Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and...
Transcript of Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and...
![Page 1: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/1.jpg)
Graph Signal Processing: An Introductory Overview
Antonio Ortega
Signal and Image Processing InstituteDepartment of Electrical EngineeringUniversity of Southern California
Los Angeles, California
May 25, 2016
![Page 2: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/2.jpg)
Acknowledgements
I Collaborators
- Sunil Narang (Microsoft), Godwin Shen (Northrop-Grumman),Eduardo Martınez Enrıquez (Univ. Carlos III, Madrid)
- Akshay Gadde, Jessie Chao, Aamir Anis, Yongzhe Wang,Eduardo Pavez, Hilmi Egilmez, Joanne Kao (USC)
- Marco Levorato (UCI), Urbashi Mitra (USC), Salman Avestimehr(USC), Aly El Gamal (USC/Purdue), Eyal En Gad (USC), NiccoloMichelusi (USC/Purdue), Gene Cheung (NII), Pierre Vandergheynst(EPFL), Pascal Frossard (EPFL), David Shuman (MacalasterCollege), Yuichi Tanaka (TUAT), David Taubman (UNSW).
I Funding
- NASA (AIST-05-0081), NSF (CCF-1018977, CCF-1410009,CCF-1527874)
- MERL, Samsung, LGE, Google- Sabbatical: Japan Society for Promotion of Science (JSPS), UNSW,
NII, TUAT
![Page 3: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/3.jpg)
Goals
I Some background and basic concepts
I A bit of history and what’s going on now (at the workshop!)
I FAQs and challenges
I Disclaimers
I Partial overview!I More questions than answers
![Page 4: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/4.jpg)
Goals
I Some background and basic concepts
I A bit of history and what’s going on now (at the workshop!)
I FAQs and challenges
I Disclaimers
I Partial overview!
I More questions than answers
![Page 5: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/5.jpg)
Goals
I Some background and basic concepts
I A bit of history and what’s going on now (at the workshop!)
I FAQs and challenges
I Disclaimers
I Partial overview!I More questions than answers
![Page 6: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/6.jpg)
Outline
Why GSP?
Basic Concepts
A bit of history and what’s going on now
FAQs and challenges
Conclusions
![Page 7: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/7.jpg)
This is the 2nd GSP workshop
I First workshop was held with IEEE GlobalSIP, Dec 2013.
I Much more activity in this field!
I More than double the number of presentations, 1 day vs 3 days.
![Page 8: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/8.jpg)
Graph signal processing: why now?
1796 Philadelphia roadmap, Library of Congress
Standard questions: What is the shortest path? What is safest path?
![Page 9: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/9.jpg)
Graph signal processing: why now?
1796 Philadelphia roadmap, Library of Congress
Standard questions: What is the shortest path? What is safest path?
![Page 10: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/10.jpg)
Graph signal processing: why now?
I Going from physical graphs to information graphs:
I From:
I Roads and railI Telephone Networks
I To:
I WebI Online social network
I Information links were always there but they were not an obvious graph
I encyclopedia vs wikipedia
I Sensing technology allows us to measure “on a graph”
I Where is GSP being used?
I Physical networksI Information networksI Regular signals
![Page 11: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/11.jpg)
Graph signal processing: why now?
I Going from physical graphs to information graphs:
I From:
I Roads and railI Telephone Networks
I To:
I WebI Online social network
I Information links were always there but they were not an obvious graph
I encyclopedia vs wikipedia
I Sensing technology allows us to measure “on a graph”
I Where is GSP being used?
I Physical networksI Information networksI Regular signals
![Page 12: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/12.jpg)
Graph signal processing: why now?
I Going from physical graphs to information graphs:
I From:
I Roads and railI Telephone Networks
I To:
I WebI Online social network
I Information links were always there but they were not an obvious graph
I encyclopedia vs wikipedia
I Sensing technology allows us to measure “on a graph”
I Where is GSP being used?
I Physical networksI Information networksI Regular signals
![Page 13: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/13.jpg)
Graph signal processing: why now?
I Going from physical graphs to information graphs:
I From:
I Roads and railI Telephone Networks
I To:
I WebI Online social network
I Information links were always there but they were not an obvious graph
I encyclopedia vs wikipedia
I Sensing technology allows us to measure “on a graph”
I Where is GSP being used?
I Physical networksI Information networksI Regular signals
![Page 14: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/14.jpg)
Graph signal processing: why now?
Lambert, 1765, Playfair ca. 1820
See (E. Tufte, The Visual Display of Quantitative Information, ’83)
Do we know how to think about and visualize graph signals?
![Page 15: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/15.jpg)
Graph signal processing: why now?
Lambert, 1765, Playfair ca. 1820
See (E. Tufte, The Visual Display of Quantitative Information, ’83)Do we know how to think about and visualize graph signals?
![Page 16: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/16.jpg)
Outline
Why GSP?
Basic Concepts
A bit of history and what’s going on now
FAQs and challenges
Conclusions
![Page 17: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/17.jpg)
Multiple algebraic representations
1
2
3
4
5
6
7
8
9
10
1112
13
14
15
16
17
18
19
20
21 22 23
24 25 26
27
I Graph G = (V,E ,w).
I Adjacency A, aij , aji = weights oflinks between i and j (could bedifferent if graph is directed.)
I Degree D = diag{di}, in case ofundirected graph.
I Various algebraic representations
I normalized adjacency 1|λmax |A
I Laplacian matrix L = D− A.I Symmetric normalized
Laplacian L = D−1/2LD−1/2
I Graph Signalf = {f (1), f (2), ..., f (N)}
I Discussion:
1. Undirected graphs easier to work with2. Some applications require directed graphs3. Graphs with self loops are useful
![Page 18: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/18.jpg)
Graph spectrum, GFT
I Different results/insights for different choices of operator
I Laplacian L = D− A = UΛU′
I Eigenvectors of L : U = {uk}k=1:N
I Eigenvalues of L : diag{Λ} = λ1 ≤ λ2 ≤ ... ≤ λN
I Eigen-pair system {(λk , uk)} provides Fourier-like interpretation —Graph Fourier Transform (GFT)
![Page 19: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/19.jpg)
Eigenvectors of graph Laplacian
(a) λ = 0.00 (b) λ = 0.04 (c) λ = 0.20
(d) λ = 0.40 (e) λ = 1.20 (f) λ = 1.49
I Basic idea: increased variation on the graph, e.g., ftLf, as frequencyincreases
![Page 20: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/20.jpg)
Graph Transforms and Filters
Input Signal Transform Output Signal Processing/
Analysis
I Properties
I InvertibleI Critically sampled/overcompleteI Orthogonal/near orthogonal/frames
I What makes these “graph transforms”?
I Frequency interpretation
I Operation is diagonalized by U
I Vertex localization: polynomial of the operator (A, L, etc)
I Polynomial degree is small (vs a polynomial of degree N − 1).
![Page 21: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/21.jpg)
Frequency interpretation
I Spectral Wavelet transforms (Hammond et al, CHA’09):
Design spectral kernels: h(λ) : σ(G)→ R.
Th = h(L) = Uh(Λ)Ut
h(Λ) = diag{h(λi )}
I Analogy: FFT implementation of filters
![Page 22: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/22.jpg)
Vertex Localization: SGWT
I Polynomial kernel approximation:
h(λ) ≈K∑
k=0
akλk
Th ≈K∑
k=0
akLk
I Note that A and L are both 1-hop operations K -hop localized: no spectraldecomposition required.
![Page 23: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/23.jpg)
Summary
I Different types of graphs (directed, undirected, with/without self loops)
I Multiple algebraic representations of graphs (A,L, ...)
I Their eigenvectors/eigenvalues induce a notion of variation
I The corresponding operators can be viewed as “shifts” or “elementaryoperators”
I Polynomials of these operators represent “local” processing on the graph
I Graph filtering: polynomial/diagonal in vertex/frequency
![Page 24: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/24.jpg)
Outline
Why GSP?
Basic Concepts
A bit of history and what’s going on now
FAQs and challenges
Conclusions
![Page 25: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/25.jpg)
Graphs Signal Processing: a bit of history
Key point: many threads lead to graph signal processing
I spectral graph theory
I image processing
I semi-supervised learning
I block transforms
I vertex domain transforms
I frequency domain transforms
![Page 26: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/26.jpg)
Spectral graph theory
I Results emerging in ’50s and ’60s linking algebraic graph structure tograph properties, initial work in Math, later interest in CS
I Classic works
I (Cvetkovic, Doobs and Sachs, ’80)I (Chung, ’96)I (Spielman, ’01)
I Primary concerns are linking spectrum and graph properties, no signals
I Link to GSP is weaker than it should be: because we are interested inworking on arbitrary graphs (more on this later)
Linking graph spectrum to graph structure
![Page 27: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/27.jpg)
DCT and KLT
I (Ahmed, Natarajan, Rao, T. on Computers, ’74) Optimality of DCT forhigh correlation random vectors (close to 1)
I (Strang, SIAM’99) Graph interpretation (eigenvectors of line graphs withweight one), connection to DST
I (Puschel & Moura, SIAM’03) Generalization,
I (Puschel & Moura, TSP’08) General Algebraic signal processingperspective:
I DCT as basis for a signal space of finite signals under differentboundary conditions (Sandryhaila & Moura, ’14) (see afternoon talk!)
I (Shen et al, PCS’10) regular graphs with irregular weights (use GFT of thegraph)
I (Zhang, Florencio & Chou, SPL’13), General case of graphs obtained fromprecision matrices corresponding to Gauss Markov Random Fields.
Eigenvectors of graphs with regular connectivity and unequal weights/self-loops
![Page 28: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/28.jpg)
Image processing
I (Wu & Leahy, PAMI, ’93), (Shi & Malik, PAMI, ’00): graph cuts forimage segmentation, smaller edge weights across image boundaries
I (Tomasi & Manduchi, ’98) Bilateral filtering, filter weights function ofpixel and photometric distances
I (Elmoataz, Lezoray, Bougleux, TIP’08), (Osher et al, SIAM’07) GraphLaplacians for regularization
I (Milanfar, SPM’13) Various signal dependent image filters from a graphperspective
Weighted graphs with edges a function of pixel distance and intensitydifferences
![Page 29: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/29.jpg)
Semi-supervised learning
I Learning from labeled and unlabeled training data
I Estimate labels for unlabeled dataI Decide what data to labelI Consider kNN data graph
I (Belkin, Niyogi, ’03 NIPS), (Zhou et al, NIPS’04), (Smola & Kondor,COLT’03), (Zhu et al, ML’03) Regularization on graphs, semi-supervisedlearning, label propagation
I Generally use L to favor smooth signals on the graph
I (Anis et al, KDD’14) Graph signal sampling interpretation
I Why should a “label” signal be smooth?
Graphs connecting datapoints in feature space (e.g., kNN), labels should beslow varying
![Page 30: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/30.jpg)
Graph Transforms: vertex domain approaches
I (Schroeder & Sweldens, ’95), (DeRose & Salesin, ’95) Transforms forattributes defined on meshes, often use lifting based techniques
I Network graphs, (Crovella& Kolaczyk, INFOCOM’03), graphs witharbitrary connectivity, analysis tool, overcomplete
I Sensor networks, (Baraniuk et al, IPSN’06), (Wang & Ramchandran, ’06),(Ciancio, et al, IPSN’06) (Shen & Ortega, IPSN’08, TSP’10)
I Emphasis on distributed operation, approaches sometimes usestructure (e.g., trees, tesselations)
I Graph lifting (Narang & Ortega, APSIPA’09), (Janson et al, RoyalStatistical Society’09)
I Even/odd assignment in regular signals correspond to bipartiteapproximation of a graph
Vertex domain approaches require graph partitions
![Page 31: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/31.jpg)
Graph Transforms: frequency domain approaches
I Diffusion wavelets (Coiffman & Maggioni’06):
I Use successive applications of a diffusion to create subspaces of lowergraph frequency content (and less localization in vertex domain)
I Eigendecomposition of powers of LI No exact localization in vertex domain
I Spectral Graph Wavelets (Hammond et al, CHA ’09)
I Spectral domain design (kernel having desirable properties and itsscalings)
I Polynomial approximation for localization, no need to explicitfrequency decomposition
I Nice vertex/frequency interpretation, overcomplete
I Filterbanks (Narang & Ortega, TSP’12, TSP’13)
I Critically sampled, orthogonal/bi-orthogonal solutionsI Exact solutions, only bi-partite graphs
Different trade-offs possible depending on whether critical sampling is required
![Page 32: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/32.jpg)
Sampling
I Irregular sampling in regular domains, e.g., properties that guaranteereconstructions (Grochenig, 92), (Aldroubi & Grochenig, ’01)
I Focus is on reconstruction based on regular domain properties(frequency)
I Optimality conditions for combinatorial graphs (Pesenson’08)
I Various approaches for sample set selection and reconstruction (Anis et al,’14), (Shomorony & Avestimehr, 14), (Chen et al, ’14)
Selection of nodes that are most informative
![Page 33: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/33.jpg)
Conference topics
I Distributed processing
I Graph learning
I Filtering
I Fundamentals
I Sampling
I Statistical Graph Signal Processing
I Applications
![Page 34: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/34.jpg)
Graph Filter Design
I Classic problem in DSP
I Goal: Design filters with different properties in terms of localization,orthogonality, etc.
I Different types of filters:
I Moving average, graph-temporalI Graph diffusionI graph-temporalI lifting approachesI representations using dictionaries
![Page 35: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/35.jpg)
Graph Learning
I Goal: learn a graph from data
I Multiple cases: covariance,propagating graph signals, etc.
I Examples: estimate a sparse inverse covariance (precision) matrix,estimate a Laplacian that makes a observed data smooth on average
I Question: what is the advantage of using a graph vs PCA?
I Interpretation, approximate KLT with polynomial graph operations
![Page 36: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/36.jpg)
Sampling
I Goal: decide which graph signal samples (values associated to vertices)should be observed so that we can reconstruct the others
I Assumptions about smoothness of signal
I (almost) any random sampling works when signals are exactly bandlimitedand noise free
I Noise and non-bandlimited behavior make things complicated
I Robust sampling methods (randomize, iterative, with/withoutknowledge of the GFT)
I New criteria for signals to be sampled (piecewise smooth)I Distributed sampling
![Page 37: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/37.jpg)
Statistical GSP
I Random signals on graphs
I Definition of Stationary Graph Signals
I Time varying signals over graphs
I PCA on graphs
![Page 38: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/38.jpg)
Applications
I Image regularization
I Compression
I Computer vision applications (e.g., motion analysis)
I Origin-Destination traffic matrices
I Tracing of outbreaks
I Wireless network optimization
I Brain connectivity
![Page 39: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/39.jpg)
Outline
Why GSP?
Basic Concepts
A bit of history and what’s going on now
FAQs and challenges
Conclusions
![Page 40: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/40.jpg)
Scale
I Can we really apply our tools to large scale datasets?
I Facebook 1G+ nodes
I How to interpret results in a large scale graph
I How to interpret locality:
I Graph diameter vs average?I We often do not consider what the real footprint is for a polynomial
of degree K .
I Large scale implementation
I ParallelizationI GraphLab
I Basic primitive: each node communicating with its neighborsI Algorithms to distribute nodes across processors to preserve
localityI Example: node requests information from neighbors and
computes an output (Lx) or apply this recursively (L(Lx)), i.e.every node stores Lx and then L2x, etc.
I Should our community contribute?
![Page 41: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/41.jpg)
How to choose a graph
I In some applications graph is given (e.g., social networks)
I In some it is a function of some known information (e.g., distance in asensor network)
I How to select weights?I e.g., bilateral filter, etcI are there optimality results?
I Designing graphs from data
I Sparse inverse covariance: why is a graph representation of a datasetbetter?
I Advantages vs other methods
![Page 42: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/42.jpg)
Shifts and localization
I Most current filtering schemes use an operator (Laplacian or Adjacency)
I Should it be considered a “shift” (note that sometimes signals vanish afterbeing shifted).
I The effect of a shift depends on the eigenvalue associated with it: graphswith same eigenvectors, but different eigenvalues? L =
∑i λiuiu
ti ,
different graph connectivity, but same frequency interpretations (Gavili &Zhang, Arxiv’15)
I Classes of equivalent graphs?
I Bounds on frequency-vertex domain localization
I Specialize these bounds to specific graph types
I Complicated because of properties of shift
I Several contributions in this workshop
![Page 43: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/43.jpg)
Need to consider special characteristics of graphs
I Example: how to deal with high multiplicity eigenvalues (high dimensionalsubspace with the same graph frequency) (Zeng et al, ICASSP’16)
I How to assess the impact of “removing” edges?
I What is the best way to approximate a graph?
I Graph reductions/simplifications
I More generally there could be interesting results that apply only to certainclasses of graphs?
I bipartite (Narang & Ortega, ’11), circulant (Ekambaran et al, ’13),M-block cyclic (Teke & Vaidyanathan, ’16)
![Page 44: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/44.jpg)
Datasets and community
I Should we have a set of standard datasets?
I Matlab code: GSP Toolbox EPFL (Perraudin & Paratte)
I Anything else we should do?
![Page 45: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/45.jpg)
What is the killer app?
I Understand in what cases a graph-based approach is better than directly
working with signals in RN.
I Many cases
I Graph is given (web, social network)I Irregular measurements (sensor networks)I Graph approaches are an alternative (e.g., images)I Data driven methods (e.g., machine learning).I Large scale systems (e.g., finite state machines)
I GSP methods are closely linked to existing approaches
I New perspectives on existing topics
I perhaps an emerging new way to understand problems
![Page 46: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/46.jpg)
Outline
Why GSP?
Basic Concepts
A bit of history and what’s going on now
FAQs and challenges
Conclusions
![Page 47: Graph Signal Processing: An Introductory Overviewgsp16/ortega.pdf · Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering ... 1 day vs 3 days.](https://reader033.fdocuments.in/reader033/viewer/2022041901/5e60538cc161b14656718b3e/html5/thumbnails/47.jpg)
Conclusions
I GSP has deep roots in the Signal Processing community
I A lot of progress, interesting results
I Many open questions!
I Outcomes
I Work with massive graph-datasets: potential benefits of localized“frequency” analysis
I Novel insights about traditional applications (image/video processing)I Promising results in machine learning, image processing, among other
areas
I To get started (Shuman et al, SPM’13), (Sandryhaila & Moura, SPM’14)
I Enjoy the workshop!