, P.: Stationary signal processing on graphs. IEEE. Trans. Signal Process., 65(13) (2017), 34623477.
[68]Gadde, A.; Ortega, A.: A probabilistic interpretation of sampling theory of graph signals, in Proc. IEEE Int. Conf. Acc., Speech, and Signal Processing, April 2015, 32573261, Brisbane, QLD, Australia.
[69]Zhang, C.; Florêncio, D.; Chou, P.A.: Graph signal processing–a probabilistic framework. Microsoft Res., Redmond, WA, USA, Technical Report MSR-TR-2015-31, 2015.
[70]Pesenson, I.: Variational splines and Paley-Wiener spaces on combinatorial graphs. Constr. Approx., 29(1) (2009), 121.
[71]Perraudin, N.; Shuman, D.; Puy, G.; Vandergheynst, P.: UNLocBoX: a matlab convex optimization toolbox using proximal splitting methods, 2014. ArXiv preprint arXiv:1402.0779.
[72]Perraudin, N.; Paratte, J.; Shuman, D.; Kalofolias, V.; Vandergheynst, P.; Hammond, D.K.: GSPBOX: a toolbox for signal processing on graphs, 2014. ArXiv preprint arXiv:1408.5781.
[73]Puy, G.; Tremblay, N.; Gribonval, R.; Vandergheynst, P.: Random sampling of bandlimited signals on graphs. Appl. Comput. Harmon. Anal., vol. 44, 2018, 446475.
[74]Chen, S.; Varma, R.; Singh, A.; Kovacević, J.: Signal recovery on graphs: Random versus experimentally designed sampling, in IEEE Int. Conf. Sampling Theory and Applications (SampTA) 2015, 2015, 337341, Washington, DC, USA.
[75]Anis, A.; Gadde, A.; Ortega, A.: Efficient sampling set selection for bandlimited graph signals using graph spectral proxies. IEEE Trans. Signal Process., 64(14) (2016), 37753789.
[76]Chen, S.; Varma, R.; Singh, A.; Kova{č}ević, J.: Signal recovery on graphs: Fundamental limits of sampling strategies. IEEE Trans. Signal Info. Process. Networks, 2(4) (2016), 539554.
[77]Strang, G.: The discrete cosine transform. SIAM Review, 41(1) (1999), 135147.
[78]Pinsky, M.A.: Introduction to Fourier Analysis and Wavelets, Vol. 102 of the Graduate Studies in Mathematics, Am. Math. Soci., 2002.
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