The problem of neural network association is to retrieve a previously memorized pattern from its noisy version using a network of neurons. An ideal neural network should include three components simultaneously: a learning algorithm, a large pattern retrieval capacity and resilience against noise. Prior works in this area usually improve one or two aspects at the cost of the third. Our work takes a step forward in closing this gap. More specifically, we show that by forcing natural constraints on the set of learning patterns, we can drastically improve the retrieval capacity of our neural network. Moreover, we devise a learning algorithm whose role is to learn those patterns satisfying the above mentioned constraints. Finally we show that our neural network can cope with a fair amount of noise.
Compressed sensing aims to undersample certain high-dimensional signals yet accurately reconstruct them by exploiting signal characteristics. Accurate reconstruction is possible when the object to be recovered is sufficiently sparse in a known basis. Currently, the best known sparsity–undersampling tradeoff is achieved when reconstructing by convex optimization, which is expensive in important large-scale applications. Fast iterative thresholding algorithms have been intensively studied as alternatives to convex optimization for large-scale problems. Unfortunately known fast algorithms offer substantially worse sparsity–undersampling tradeoffs than convex optimization. We introduce a simple costless modification to iterative thresholding making the sparsity–undersampling tradeoff of the new algorithms equivalent to that of the corresponding convex optimization procedures. The new iterative-thresholding algorithms are inspired by belief propagation in graphical models. Our empirical measurements of the sparsity–undersampling tradeoff for the new algorithms agree with theoretical calculations. We show that a state evolution formalism correctly derives the true sparsity–undersampling tradeoff. There is a surprising agreement between earlier calculations based on random convex polytopes and this apparently very different theoretical formalism. The goal of the sparse approximation problem is to approximate a target signal using a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, to the circumstances in which individual methods tend to perform well, and to the theoretical guarantees available. Many fundamental questions in electrical engineering, statistics, and applied mathematics can be posed as sparse approximation problems, making these algorithms versatile and relevant to a plethora of applications.