Operations Research and Signal Processing on graphs : case of
sampling and perfect reconstructing tasks
Pascal Bianchi1 , Valeria Borodin2 , Faicel Hnaien3 , Nacima Labadie3 , Hichem Snoussi4
1
2
Institut Mines-Télécom, Télécom ParisTech, CNRS LTCI, 46 rue Barrault, 75013 Paris, France
[email protected]
École des Mines de Saint-Étienne (CMP), CNRS UMR 6158 LIMOS, 13541, Gardanne, France
[email protected]
3
Laboratory of Industrial Systems Optimization (LOSI),
4
Laboratory of Systems Modelling and Dependability (LM2S), Charles Delaunay Institute,
University of Technology of Troyes, 12 rue Marie Curie - CS 42060, 10004 Troyes, France
{faicel.hnaien, nacima.labadie, hichem.snoussi}@utt.fr
Mots-clés : signal processing, perfect reconstruction, cutt-off frequency, operations research,
maximum cut problem
1
Introduction
For many environmental monitoring applications, a faithful mapping of a spatio-temporal
phenomena expanded over a physical limited location represents a primordial task. Recent
years have witnessed a remarkable expansion of signal processing on graphs. Researches in this
emerging domain attracts more and more interest, especially when Operations Research tools
can be used to solve the different problems that are raised. In this context, the current paper
deals with the problem of perfect signal reconstruction from a sampled graph signal, which
entails the resolution of integer linear models.
2
Problem and resolution approach
Signal processing on graphs generalizes classical discrete signal processing to signals with
an intrinsic complex or irregular structure. Notwithstanding that graph signal processing is
nowadays in its infancy [4], the notion of cut-off frequency for the band-limited graph signals
that can be reconstructed from a given set of samples (i.e., graph vertices or edges) was already
defined by [3]. More precisely, the notion of frequency is introduced via the eigenvalues and
eigenvectors of the graph Laplacian.
Let us focus on analyzing signals defined on an undirected connected and weighted graph
G = (V, E, W ), which consists of a finite set of vertices V (|V | = n) and a set of weighted
edges E (|E| = m). The weights associated to the edges closely depend on the applications. In
order to determine the weight w(i, j) of an edge linking vertices i and j, various methods can
be found in the spectral graph theory (for more details, see for instance [4, 5]).
The Fourier transform is a mathematical operation that converts a signal into its constituent
frequencies. In classical signal processing, given a band-limited signal and a sampling rate fs ,
the Nyquist-Shannon theorem provides the condition for unique reconstruction of the signal
from its samples as : fmax < fs /2. On the other hand, in graph signals, the notion of frequency
is expressed via the eigenvalues and eigenvectors of the graph Laplacian L , since these latter
can be regarded as defining a frequency domain for graph signals on G. Remember that,
L = D − W , where D = diag{d1 , d2 , · · · , dn } is the degree matrix of the graph G and W
represents the weighted adjacency matrix of G.
Based on the fact that the eigenvalues and eigenvectors of L give a spectral interpretation
for a graph signal, similar to the Fourier transform in classical signal processing. The graph
Fourier transform (GFT) of a signal f is defined as its projection onto the eigenvectors of the
graph Laplacian, i.e. :
f̃f (λi ) =< f , u i >,
u1 , u 2 , · · · , u n , }
where λ1 , · · · , λn (0 = λ1 ≤ λ2 , · · · , λn ≤ 2) represent the eigenvalues of L , and {u
L
is the set of orthonormal eigenvectors of .
Hence, similar to the classical signal processing setting, where the Fourier transform decomposes a signal into the frequency domain, the GFT translates a graph signal f into the
u1 , u 2 , · · · , u n , }. Low frequencies of eigenvalues λi corresponds to an eigenvector that
basis {u
varies slowly across the graph, i.e. if two adjacent vertices are connected by an edge with a
large weight, the values of the eigenvector at those locations are quite identical. Conversely,
the eigenvectors associated with higher eigenvalues present important oscillations across the
graph [3, 4]. In this context, the cut-off frequency ωc (S) of a set S is ω, such that S is a
ui : λi ) (i.e. for the subspace of Rn
uniqueness set for the Paley-Wiener space P Wω = span(u
with all ω-band-limited signals) [3].
In addition to the introduction of the notion of cut-off-frequency and the derivation of the
sampling theorem for signal on graphs, [1, 3] posed two optimization problems related to finding
optimal sampling sets, namely : (i) for a given ω find the smallest set S for which ωc (S) ≥ ω,
and (ii) for a given cardinality of the set S, find the maximum cut-off frequency. By analyzing
the obtained results, the relevance of the notion of cut-off frequency is questionable, since it
seems to not be a robust metric for choosing the best sampling set.
Being in line with the above described state-of-the art progress made in the framework
sampling on signal graphs, we propose to reformulate the second optimization problem as a
maximum-cut problem. More specifically, instead of seeking for the maximum cut-off frequency
for a given size of the set S, we search a subset S ⊂ V , which maximizes the total weight of
the edges between the subset S and its complement. This way of dealing with this problem
offers two significant advantages : (i) firstly, for a given size m of S : ωc (S) = λm , and (ii)
secondly, by virtue of its objective function, the optimization problem thus defined is looking
for the most appropriate sampling set.
Although it has a practical interest, the maximum cut problem is a well-known NP-hard
problem in combinatorial optimization [2]. In that vein, our future research efforts will be
dedicated to circumventing the limitation of computational complexity for making the problem
tractable even for large-scale instances.
Références
[1] Anas Anis, Akshay Gadde, and Antonio Ortega. Towards a sampling theorem for signals
on arbitrary graphs. In Acoustics, Speech and Signal Processing (ICASSP), 2014 IEEE
International Conference on, pages 3864–3868. IEEE, 2014.
[2] Christos Papadimitriou and Mihalis Yannakakis. Optimization, approximation, and complexity classes. In Proceedings of the twentieth annual ACM symposium on Theory of
computing, pages 229–234. ACM, 1988.
[3] Han Shomorony et al. Sampling large data on graphs. In Signal and Information Processing
(GlobalSIP), 2014 IEEE Global Conference on, pages 933–936. IEEE, 2014.
[4] David Shuman, Sunil K Narang, Pascal Frossard, Antonio Ortega, Pierre Vandergheynst,
et al. 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.
[5] Xiaofan Zhu and M. Rabbat. Approximating signals supported on graphs. In Acoustics,
Speech and Signal Processing (ICASSP), 2012 IEEE International Conference on, pages
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