On Kolmogorov's superpositions and Boolean functions

The paper overviews results dealing with the approximation capabilities of neural networks, as well as bounds on the size of threshold gate circuits. Based on an explicit numerical (i.e., constructive) algorithm for Kolmogorov's superpositions they will show that for obtaining minimum size neutral networks for implementing any Boolean function, the activation function of the neurons is the identity function. Because classical AND-OR implementations, as well as threshold gate implementations require exponential size (in the worst case), it will follow that size-optimal solutions for implementing arbitrary Boolean functions require analog circuitry. Conclusions and several comments on the required precision are ending the paper

On automatic synthesis of analog/digital circuits

The paper builds on a recent explicit numerical algorithm for Kolmogorov`s superpositions, and will show that in order to synthesize minimum size (i.e., size-optimal) circuits for implementing any Boolean function, the nonlinear activation function of the gates has to be the identity function. Because classical and--or implementations, as well as threshold gate implementations require exponential size, it follows that size-optimal solutions for implementing arbitrary Boolean functions can be obtained using analog (or mixed analog/digital) circuits. Conclusions and several comments are ending the paper

A Theory of Networks for Appxoimation and Learning

Learning an input-output mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multi-dimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nolinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. We develop a theoretical framework for approximation based on regularization techniques that leads to a class of three-layer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the well-known Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods such as Parzen windows and potential functions and to several neural network algorithms, such as Kanerva's associative memory, backpropagation and Kohonen's topology preserving map. They also have an interesting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data

How to build VLSI-efficient neural chips

This paper presents several upper and lower bounds for the number-of-bits required for solving a classification problem, as well as ways in which these bounds can be used to efficiently build neural network chips. The focus will be on complexity aspects pertaining to neural networks: (1) size complexity and depth (size) tradeoffs, and (2) precision of weights and thresholds as well as limited interconnectivity. They show difficult problems-exponential growth in either space (precision and size) and/or time (learning and depth)-when using neural networks for solving general classes of problems (particular cases may enjoy better performances). The bounds for the number-of-bits required for solving a classification problem represent the first step of a general class of constructive algorithms, by showing how the quantization of the input space could be done in O (m{sup 2}n) steps. Here m is the number of examples, while n is the number of dimensions. The second step of the algorithm finds its roots in the implementation of a class of Boolean functions using threshold gates. It is substantiated by mathematical proofs for the size O (mn/{Delta}), and the depth O [log(mn)/log{Delta}] of the resulting network (here {Delta} is the maximum fan in). Using the fan in as a parameter, a full class of solutions can be designed. The third step of the algorithm represents a reduction of the size and an increase of its generalization capabilities. Extensions by using analogue COMPARISONs, allows for real inputs, and increase the generalization capabilities at the expense of longer training times. Finally, several solutions which can lower the size of the resulting neural network are detailed. The interesting aspect is that they are obtained for limited, or even constant, fan-ins. In support of these claims many simulations have been performed and are called upon

When constants are important

In this paper the authors discuss several complexity aspects pertaining to neural networks, commonly known as the curse of dimensionality. The focus will be on: (1) size complexity and depth-size tradeoffs; (2) complexity of learning; and (3) precision and limited interconnectivity. Results have been obtained for each of these problems when dealt with separately, but few things are known as to the links among them. They start by presenting known results and try to establish connections between them. These show that they are facing very difficult problems--exponential growth in either space (i.e. precision and size) and/or time (i.e., learning and depth)--when resorting to neural networks for solving general problems. The paper will present a solution for lowering some constants, by playing on the depth-size tradeoff

2D neural hardware versus 3D biological ones

This paper will present important limitations of hardware neural nets as opposed to biological neural nets (i.e. the real ones). The author starts by discussing neural structures and their biological inspirations, while mentioning the simplifications leading to artificial neural nets. Going further, the focus will be on hardware constraints. The author will present recent results for three different alternatives of implementing neural networks: digital, threshold gate, and analog, while the area and the delay will be related to neurons' fan-in and weights' precision. Based on all of these, it will be shown why hardware implementations cannot cope with their biological inspiration with respect to their power of computation: the mapping onto silicon lacking the third dimension of biological nets. This translates into reduced fan-in, and leads to reduced precision. The main conclusion is that one is faced with the following alternatives: (1) try to cope with the limitations imposed by silicon, by speeding up the computation of the elementary silicon neurons; (2) investigate solutions which would allow one to use the third dimension, e.g. using optical interconnections

09391 Abstracts Collection -- Algorithms and Complexity for Continuous Problems

From 20.09.09 to 25.09.09, the Dagstuhl Seminar 09391 Algorithms and Complexity for Continuous Problems was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Combined optimization algorithms applied to pattern classification

Accurate classification by minimizing the error on test samples is the main goal in pattern classification. Combinatorial optimization is a well-known method for solving minimization problems, however, only a few examples of classifiers axe described in the literature where combinatorial optimization is used in pattern classification. Recently, there has been a growing interest in combining classifiers and improving the consensus of results for a greater accuracy. In the light of the "No Ree Lunch Theorems", we analyse the combination of simulated annealing, a powerful combinatorial optimization method that produces high quality results, with the classical perceptron algorithm. This combination is called LSA machine. Our analysis aims at finding paradigms for problem-dependent parameter settings that ensure high classifica, tion results. Our computational experiments on a large number of benchmark problems lead to results that either outperform or axe at least competitive to results published in the literature. Apart from paxameter settings, our analysis focuses on a difficult problem in computation theory, namely the network complexity problem. The depth vs size problem of neural networks is one of the hardest problems in theoretical computing, with very little progress over the past decades. In order to investigate this problem, we introduce a new recursive learning method for training hidden layers in constant depth circuits. Our findings make contributions to a) the field of Machine Learning, as the proposed method is applicable in training feedforward neural networks, and to b) the field of circuit complexity by proposing an upper bound for the number of hidden units sufficient to achieve a high classification rate. One of the major findings of our research is that the size of the network can be bounded by the input size of the problem and an approximate upper bound of 8 + √2n/n threshold gates as being sufficient for a small error rate, where n := log/SL and SL is the training set

Simplified neural networks algorithms for function approximation and regression boosting on discrete input spaces

Function approximation capabilities of feedforward Neural Networks have been widely investigated over the past couple of decades. There has been quite a lot of work carried out in order to prove 'Universal Approximation Property' of these Networks. Most of the work in application of Neural Networks for function approximation has concentrated on problems where the input variables are continuous. However, there are many real world examples around us in which input variables constitute only discrete values, or a significant number of these input variables are discrete. Most of the learning algorithms proposed so far do not distinguish between different features of continuous and discrete input spaces and treat them in more or less the same way. Due to this reason, corresponding learning algorithms becomes unnecessarily complex and time consuming, especially when dealing with inputs mainly consisting of discrete variables. More recently, it has been shown that by focusing on special features of discrete input spaces, more simplified and robust algorithms can be developed. The main objective of this work is to address the function approximation capabilities of Artificial Neural Networks. There is particular emphasis on development, implementation, testing and analysis of new learning algorithms for the Simplified Neural Network approximation scheme for functions defined on discrete input spaces. By developing the corresponding learning algorithms, and testing with different benchmarking data sets, it is shown that comparing conventional multilayer neural networks for approximating functions on discrete input spaces, the proposed simplified neural network architecture and algorithms can achieve similar or better approximation accuracy. This is particularly the case when dealing with high dimensional-low sample cases, but with a much simpler architecture and less parameters. In order to investigate wider implications of simplified Neural Networks, their application has been extended to the Regression Boosting frame work. By developing, implementing and testing with empirical data it has been shown that these simplified Neural Network based algorithms also performs well in other Neural Network based ensembles.EThOS - Electronic Theses Online ServiceGBUnited Kingdo