EE 599 Graph Signal Processing

EE 599, Graph Signal Processing, Fall 2015

Course Description: Theory and applications of emerging tools for signal processing on graphs, including a review of spectral graph theory and newly developed ideas filtering, downsampling, multiresolution decompositions and wavelet transforms"

Prerequisites: EE 483, Introduction to Digital Signal Processing and EE 441, Applied Linear Algebra for Engineering, or equivalent courses. Please note that the course will assume some knowledge of standard DSP concepts as well as of some basic linear algebra. If you took these two courses some time ago it would be a good idea to review some of the key material early in the semester

Background: Graphs have long been used in a wide variety of problems, such analysis of social networks, machine learning, network protocol optimization, decoding of LDPCs or image processing. Techniques based on spectral graph theory provide a "frequency" interpretation of graph data and have proven to be quite popular in many of these applications. In the last few years, a growing amount of work has started extending and complementing spectral graph techniques, leading to the emergence of "Graph Signal Processing" as a broad research field. A common characteristic of this recent work is that it considers the data attached to the vertices as a "graph-signal" and seeks to create new techniques (filtering, sampling, interpolation), similar to those commonly used in conventional signal processing (for audio, images or video), so that they can be applied to these graph signals.

Goals: In this class we provide an overview of this emerging area. The course is aimed at graduate students who have already completed basic coursework in the general areas of signal processing, communications and controls. We start with a review of core concepts, including a review of relevant linear algebra and signal processing concepts. This will be followed by a discussion of advanced topics, focusing on how well established concepts in signal processing are being extended to graph signals (most of this work has taken place in the last 10 years). Finally, we will study specific applications of graph signal processing methods.

Instructor

Antonio Ortega

Signal and Image Processing Institute
Department of Electrical Engineering
University of Southern California
3740 McClintock Ave., EEB 436
Los Angeles, CA 90089-2564

Tel: (213) 740-2320
Fax: (213) 740-4651
Email: antonio DOT ortega AT sipi DOT usc DOT edu

Schedule

• Lectures Tuesday and Thursday, 11:00am-12:20pm, WPH 106
• Office hours Tuesday and Thursday, 2-3:30pm, EEB 436, and by appointment.
• Midterm , Thursday, Nov 5, 11am-12:20pm, in class, room WPH 106.
• Final There will be no final exam

Grading

Class participation and homework (20%), Midterm (40%), Project (40%). The final project reports will be due on TBD.

Material Covered (Subject to Change)

• Week 1: Introduction -- Why Graph Signal Processing: concepts, applications and challenges
• Week 2: Introduction to graph concepts -- Linear algebra review
• Week 3: Spectral graph theory -- Orthogonal transforms review
• Week 4:Frequency interpretation -- Nodal Theorems
• Week 5: Graph filtering -- Vertex and Spectral interpretations
• Week 6:Advanced Topic 1: Shift invariance, localization and uncertainty principles
• Week 7: Advanced Topic 2: Downsampling
• Week 8:Advanced Topic 3: Wavelets
• Week 9:Advanced Topic 4: Multiresolution and graph approximation
• Week 10:Advanced Topic 5: Directed Graphs --- Midterm
• Week 11:Application 1: Image Processing
• Week 12:Application 2: Sensor Networks
• Week 13:Application 3: Machine Learning
• Week 14:Application 4: Finite State Machines
• Week 15: Project Discussions and Presentations

Texbooks

• No required textbook. The reference material will include textbooks as well as a number of recent articles

References

Partial list -- more to be added during the semester

• F. R. Chung, Spectral graph theory, volume92, AMS Bookstore, 1997.
• D. M. Cvetkovic, P. Rowlinson, and S. Simic, An introduction to the theory of graph spectra . Cambridge University Press Cambridge, 2010.
• D. K. Hammond, P. Vandergheynst, and R. Gribonval. Wavelets on graphs via spectral graph theory. Applied and Computational Harmonic Analysis , 30(2):129--150, 2011.
• P. Milanfar. A tour of modern image filtering: new insights and methods, both practical and theoretical. Signal Processing Magazine, IEEE , 30(1):106--128, 2013.
• S. K. Narang and A. Ortega. Perfect reconstruction two-channel wavelet filter banks for graph structured data. Signal Processing, IEEE Transactions on , 60(6):2786--2799, 2012.
• A. Sandryhaila and J. M. Moura. Discrete signal processing on graphs. IEEE transactions on signal processing , 61(5-8):1644--1656, 2013.
• 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. Signal Processing Magazine, IEEE , 30(3):83--98, 2013.
• D. Spielman, Spectral graph theory, Lecture Notes, Yale University, 2009.

Resources

The Graph Signal Processing Toolbox

Lectures and Schedule

• For an outline of topics, see above, detailed listing of topics will be updated throughout the semester.
• Lecture 1 (8/25/15)
  • Introduction: Graph Signal Processing
  • Why is the structure of the graph important
  • Examples of graphs in several applications
• Lecture 2 (8/27/15)
  • Graphs; Definitions, types of graphs
  • The adjacency matrix and the Laplacian
  • Signal variation on a graph and frequency
• Lecture 3 (9/1/15)
  • Signal variation on a graph and frequency (cont'd)
  • Graph Filtering
• Lecture 4 (9/3/15)
  • Linear Algebra Review: spaces, inner products, orthogonality, bases and subspaces
• Lecture 5 (9/10/13)
  • Eigenvalues and eigenvectors -- Interpretation -- Circular convolution
  • Linear Operators based on Polynomials of Adjancency and Laplacian Matrices
  • Eigenvectors and Eigenvalues of graphs.
• Lecture 6 (9/12/13)
  • Perron-Frobenius Theorem, Rayleigh's Quotient
  • Examples
• No lecture on 9/17/13
• Lecture 7 (9/19/13)
  • Presentation and discussion of graph examples from Homework 1
• Lecture 8 (9/24/13)
  • Presentations continued
  • Bipartite graphs
  • Graph Laplacian, Symmetric Normalized Laplacian, Random Walk Laplacian
• Lecture 9 (9/26/13)
  • General bounds on eigenvalues
  • Fiedler eigenvector and its application to graph bisection
• Lecture 10 (10/1/13)
  • Eigenvalue bounds
  • Summary of Graph Laplacians
• Lecture 11 (10/3/13)
  • Spectral decomposition of a graph signal: spectral filtering
  • Interpretation
  • Nodal domains: definition
• Lecture 12 (10/8/13)
  • Results on nodal domains
  • Vertex domain filtering
  • Polynomials of Graph Laplacian and localized filtering
• Lecture 13 (10/10/13)
  • Discussion of IIR, FIR filtering on a graph
  • Time frequency localization for regular signals
• Lecture 14 (10/15/13)
  • Time frequency localization in Graphs
  • Agaskar and Lu (2013)
• Lecture 15 (10/17/13)
  • Bounds on Graph Signal Localization
  • Motivation of downsampling on graphs
• Lecture 16 (10/22/13)
  • Downsampling regular signals
  • Results for bipartite graphs
  • Open questions
• Lecture 17 (10/24/13)
  • Review for the midterm
• No lecture on 10/29/13
• Midterm (10/31/13)
• Lecture 18 (11/5/13)
  • Introduction to wavelets
  • Two channel filterbanks
• Lecture 19 (11/7/13)
  • Lifting based solutions
  • Time-frequency trade-off, time, frequency localization
• Lecture 20 (11/12/13)
  • Multiresolution Analysis
  • Diffusion Wavelets
• Lecture 21 (11/14/13)
  • Diffusion Wavelets Construction
  • Continuous time wavelet transform
• Lecture 22 (11/19/13)
  • Spectral Graph Wavelets
  • Graph Filterbanks
• Lecture 23 (11/21/13)
• Lecture 24 (11/26/13)
• Lecture 25 (12/3/13)
• Lecture 26 (12/5/13)
• Project presentations (12/12/13)

Projects

• Individual project requirements: TBD

Statement for Students with Disabilities

Any student requesting academic accommodations based on a disability is required to register with Disability Services and Programs (DSP) each semester. A letter of verification for approved accommodations can be obtained from DSP. Please be sure the letter is delivered to me (or to TA) as early in the semester as possible. DSP is located in STU 301 and is open 8:30 a.m.--5:00 p.m., Monday through Friday. The phone number for DSP is (213) 740-0776.

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