No abstract

The key point in design of radial basis function networks is to specify the number and the locations of the centers. Several heuristic hybrid learning methods, which apply a clustering algorithm for locating the centers and subsequently a linear leastsquares method for the linear weights, have been previously suggested. These hybrid methods can be put into two groups, which will be called as input clustering (IC) and input-output clustering (IOC), depending on whether the output vector is also involved in the clustering process. The idea of concatenating the output vector to the input vector in the clustering process has independently been proposed by several papers in the literature although none of them presented a theoretical analysis on such procedures, but rather demonstrated their effectiveness in several applications. The main contribution of this paper is to present an approach for investigating the relationship between clustering process on input-output training samples and the mean squared output error in the context of a radial basis function netowork (RBFN). We may summarize our investigations in that matter as follows: 1) A weighted mean squared input-output quantization error, which is to be minimized by IOC, yields an upper bound to the mean squared output error. 2) This upper bound and consequently the output error can be made arbitrarily small (zero in the limit case) by decreasing the quantization error which can be accomplished through increasing the number of hidden units.

Polar codes are known as the first provable code construction to achieve Shannon capacity for arbitrary symmetric binary-input channels. Although, there exist efficient sub-optimal decoders with reduced complexity for polar codes, the complexity of the optimum ML decoder increases exponentially. Hence the optimum decoder is infeasible for the practical implementation of polar coding. In this paper, our motivation is about developing efficient ML decoder with reduced complexity. In this purpose, polar code based sphere decoding algorithm is proposed with the optimal performance. Additionally, proposed technique exploits two properties of polar coding to reduce decoding complexity. By this way, the reduced complexity of optimal decoding is only cubic, not exponential.

In this paper, we propose efficient maximumlikelihood (ML) decoding for binary Kronecker product-based (KPB) codes. This class of codes, have a matrix defined by the n-fold iterated Kronecker product Gn = F ⊗n of a binary upper-triangular kernel matrix F, where some columns are suppressed given a specific puncturing pattern. Polar and Reed-Muller codes are well known examples of such KPB codes.The triangular structure of Gn enables to perform ML decoding as a binary tree search for the closest codeword to the received point. We take advantage of the highly regular fractal structure of Gn and the "tree folding" technique to design an efficient ML decoder, enabling to decode relatively longer block lengths than with a standard binary tree search. The tree κfolding operation transforms the binary tree with N levels into a non-binary tree with N/2 κ levels, where the search can be significantly accelerated by a suitable ordering of the branch metrics. For a given κ we can find ( n κ ) different folding which lead to decoders with different complexity, for a given code.Using the proposed folded tree decoder, we provide exact ML performances of some Reed-Muller and polar codes over a binary AWGN channel for the block length up to 256.

Abstract-Polar codes are the first explicit class of codes that are provably capacity-achieving under the successive cancelation (SC) decoding. As a suboptimal decoder, SC has quasi-linear complexity N (1 + log N ) in the code length N . In this paper, we propose a new non-binary SC decoder with reduced complexity) based on the folding operation, which was first proposed in [11] to implement folded tree maximum-likelihood decoding of polar codes. Simulation results for the additive white Gaussian noise channel show that folded SC decoders can achieve the same error performance of standard SC by suitable selecting the folding of the polar code.