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.
Suppose we are given a vector f in a class FsubeRopfN , e.g., a class of digital signals or digital images. How many linear measurements do we need to make about f to be able to recover f to within precision epsi in the Euclidean (lscr2) metric? This paper shows that if the objects of interest are sparse in a fixed basis or compressible, then it is possible to reconstruct f to within very high accuracy from a small number of random measurements by solving a simple linear program. More precisely, suppose that the nth largest entry of the vector |f| (or of its coefficients in a fixed basis) obeys |f|(n)lesRmiddotn-1p , where R>0 and p>0. Suppose that we take measurements yk=langf# ,Xkrang,k=1,...,K, where the Xk are N-dimensional Gaussian vectors with independent standard normal entries. Then for each f obeying the decay estimate above for some 0<p<1 and with overwhelming probability, our reconstruction ft, defined as the solution to the constraints yk=langf# ,Xkrang with minimal lscr1 norm, obeys parf-f#parlscr2lesCp middotRmiddot(K logN)-r, r=1 p-1 2. There is a sense in which this result is optimal; it is generally impossible to obtain a higher accuracy from any set of K measurements whatsoever. The methodology extends to various other random measurement ensembles; for example, we show that similar results hold if one observes a few randomly sampled Fourier coefficients of f. In fact, the results are quite general and require only two hypotheses on the measurement ensemble which are detailed Vector subspaces have been suggested for representations of structured information. In the theory of associative memory and associative information processing, the projection principle and subspaces are used in explaining the optimality of associative mappings and novelty filters. These formalisms seem to be very pertinent to neural networks, too. Based on these operations, the subspace method has been developed for a practical pattern-recognition algorithm. The method is reviewed, and some recent results on image analysis are given. > A new modification of the subspace pattern recognition method, called the dual subspace pattern recognition (DSPR) method, is proposed, and neural network models combining both constrained Hebbian and anti-Hebbian learning rules are developed for implementing the DSPR method. An experimental comparison is made by using our model and a three-layer forward net with backpropagation learning. The results illustrate that our model can outperform the backpropagation model in suitable applications.