using an integration by part w.r.t w kl (the boundary terms can be shown to vanish), then Bayes formula, and finally (153), one obtains References Arash A, Information Theory, pp.2454-2458, 2008. ,
Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices, Annals of Probability, pp.1643-1697, 2005. ,
A nearly tight sum-of-squares lower bound for the planted clique problem, 2016. ,
The adaptive interpolation method: a simple scheme to prove replica formulas in bayesian inference. Probability Theory and Related Fields, 2018. ,
The mutual information in random linear estimation, 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp.625-632, 2016. ,
Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula, Advances in Neural Information Processing Systems, vol.29, pp.424-432, 2016. ,
URL : https://hal.archives-ouvertes.fr/cea-01568705
Universal sparse superposition codes with spatial coupling and gamp decoding, 2017. ,
Mutual information and optimality of approximate message-passing in random linear estimation, 2017. ,
The dynamics of message passing on dense graphs, with applications to compressed sensing, IEEE Trans. on Information Theory, vol.57, issue.2, pp.764-785, 2011. ,
A nonparametric view of network models and newmangirvan and other modularities, Proceedings of the National Academy of Sciences, vol.106, issue.50, pp.21068-21073, 2009. ,
A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, vol.20, issue.4, pp.1956-1982, 2010. ,
Dynamics and termination cost of spatially coupled mean-field models, Phys. Rev. E, vol.89, p.12102, 2014. ,
URL : https://hal.archives-ouvertes.fr/hal-00996993
Exact matrix completion via convex optimization, Foundations of Computational mathematics, vol.9, issue.6, pp.717-772, 2009. ,
Statistical-computational tradeoffs in planted problems and submatrix localization with a growing number of clusters and submatrices, 2014. ,
A direct formulation for sparse pca using semidefinite programming, SIAM review, vol.49, issue.3, pp.434-448, 2007. ,
Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications, Physical Review E, vol.84, issue.6, p.66106, 2011. ,
URL : https://hal.archives-ouvertes.fr/hal-00661643
Information-theoretically optimal sparse pca, IEEE International Symposium on, pp.2197-2201, 2014. ,
Asymptotic mutual information for the two-groups stochastic block model, 2015. ,
High-Dimensional Inference on Dense Graphs with Applications to Coding Theory and Machine Learning, EPFL IC School, 2018. ,
Finite-range spin glasses in the kac limit: free energy and local observables, Journal of Physics A: Mathematical and General, vol.37, issue.30, p.7433, 2004. ,
Spatial coupling as a proof technique and three applications, IEEE Transactions on Information Theory, vol.62, issue.10, pp.5281-5295, 2016. ,
An introduction to mean field spin glass theory: methods and results, Mathematical Statistical Physics, pp.243-271, 2005. ,
The thermodynamic limit in mean field spin glass models, Commun. Math. Phys, vol.230, issue.1, pp.71-79, 2002. ,
Mutual information and minimum mean-square error in gaussian channels, IEEE Trans. on Information Theory, vol.51, 2005. ,
Submatrix localization via message passing, 2015. ,
Coupled graphical models and their thresholds, IEEE Information Theory Workshop (ITW), 2010. ,
State evolution for general approximate message passing algorithms, with applications to spatial coupling, Journal of Information and Inference, vol.2, issue.2, pp.115-144, 2013. ,
, , vol.7, 2004.
On consistency and sparsity for principal components analysis in high dimensions, Journal of the American Statistical Association, 2012. ,
Stochastic blockmodels and community structure in networks, Physical Review E, vol.83, issue.1, p.16107, 2011. ,
Matrix completion from a few entries, IEEE International Symposium on, pp.324-328, 2009. ,
Exact solution of the gauge symmetric p-spin glass model on a complete graph, Journal of Statistical Physics, vol.136, issue.2, pp.205-230, 2009. ,
Mutual information in rank-one matrix estimation, 2016. ,
Threshold saturation via spatial coupling: Why convolutional ldpc ensembles perform so well over the bec, IEEE Trans. on Inf. Th, vol.57, 2011. ,
Fundamental limits of symmetric low-rank matrix estimation, 2017. ,
URL : https://hal.archives-ouvertes.fr/hal-01648368
Mmse of probabilistic low-rank matrix estimation: Universality with respect to the output channel, 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp.680-687, 2015. ,
URL : https://hal.archives-ouvertes.fr/cea-01222294
Phase transitions in sparse pca, IEEE International Symposium on, pp.1635-1639, 2015. ,
URL : https://hal.archives-ouvertes.fr/cea-01140712
Community detection thresholds and the weak ramanujan property, Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pp.694-703, 2014. ,
Low-rank matrix reconstruction and clustering via approximate message passing, Advances in Neural Information Processing Systems, pp.917-925, 2013. ,
Iterative estimation of constrained rank-one matrices in noise, 2012 IEEE International Symposium on, pp.1246-1250, 2012. ,
Performance of a community detection algorithm based on semidefinite programming, Journal of Physics: Conference Series, vol.699, p.12015, 2016. ,
Spectral clustering of graphs with the bethe hessian, Advances in Neural Information Processing Systems, pp.406-414, 2014. ,
URL : https://hal.archives-ouvertes.fr/cea-01140852
Matrix completion from fewer entries: Spectral detectability and rank estimation, Advances in Neural Information Processing Systems, pp.1261-1269, 2015. ,
URL : https://hal.archives-ouvertes.fr/cea-01222302
A simple proof of maxwell saturation for coupled scalar recursions, IEEE Trans. on Inf. Theory, vol.60, issue.11, pp.6943-6965, 2014. ,
Journal of computational and graphical statistics, vol.15, pp.265-286, 2006. ,