MAT 280 Harmonic Analysis on Graphs & Networks Reference Page (Fall 2019)
The general introductory references For general introduction to graphs and networks and significant applications:
F. Chung and L. Lu: Complex Graphs and Networks ,
No. 107 in CBMS Regional Conference Series in Mathematics, Amer. Math.
Soc., 2006.
D. Easley and J. Kleinberg: Networks, Crowds, and Markets: Reasoning and a Highly Connected World , Cambridge Univ. Press, 2010.
L. Lovász: Large Networks and Graph Limits , Colloquium Publications, Vol. 60, Amer. Math. Soc., 2012.
M. Newman: Networks , 2nd Ed., Oxford Univ. Press, 2018.
For Laplacians on graphs:
D. Cvetković, P. Rowlinson, and S. Simić: An Introduction to the Theory of Graph Spectra , Vol. 75, London Mathematical Society Student Texts, Cambridge Univ. Press, 2010.
F. R. K. Chung: Spectral Graph Theory , AMS, 1997. A. E. Brouwer and W. H. Haemers: Spectra of Graphs ,
Springer, 2012. R. B. Bapat: Graphs and Matrices , 2nd Ed., Universitext, Springer, 2014.
D. Spielman: "Spectral graph theory," in Combinatorial Scientific Computing (O. Schenk, ed.), Chap.18, pp.495-524, CRC Press, 2012.
For Laplacians on Euclidean domains:
W. A. Strauss: Partial Differential Equations: An Introduction , 2nd Ed., Chap. 10 & 11, John Wiley & Sons, 2008. R. Courant and D. Hilbert: Methods of Mathematical Physics, Vol. I , Chap. V, VI, & VII, First English Edition, John Wiley and Sons, 1953. Republished as Wiley Classics Library in 1989.
For Laplacians on Riemannian manifolds:
For Graph Signal Processing:
For Spectral Clustering:
Lecture 1: Overture: Why Graphs & Networks? Our Course Plan
Some references sited in this lecture are:
F. Chung and L. Lu: Complex Graphs and Networks , Chap. 1 , Amer. Math. Soc., 2006.
E. Bullmore and O. Sporns: Complex brain networks: graph theoretical analysis of structural and functional systems, Nature Reviews Neuroscience , vol. 10, pp. 186-198, 2009.
N. Saito and E. Woei: Analysis of neuronal dendrite patterns using eigenvalues of graph Laplacians, Japan SIAM Letters , vol. 1, pp. 13-16, 2009.
A. Buades, B. Coll, and J. M. Morel: Image denoising methods. A new nonlocal principle. SIAM Review , vol. 52, no. 1, pp. 113-147, 2010.
A. Singer, Y. Shkolnisky, and B. Nadler: Diffusion interpretation of nonlocal neighborhood filters for signal denoising, SIAM J. Imaging Sciences , vol. 2, no. 1, pp. 118-139, 2009.
S. I. Daitch, J. A. Kelner, and D. A. Spielman: Fitting a Graph to Vector Data, Proc. 26th Intern. Conf. Machine Learning , pp. 201-208, 2009.
Lecture 2: Prelude to Harmonic Analysis on Graphs:
Laplacian Eigenfunctions on General Shape Domains in R d
History of Laplacian eigenvalue problems, in particular, prehistory of
spectral geometry can be found in
M. Kac: "Can one hear the shape of a drum?" Amer. Math. Monthly , vol. 73, no. 4, part 2, pp. 1-23, 1966. M. H. Protter:
"Can one hear the shape of a drum? Revisited" SIAM Review , vol. 29, no. 2, pp. 185-197, 1987.
For the derivation of the volume of the unit ball and the surface area of the unit
sphere in R d G. B. Folland: Introduction to Partial Differential Equations ,
2nd Ed., Princeton Univ. Press, 1995.
For further information about the applications of my integral operator
commuting with the Laplacian, see, e.g.,
M. F. Beg, P. R. Raamana, S. Barbieri, and L. Wang: "Comparison of four shape features for detecting hippocampal shape changes in early Alzheimer's," Stat Methods in Medical Research , vol. 22, no. 4, pp. 439-462, 2013. T. DelSole and M. K. Tippett: "Laplacian Eigenfunctions for Climate Analysis," Journal of Climate , vol. 28, no. 18, pp. 7420-7436, 2015.
The fast algorithms for the Laplacian eigenvalues and eigenfunctions
via commuting integral operators and the Fast Multipole Methods, see
Allen Xue's Ph.D. dissertation.
Of course, it is indispensable to check the following ground breaking papers:
V. Rokhlin: "Rapid solution of integral equations of classical potential theory," J. Comput. Phys. , vol. 60, no. 2, pp. 187-207, 1985.
L. Greengard and V. Rokhlin: "A fast algorithm for particle simulations," J. Comput. Phys. , vol. 73, no. 2, pp. 325-348, 1987. S. Chandrasekaran, M. Gu, and T. Pals: "A fast ULV decomposition solver for hierarchically semiseparable representations," SIAM J. Matrix Anal. Appl. , vol. 28, no. 3, pp. 603-622, 2006.
Lecture 3: Basics of Graph Theory: Graph Laplacians
This lecture is based on the following material:
J. H. van Lint and R. M. Wilson: A Course in Combinatorics , 2nd Ed.,
Chap. 31, Cambridge Univ. Press, 2001.
H. Urakawa: "Spectral geometry and graph theory," Ouyou Suuri (Bulletin of the Japan Society of Industrial and Applied Mathematics) , vol. 12, no. 1, pp.29-45, 2002.
F. R. K. Chung: Spectral Graph Theory , Sec. 1.1-1.3, AMS, 1997.
R. B. Bapat: Graphs and Matrices , 2nd Ed., Universitext, Chap.1-4, Springer, 2014. U. von Luxburg: "A tutorial on spectral clustering," Statistics and Computing ,
vol. 17, no. 4, pp.395-416, 2007.
I explained that the graph Laplacian of a simple path graph is the same as the matrix used in the Discrete Cosine Transform Type II (DCT-II).
More precisely, the eigenvectors of that graph Laplacian are the DCT-II
basis vectors. Subtle changes in the discrete boundary condition generates
matrices whose eigenvectors are DCT-I, III, IV, ... More about this interesting
relationship between the eigenvectors of discrete boundary value problems
and the various types of DCT and Discrete Sine Transform (DST) can be
found in
G. Strang: "The discrete cosine transform," SIAM Review , vol. 41, no. 1, pp. 135-147, 1999.
The other references that you may find interesting or that we touched upon in this lecture are:
J. Dodziuk: "Difference equations, isoperimetric inequality and transience of certain random walks," Trans. Amer. Math. Soc. , vol. 284, no. 2. pp. 787-794, 1984. N. Masuda, M. A. Porter, and R. Lambiotte: "Random walks and diffusion on networks," Physics Report , vol. 716-717, pp. 1-58, 2017.
E. Deza and M. M. Deza: Encyclopedia of Distances , 4th Ed., Springer, 2016. L. Lovász: "Discrete and continuous: two sides of the same?" Geom. Funct. Anal. , Special Volume, Part I, pp. 359–382, 2000. L. Lovász: "Discrete analytic functions: an exposition," Surv. Differ. Geom. , vol. 9, pp.241-273, 2004. I. Benjiamini and L. Lovász: "Harmonic and analytic functions on graphs," J. Geom. , vol. 76, no. 1-2, pp. 3-15, 2003. I. Benjiamini and L. Lovász: "Global information from local observation," Proc. 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS) , pp. 701-710, 2002.
Lecture 4: Graph Laplacian Eigenvalues
Books by Cvetković et al., Chung, Bapat, Brouwer and W. H. Haemers
contain relevant chapters on graph Laplacian eigenvalues.
But there are many good survey papers on graph Laplacian eigenvalues. Here are
some representative ones.
R. Merris: "Laplacian matrices of graphs: A survey," Lin. Alg. Appl. , vol. 197/198, pp. 143-176, 1994. R. Merris: "A survey of graph Laplacians," Linear and Multilinear Algebra , vol. 39, pp. 19-31, 1995. R. Grone, R. Merris, and V. S. Sunder: "The Laplacian spectrum of a graph," SIAM J. Matrix Anal. , vol. 11, no. 2, pp. 218-238, 1990. R. Grone and R. Merris: "The Laplacian spectrum of a graph II," SIAM J. Discrete Math. , vol. 7, no. 2, pp. 221-229, 1994. B. Mohar: "Laplace eigenvalues of graphs---A survey," Discrete Math. , vol. 109, pp. 171-183, 1992.
The books on human vision I referred to during my lecture are:
D. Hubel: Eye, Brain, and Vision , Scientific American Library, 1995.
D. Marr: Vision: A Computational Investigation into the Human Representation and Processing of Visual Information , W. H. Freeman & Co., 1982, republished by The MIT Press, 2010.
Lecture 5: Graph Laplacian Eigenvalues II
On isospectrality and spectral characterization of graphs:
A. E. Brouwer and W. H. Haemers: Spectra of Graphs , Chap. 14, Springer, 2012. J. Tan: "On isospectral graphs," Interdisciplinary Information Sciences ,
vol. 4, no. 2, pp. 117-124, 1998. H. Fujii and A. Katsuda: "Isospectral graphs and isoperimetric constants," Discrete Math. , vol. 207, pp. 33-52, 1999. G. R. Omidi and K. Tajbakhsh: "Starlike trees are determined by their Laplacian spectrum," Lin. Alg. Appl. , vol. 422, pp. 654-658, 2007. R. Boulet: "The centipede is determined by its Laplacian spectrum," C. R. Acad. Sci. Paris, Ser. I , vol. 346, no. 13-14, pp. 711-716, 2008.
For our analysis on dendritic trees including the phase transition phenomena and morphological feature extraction from dendritic trees using their graph
Laplacians, see:
N. Saito and E. Woei: "Analysis of neuronal dendrite patterns using eigenvalues of graph Laplacians," Japan SIAM Letters , vol. 1, pp. 13-16, 2009.
Y. Nakatsukasa, N. Saito, and E. Woei: "Mysteries around graph Laplacian eigenvalue 4," Lin. Alg. Appl. , vol. 438, no. 8, pp.3231-3246, 2013. N. Saito and E. Woei: "Tree simplification and the 'plateaux' phenomenon of graph Laplacian eigenvalues," Lin. Alg. Appl. , vol. 481, pp. 263-279, 2015.
Lecture 6: Graph Laplacian Eigenfunctions
The basic reference here is:
T. Bıyıkoğlu, J. Leydold, and P. F. Stadler: Laplacian Eigenvectors of Graphs , Lecture Notes in Mathematics, vol. 1915, Springer, 2007.
R. Merris: "Laplacian graph eigenvectors," Lin. Alg. Appl. , vol. 278, pp. 221-236, 1998.
For the classical nodal domain theorem in Euclidean domains, see:
R. Courant: "Ein allgemeiner Satz zur Theorie der Eigenfunktionen selbstadjungierter Differentialausdrücke," Nachr. Ges. Göttingen (math.-phys. Kl.) ,
pp. 81-85, 1923. W. A. Strauss: Partial Differential Equations: An Introduction ,
2nd Ed., Sec. 10.4, John Wiley & Sons, 2008.
R. Courant and D. Hilbert: Methods of Mathematical Physics , Vol. I,
Sec. V.5, VI.6, Wiley-Interscience, 1953.
N. Saito: MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations,
Lecture 7: Nodal Sets , Spring 2007.
For more about Ernst Chladni and the Chladni Plates, see:
H.-J. Stöckmann:
"Chladni meets Napoleon," The European Physical Journal - Special Topics ,
vol. 145, no. 1, pp. 15-23, 2007.
http://www.cymascope.com/cyma_research/history.html .Wikipedia contributors, "Cymatics," Wikipedia, The Free Encyclopedia, 2019. Cymatics: Chladni Plate - Sound, Vibration and Sand, YouTube video, Nov. 12, 2014.
For the Krein-Rutman theorem (the Perron-Frobenius theorem for compact
operators) that I briefly mentioned in the lecture, see:
Y. Du: Order structure and topological methods in nonlinear partial differential equations. Vol. 1. , Chap. 1 , World Scientific Pub. Co., 2006.
Lecture 7: Graph Laplacian Eigenfunctions II: Spectral Clustering The normalized cut and its application to image segmentation started with
the following ground-breaking paper:
J. Shi and J. Malik: "Normalized cuts and image segmentation,"
IEEE Trans. Pattern Anal. Machine Intell. , vol. 22, no. 8, pp. 888-905,
2000.
Y. Weiss: "Segmentation using eigenvectors: a unifying view,"
in Proc. the Seventh IEEE International Conference on Computer Vision ,
vol. 2, pp. 975-982, 1999.
R. Kannan, S. Vempala, and A. Vetta: "On clusterings: good, bad and spectral,"
J. Assoc. Comput. Mach. vol. 51 no. 3, pp. 497-515, 2004.
Lecture 8: Graph Laplacian Eigenfunctions III: Analysis of Localization/Phase Transition Phenomena
This lecture is based on our own paper:
Y. Nakatsukasa, N. Saito, and E. Woei: "Mysteries around the graph Laplacian eigenvalue 4,"
Lin. Alg. Appl. , vol.438, pp.3231-3246, 2013. N. Saito and E. Woei: "Tree simplification and the 'plateaux' phenomenon of graph Laplacian eigenvalues," Lin. Alg. Appl. , vol.481, pp.263-279, 2015.
The following papers were also referred to in the lecture:
K. C. Das: "Some spectral properties of the Laplacian matrix
of starlike trees," Italian J. Pure Appl. Math. , vol. 21, pp. 197-210, 2007.
R. Grone and R. Merris: "The Laplacian spectrum of a graph II," SIAM J. Discrete Math. , vol. 7, no. 2, pp. 221-229, 1994.
J. M. Guo: "The k th Laplacian eigenvalue of a tree,"
J. Graph Theor. , vol. 54, no. 1, pp. 51-57, 2007.
R. L. Burden and G. W. Hedstrom: "The distribution of the eigenvalues of the discrete Laplacian," BIT , vol. 12, no. 4, pp. 475-488, 1972.
Lecture 9: Graph Construction from Given Datasets
This lecture refers to the following papers:
U. von Luxburg: "A tutorial on spectral clustering,"
Statistics and Computing , vol. 17, no. 4, pp.395-416, 2007. M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars: Computational Geometry: Algorithms and Applications , 3rd Ed., Chap. 9, Springer, 2008.
R. L. Graham and P. Hell: "On the history of the minimum spanning tree problem,"
Annals of the History of Computing , vol.7, no.1, pp.43-57, 1985.
S. I. Daitch, J. A. Kelner, and D. A. Spielman: "Fitting a graph to vector data,"
Proc. 26th Intern. Conf. Machine Learning , pp.201-208, 2009.
Lecture 10: Distances on Graphs
For the A* search algorithm to compute the
shortest path distances, see:
P. E. Hart, N. J. Nilsson, and B. Raphael: "A formal basis for the heuristic determination of minimum cost paths,"
IEEE Trans. Syst. Sci. Cybern. , vol. 4, no. 2, pp. 100-107, 1968. Wikipedia contributors: "A* search algorithm," Wikipedia, The Free Encyclopedia , accessed Oct. 29, 2019.
For the resistance and commute-time distances, see, e.g.,
F. Fouss, A. Pirotte, J.-M. Renders, and M. Saerens:
"Random-walk computation of similarities between nodes of a graph with application to collaborative recommendation,"
IEEE Trans. Knowledge and Data Engineering , vol. 19, no. 3, pp. 355-369, 2007.
U. von Luxburg: "A tutorial on spectral clustering,"
Statistics and Computing , vol. 17, no. 4, pp.395-416, 2007; in particular,
Sec. 6.
D. J. Klein and M. Randić: "Resistance distances," J. Math. Chem. ,
vol. 12, no. 1, pp. 81-95, 1993.
W. Xiao and I. Gutman: "Resistance distance and Laplacian spectrum,"
Theor. Chem. Acc. , vol. 110, no. 4, pp. 284-289, 2003.
V. Gurvich: "Metric and ultrametric spaces of resistances,"
Discr. Appl. Math. , vol. 158, no. 14, pp. 1496-1505, 2010.
R. B. Bapat: Graphs and Matrices , 2nd Ed., Universitext, Chap. 9 & 10, Springer, 2014.
For the excellent read on the random walks on graphs and networks, see:
P. G. Doyle and J. L. Snell: Random Walks and
Electric Networks , The Carus Mathematical Monographs, no. 22,
Math. Assoc. Amer., 1984. Also available via arXiv:math/0001057v1.
M. Meila and J. Shi: "Learning segmentation by random walks," in Advances in Neural Information Processing Systems 13 (T. K. Leen, T. G. Dietterich, and V. Tresp, eds.), pp.873-879, MIT Press, 2001. N. Masuda, M. A. Porter, and R. Lambiotte: "Random walks and diffusion on networks," Physics Report , vol. 716-717, pp. 1-58, 2017.
For a simple and useful explanation of the singular value decomposition, see, e.g., my MAT 167 lecture notes:
N. Saito: "Singular Value Decomposition (SVD) I", Lecture 13, MAT 167, Spring, 2017. N. Saito: "SVD II", Lecture 14, MAT 167, Spring, 2017. N. Saito: "More about SVD!", Lecture 15, MAT 167, Spring, 2017.
Lecture 11: Distances on Graphs II: Applications of Commute-Time Distances This lecture is based on the following paper:
Y. Deng, R. Wang, and Z. Zhang: "Commute time guided transformation for feature extraction," Computer Vision and Image Understanding , vol. 116, no. 4, pp. 473-483, 2012.
X. He, S. Yan, Y. Hu, P. Niyogi, H.-J. Zhang: "Face recognition using Laplacianfaces," IEEE Trans. Pattern Anal. Mach. Intell. , vol. 27 no. 3, pp. 328-340, 2005.
The following papers of my own may be useful for understanding PCA,
MDS, and the related embeddings/dimension reduction techniques:
N. Saito: "Image approximation and modeling via least statistically dependent bases," Pattern Recognition , vol. 34, no. 9, pp. 1765-1784, 2001.
L. Lieu and N. Saito: "Signal ensemble classification using low-dimensional embeddings and Earth Mover's Distance," in Wavelets and Multiscale Analysis: Theory and Applications (J. Cohen and A. I. Zayed, eds.), Chap.11, pp.227-256, Birkhäuser, 2011.
For construction of sparse graphs and their applications, see:
B. Cheng, J. Yang, S. Yan, Y. Fu, and T. S. Huang:
"Learning with ℓ1 -graph for image analysis,"
IEEE Trans. Image Process. , vol. 19 no. 4, pp. 858-866, 2010.
H. Cheng, Z. Liu, and J. Yang: "Sparsity induced similarity measure for label propagation," Proceedings of the International Conference on Computer Vision , 2009, pp. 317-324. C. Weaver and N. Saito: "Improving sparse representation-based classification using local principal component analysis," in Computational Intelligence for Pattern Recognition (W. Pedrycz and S.-M. Chen, eds.), Studies in Computational Intelligence, vol.777, pp.165-206, Springer, 2018.
For the relationship between SVD and PCA, see, e.g.,
N. Saito: "SVD, Least Squares Problems, and PCA", Lecture 17, MAT 167, Spring, 2017. N. Saito: "PCA & SVD", Lecture 18, MAT 167, Spring, 2017.
Lecture 12: Applications of Sparse Graphs: Signal Classification Using Sparse Representation The other key papers cited in our paper above are:
J. Wright, A. Y. Yang, A. Ganesh, S. S. Sastry, and Y. Ma : "Robust face recognition via sparse representation," IEEE Trans. Pattern Anal. Mach. Intell. , vol.31, no.2, pp.210-227, 2009. A.Singer and H.-T. Wu: "Vectordiffusion maps and the connection Laplacian," Commun. Pure Appl. Math. , vol.65, no.8, pp.1067-1144, 2012. D. L. Donoho and Y. Tsaig: "Fast solution of l 1 IEEE Trans. Inform. Theory , vol.54, no.11, pp.4789-4812, 2008.
Lecture 13: Distances on Graphs III: From Commute-Time Distance to Diffusion Distance For the diffusion distance, the main article is:
R. R. Coifman and S. Lafon: "Diffusion maps,"
Appl. Comput. Harmon. Anal. ,
vol. 21, no. 1, pp. 5-30, 2006.
diffusion geometry
and applied for hundreds of practical problems. Some of their early effort was
summarized in the following special issue of the journal, which include
the above article on the diffusion maps:
Appl. Comput. Harmon. Anal. , Special Issue on Diffusion Maps and Wavelets,
(D. Donoho and C. Chui, Eds.), vol. 21, no. 1, pp. 1-144, 2006.
Lecture 14: Dimension Reduction via PCA, Laplacian Eigenmaps, & Diffusion Maps For the comparison of various dimension reduction techniques, see, e.g.:
L. van der Maaten, E. Postma, and J. van den Herik:
"Dimensionality reduction: A comparative review," Tilburg University Technical Report , TiCC-TR 2009-005, 2009.
L. Lieu and N. Saito: "Signal ensemble classification using low-dimensional embeddings and Earth Mover's Distance," in Wavelets and Multiscale Analysis: Theory and Applications (J. Cohen and A. I. Zayed, eds.), Chap. 11, pp. 227-256, Birkhäuser, 2011.
For the geometric harmonics multiscale extensions, see:
S. Lafon, Y. Keller, R. R. Coifman:
"Data fusion and multicue data matching by diffusion maps,"
IEEE Trans. Pattern Anal. Machine Intell. , vol. 28, no. 11, pp. 1784-1797, 2006.
R. R. Coifman and S. Lafon: "Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions,"
Appl. Comput. Harmon. Anal. ,
vol. 21, no. 1, pp. 31-52, 2006.
Lecture 15: Applications of Dimension Reduction Techniques to Signal Ensemble Classification Earth Mover's Distance and its more general version
Wasserstein distances have become very popular in many applications. There are too many references on these distances. Below are just a few picks:
Y. Rubner, C. Tomasi, and L. J. Guibas: "The Earth Mover's Distance as a metric for image retrieval," Intern. J. Comput. Vision , vol. 40, no. 2, pp. 99-121, 2000.
S. Kolouri, S. R. Park, M. Thorpe, D. Slepcev, and G. K. Rohde: "Optimal Mass Transport: Signal processing and machine learning applications," IEEE Signal Processing Magazine , vol.34, no.4, pp.43-59, 2017. G. Peyré and M. Cuturi: Computational Optimal Transport , arXiv:1803.00567v3 [stat.ML], 2019.
For the Local Discriminant Basis (LDB) algorithm and its
improved version, see:
N. Saito and R. R. Coifman: "Local discriminant bases and their applications," Journal of Mathematical Imaging and Vision , vol.5, no.4, pp.337-358, 1995, Invited paper. N. Saito, R. R. Coifman, F. B. Geshwind, and F. Warner: "Discriminant feature extraction using empirical probability density estimation and a local basis library," Pattern Recognition , vol.35, pp.2841-2852, 2002.
Lecture 16: Wavelets on Graphs I For a good description of the continuous wavelet transforms, see, e.g., the following lecture notes of mine:
N. Saito: "Continuous Wavelet Transforms", Lecture 13, MAT 271, Winter, 2018. N. Saito: "Continuous Wavelet Transforms II", Lecture 14, MAT 271, Winter, 2018. S. Mallat: A Wavelet Tour of Signal Processing , 3rd Ed., Academic Press, 2009. Chap. 4.
For a good description of the concept of the frame theory, see, e.g., the following lecture notes of mine:
N. Saito: "Introductory Frame Theory", Lecture 12, MAT 271, Winter, 2018. S. Mallat: A Wavelet Tour of Signal Processing , 3rd Ed., Academic Press, 2009. Chap. 5.
There are too many wavelet-like transform constructions on graphs and networks to list them all here. The following are good surveys of some of those efforts:
D. I. Shuman, S. K. Narang, P. Frossard, A. Ortega, and P. Vandergheynst: "The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains," IEEE Signal Processing Magazine , vol.30, no.3, pp.83-98, 2013. J. Irion and N. Saito: "Applied and computational harmonic analysis on graphs and networks," in Wavelets and Sparsity XVI (M. Papadakis, V. K. Goyal, D. Van De Ville, eds.), Proc. SPIE 9597 , Paper #95971F, 2015. Invited paper.
Lectures 17/18: Wavelets on Graphs II/III: Organizing `Dual' Domains of Graphs Lecture 19: Wavelets on Graphs IV: Hierarchical Graph Laplacian Eigen Transform (HGLET) The main references of this lecture are:
J. Irion and N. Saito: "Hierarchical graph Laplacian eigen transforms," JSIAM Letters , vol.6, pp.21-24, 2014. J. Irion and N. Saito: "Applied and computational harmonic analysis on graphs and networks," in Wavelets and Sparsity XVI (M. Papadakis, V. K. Goyal, D. Van De Ville, eds.), Proc. SPIE 9597 , Paper #95971F, 2015. Invited paper. J. L. Irion: Multiscale Transforms for Signals on Graphs: Methods and Applications , Ph.D.\ dissertation, Appl.\ Math., Univ.\ California, Davis, Dec.\ 2015. J. L. Irion: Multiscale Transforms for Signals on Graphs (MTSG) Toolbox for MATLAB , 2015.
I also mentioned the concept of the Minimum Description Length (MDL) criterion for model selection, which was originally proposed by Jorma Rissanen , and is quite interesting and important in many statistical inference problems. I would suggest the following paper and books to know the MDL principle better:
J. Rissanen: Stochastic Complexity in Statistical Inquiry , World Scientific, 1989. N. Saito: "Simultaneous noise suppression and signal compression using a library of orthonormal bases and the minimum description length criterion," in Wavelets in Geophysics (E. Foufoula-Georgiou and P. Kumar, eds.), Chap.XI, pp.299-324, Academic Press, San Diego, CA, 1994. P. D. Grünwald: The Minimum Description Length Principle , MIT Press, 2007.
Lecture 20: Wavelets on Graphs V: Generalized Haar-Walsh Transform (GHWT) and its extension eGHWT The main references of this lecture are:
J. Irion and N. Saito: "The generalized Haar-Walsh transform," Proc. 2014 IEEE Workshop on Statistical Signal Processing , pp.472-475, 2014. J. Irion and N. Saito: "Applied and computational harmonic analysis on graphs and networks," in Wavelets and Sparsity XVI (M. Papadakis, V. K. Goyal, D. Van De Ville, eds.), Proc. SPIE 9597 , Paper #95971F, 2015. Invited paper. Y. Shao and N. Saito: "The extended Generalized Haar-Walsh Transform and applications," in Wavelets and Sparsity XVIII (D. Van De Ville, M. Papadakis, Y. M. Lu, eds.), Proc. SPIE 11138 , Paper #111380C, 2019.
There are many proposed methods to construct the Haar-like wavelets;
see, e.g.,
R. R. Coifman and M. Gavish: "Harmonic analysis of digital data bases,"
in Wavelets and Multiscale Analysis: Theory and Applications (J. Cohen and A. I. Zayed, eds.), Chap. 9, pp. 161-197, Birkhäuser, 2011.
M. Gavish, B. Nadler, & R. R. Coifman: "Multiscale wavelets on trees, graphs and high dimensional data: theory and applications to semi supervised learning,"
in Proc. 27th Intern. Conf. Machine Learning (J. Fürnkranz and T. Joachims, eds.), pp.367-374, Omnipress, 2010; With supplementary material.
email
me if you have any comments or questions! Go
back to MAT 280: Harmonic Analysis on Graphs & Networks Home Page