The network nullspace property for compressed sensing over networks

We study compressed sensing of graph signals defined over complex networks. In particular, we propose and analyze a convex optimization method for recovering smooth graph signals from a small number of samples. Assuming the true underlying graph signal to be constant over well connected subset of nodes (clusters), we derive a sufficient condition on the sampling set and network structure such that the proposed convex method is accurate. This condition, which we coin the network nullspace property, characterizes which nodes of the graph should be sampled in order to retain the full information about the underlying graph signal.

[1]  Alexander J. Smola,et al.  Trend Filtering on Graphs , 2014, J. Mach. Learn. Res..

[2]  José M. F. Moura,et al.  Big Data over Networks , 2016 .

[3]  José M. F. Moura,et al.  Big Data Analysis with Signal Processing on Graphs: Representation and processing of massive data sets with irregular structure , 2014, IEEE Signal Processing Magazine.

[4]  Guigang Zhang,et al.  Deep Learning , 2016, Int. J. Semantic Comput..

[5]  Jelena Kovacevic,et al.  Discrete Signal Processing on Graphs: Sampling Theory , 2015, IEEE Transactions on Signal Processing.

[6]  Frank Harary,et al.  Graph Theory , 2016 .

[7]  Alexander Zien,et al.  Semi-Supervised Learning , 2006 .

[8]  Vladimir Kolmogorov,et al.  "GrabCut": interactive foreground extraction using iterated graph cuts , 2004, ACM Trans. Graph..

[9]  Alexander Jung,et al.  Scalable graph signal recovery for big data over networks , 2016, 2016 IEEE 17th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC).

[10]  Antonin Chambolle,et al.  A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging , 2011, Journal of Mathematical Imaging and Vision.

[11]  Radford M. Neal.  Pattern Recognition and Machine Learning , 2007, Technometrics.

[12]  Yonina C. Eldar.  Sampling Theory: Beyond Bandlimited Systems , 2015 .

[13]  Alfred O. Hero,et al.  Scalable Semi-Supervised Learning over Networks using Nonsmooth Convex Optimization , 2016, ArXiv.

[14]  Andrew Blake,et al.  "GrabCut" , 2004, ACM Trans. Graph..

[15]  A. Juditsky,et al.  5 First-Order Methods for Nonsmooth Convex Large-Scale Optimization , I : General Purpose Methods , 2010 .

[16]  Alexander Jung,et al.  Efficient graph signal recovery over big networks , 2016, 2016 50th Asilomar Conference on Signals, Systems and Computers.

[17]  Yunzhang Zhu.  An Augmented ADMM Algorithm With Application to the Generalized Lasso Problem , 2017 .

[18]  Emmanuel J. Candès,et al.  NESTA: A Fast and Accurate First-Order Method for Sparse Recovery , 2009, SIAM J. Imaging Sci..

[19]  Santiago Segarra,et al.  Sampling of Graph Signals With Successive Local Aggregations , 2015, IEEE Transactions on Signal Processing.

[20]  Editors , 1986, Brain Research Bulletin.

[21]  Éva Tardos,et al.  Algorithm design , 2005 .

[22]  Alessandro Rinaldo,et al.  Sparsistency of the Edge Lasso over Graphs , 2012, AISTATS.

[23]  Holger Rauhut,et al.  A Mathematical Introduction to Compressive Sensing , 2013, Applied and Numerical Harmonic Analysis.