toward an uncertainty principle for weighted...
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EUSIPCO 2015
Toward an uncertainty principlefor weighted graphs
02/09/2015
Bastien Pasdeloup*, Réda Alami*, Vincent Gripon*, Michael Rabbat**
Signal processing on graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
1Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Generalization of classical Fourier analysis to more complex domains
Normalized Laplacian:
Signal processing on graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
1Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Generalization of classical Fourier analysis to more complex domains
Normalized Laplacian:
Assumption: normalized signals
Graph Fourier transform
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
2Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Diagonalization of
( real and symmetric)
Graph Fourier transform
Inverse graph Fourier transform
Graph domain Spectral domain
A portage of tools
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
3Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Filtering
Convolution
Translation
Modulation
Dilatation
…
Reference paper: Shuman et. al – The emerging field on signal processing on graphs – 2013
The tool that interests us here
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
4Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Heisenberg principle A signal cannot be both fully localized in thetime and frequency domains
On graphs A signal cannot be both fully localized in thegraph and spectral domains
The uncertainty principle applied to graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
5Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Reference paper: Agaskar & Lu – A spectral graph uncertainty principle – 2013
The uncertainty principle applied to graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
5Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Reference paper: Agaskar & Lu – A spectral graph uncertainty principle – 2013
Graph spread: – Around a particular node – Use of the geodesic (i.e shortest path) distance
The uncertainty principle applied to graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
5Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Reference paper: Agaskar & Lu – A spectral graph uncertainty principle – 2013
Spectral spread: – Around the smallest eigenvalue (=0) – This choice is often discussed, but corresponds to the steady state
The uncertainty principle applied to graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
5Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Reference paper: Agaskar & Lu – A spectral graph uncertainty principle – 2013
Spectral spread: – Around the smallest eigenvalue (=0) – This choice is often discussed, but corresponds to the steady state
x
0 diffusion steps
The uncertainty principle applied to graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
5Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Reference paper: Agaskar & Lu – A spectral graph uncertainty principle – 2013
Spectral spread: – Around the smallest eigenvalue (=0) – This choice is often discussed, but corresponds to the steady state
x
1 diffusion steps
The uncertainty principle applied to graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
5Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Reference paper: Agaskar & Lu – A spectral graph uncertainty principle – 2013
Spectral spread: – Around the smallest eigenvalue (=0) – This choice is often discussed, but corresponds to the steady state
x
2 diffusion steps
The uncertainty principle applied to graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
5Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Reference paper: Agaskar & Lu – A spectral graph uncertainty principle – 2013
Spectral spread: – Around the smallest eigenvalue (=0) – This choice is often discussed, but corresponds to the steady state
x
3 diffusion steps
The uncertainty principle applied to graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
5Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Reference paper: Agaskar & Lu – A spectral graph uncertainty principle – 2013
Spectral spread: – Around the smallest eigenvalue (=0) – This choice is often discussed, but corresponds to the steady state
x
k diffusion steps
The uncertainty principle applied to graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
5Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Reference paper: Agaskar & Lu – A spectral graph uncertainty principle – 2013
Spectral spread: – Around the smallest eigenvalue (=0) – This choice is often discussed, but corresponds to the steady state
x
Convergence
The uncertainty principle applied to graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
5Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Reference paper: Agaskar & Lu – A spectral graph uncertainty principle – 2013
Spectral spread: – Around the smallest eigenvalue (=0) – This choice is often discussed, but corresponds to the steady state
x
Convergence
Coincides with the observation thatdiffusion concentrates the signal in
the frequency domain, but smoothes itin the graph domain (i.e decreases thegraph Laplacian quadratic form: )
Current limitations of their work
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
6Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
– Currently addresses unweighted graphs only
– Arbitrarily uses the geodesic distance
– Does not give any hint on the choice of the reference node
Current limitations of their work
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
6Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
– Currently addresses unweighted graphs only
– Arbitrarily uses the geodesic distance
– Does not give any hint on the choice of the reference node
Addressed in this paper
A naive extension to weighted graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
7Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Idea #1: Let's do the same with a weighted adjacency matrix!
Geodesic distance is also defined for weighted graphs
The normalized Laplacian is still real and symmetric
A naive extension to weighted graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
7Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Idea #1: Let's do the same with a weighted adjacency matrix!
Geodesic distance is also defined for weighted graphs
The normalized Laplacian is still real and symmetric
This leads to a discontinuity of the graph spreadwith respect to the graph structure
Illustration of the problem
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
7Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Illustration of the problem
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
7Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Illustration of the problem
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
7Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Illustration of the problem
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
7Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Illustration of the problem
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
7Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Explanation of the discontinuity
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
8Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
?
Source of the problem
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
9Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Source of the problem
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
9Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Distance function(since we want the spread to
increase as the signal is far away)
Source of the problem
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
9Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Distance function(since we want the spread to
increase as the signal is far away)
Recall that this definition of spectral spread is acceptable
because it is minimized for completely diffused signals (that are smooth in the graph domain)
Source of the problem
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
9Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Distance function(since we want the spread to
increase as the signal is far away)
Recall that this definition of spectral spread is acceptable
because it is minimized for completely diffused signals (that are smooth in the graph domain)
Laplacian quadratic form:
Source of the problem
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
9Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Distance function(since we want the spread to
increase as the signal is far away)
Recall that this definition of spectral spread is acceptable
because it is minimized for completely diffused signals (that are smooth in the graph domain)
Laplacian quadratic form:
To be minimized (i.e localized in the spectral domain),high values in should be associated to very similar nodes
Source of the problem
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
9Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Distance function(since we want the spread to
increase as the signal is far away)
Recall that this definition of spectral spread is acceptable
because it is minimized for completely diffused signals (that are smooth in the graph domain)
To be minimized (i.e localized in the spectral domain),high values in should be associated to very similar nodes
Laplacian quadratic form:
Source of the problem
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
9Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Distance function(since we want the spread to
increase as the signal is far away)
Recall that this definition of spectral spread is acceptable
because it is minimized for completely diffused signals (that are smooth in the graph domain)
To be minimized (i.e localized in the spectral domain),high values in should be associated to very similar nodes
Laplacian quadratic form:
From now on, will be regarded as a similarity matrix
A change of distance function
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
10Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
New definition of graph spread:
– Similar to the previous definition
– We need to change the distance function so that we do not use as distances
Proposed requirements on the distance function :
1°)
2°)
3°) is continuous, and if we increase for a single edge , then does not increase
A change of distance function
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
10Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
New definition of graph spread:
– Similar to the previous definition
– We need to change the distance function so that we do not use as distances
Proposed requirements on the distance function :
1°)
2°)
3°) is continuous, and if we increase for a single edge , then does not increase
The spread should be a positive quantity
A change of distance function
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
10Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
New definition of graph spread:
– Similar to the previous definition
– We need to change the distance function so that we do not use as distances
Proposed requirements on the distance function :
1°)
2°)
3°) is continuous, and if we increase for a single edge , then does not increase
To have for and only for a signallocalized on
The spread should be a positive quantity
A change of distance function
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
10Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
New definition of graph spread:
– Similar to the previous definition
– We need to change the distance function so that we do not use as distances
Proposed requirements on the distance function :
1°)
2°)
3°) is continuous, and if we increase for a single edge , then does not increase
Similar graphs shouldhave similar spreads
The spread should be a positive quantity
To have for and only for a signallocalized on
A change of distance function
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
10Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
New definition of graph spread:
– Similar to the previous definition
– We need to change the distance function so that we do not use as distances
Proposed requirements on the distance function :
1°)
2°)
3°) is continuous, and if we increase for a single edge , then does not increase
Similar graphs shouldhave similar spreads
is considered as a matrix of similarities
The spread should be a positive quantity
To have for and only for a signallocalized on
Some compliant functions
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
11Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Inverse similarity matrix:
– Use of the squared geodesic distance with instead of :
– Simple correction of the discontinuity
– Basically, can be replaced by any positive non-increasing function
(eg. Gaussian kernel)
Diffusion distance:
– is a signal fully localized on node
Associated uncertainty curves
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
12Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Mean uncertainty curves for various graphs, for 100 random weights
Associated uncertainty curves
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
12Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Mean uncertainty curves for various graphs, for 100 random weights
softer than
Associated uncertainty curves
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
12Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Mean uncertainty curves for various graphs, for 100 random weights
softer than
Same curves order
Application to semi-localized graphs
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
13Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Conclusion
Introduction & motivation – Uncertainty on graphs – Extension to weighted graphs – Conclusion
14Bastien Pasdeloup – Toward an uncertainty principle for weighted graphs – EUSIPCO 2015 – 02/09/2015
Summary of the paper:
– Extension of the uncertainty principle to weighted graphs
– Highlight of the confusion when using as a distances matrix
– Statement of requirements on the distance functions used in
Future work:
– Study of the impact of the choice for the distance function
– Obtention of the analytical expressions of the curves
– Study of the impact of the choice of the central node (eg. to compare graphs)
– Use the uncertainty principle to guide a graph reconstruction process
EUSIPCO 2015
Toward an uncertainty principlefor weighted graphs
02/09/2015
Bastien Pasdeloup*, Réda Alami*, Vincent Gripon*, Michael Rabbat**