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**11 - 14**of**14**### Devant le jury omposé de:

"... Présentée pour obtenir le grade de do
teur de l'université de Cergy-Pontoise ..."

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Présentée pour obtenir le grade de do
teur de l'université de Cergy-Pontoise

### A Simple Message-Passing Algorithm for Compressed Sensing

"... Abstract—We consider the recovery of a nonnegative vector x from measurements y = Ax, where A ∈ {0, 1}m×n. We establish that when A corresponds to the adjacency matrix of a bipartite graph with sufficient expansion, a simple message-passing algo-rithm produces an estimate x ̂ of x satisfying ‖x−x̂‖1 ..."

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Abstract—We consider the recovery of a nonnegative vector x from measurements y = Ax, where A ∈ {0, 1}m×n. We establish that when A corresponds to the adjacency matrix of a bipartite graph with sufficient expansion, a simple message-passing algo-rithm produces an estimate x ̂ of x satisfying ‖x−x̂‖1 ≤ O(nk)‖x− x(k)‖1, where x(k) is the best k-sparse approximation of x. The algorithm performs O(n(log(n k))2 log(k)) computation in total, and the number of measurements required is m = O(k log(n k In the special case when x is k-sparse, the algorithm recovers x exactly in time O(n log(n k) log(k)). Ultimately, this work is a further step in the direction of more formally developing the broader role of message-passing algorithms in solving compressed sensing problems. I.

### Devant le jury omposé de:

, 2013

"... Codes LDPC multi-binaires hybrides et méthodes de décodage itératif ..."

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