Boltzmann Machines and Their Applications Emile H.L. Aarts and Jan H.M. Korst Philips Research Laboratories P.O. Box 80.000, 5600 JA Eindhoven, the Netherlands Abstract In this paper we present a formal model of the Boltzmann machine and a discussion of two different applications of the model, viz. (i) solving combinatorial optimization problems and (ii) carrying out learning tasks. Numerical results of computer simulations are presented to demonstrate the characteristic features of the Boltzmann machine. Keywords: Boltzmann machines, simulated annealing, combinatorial optimization, learning. 1 Introduction Many researchers believe that massive parallelism rather than raw speed of individual processors may provide the computational power required to carry out the increasingly complex calculations imposed by e.g. combinatorial optimization [3,13,15] and artificial intelligence [4,6,7]. A revo- lutionary development in the field of parallel computer architectures is based on the so-called connectionist models [5,6]. These models incorporate massive parallelism and the assumption that information can be represented by the strengths of the connections between individual computing elements. The Boltzmann machine, introduced by Hinton et al. [4,6,11], is a novel approach to connectionist models using a distributed knowledge representation and a massively parallel network of simple stochastic computing elements. The computing elements are considered as logic units having two discrete states, viz. 'on' or 'off'. The units are connected to eachother. With each connection a connection strength is associated representing a quantitative measure of the hypothesis that the two connected units are both 'on'. A consensus function assigns to a configuration of the Boltzmann machine (determined by the states of the individual units) a real number which is a quantitative measure of the amount of consensus in the Boltzmann machine with respect to the set of underlying hypotheses. The state of an individual unit (if not externally forced) is determined by a stochastic function of the states of the units it is connected to and the associated connection strengths. Maximization of the consensus function corresponds to maximization of the amount of information contained within the Boltzmann machine. Interest in the Boltzmann machine extends over a number of disciplines, i.e. future computer archi- tectures [6,23], knowledge representation in intelligent systems [2,6,23,24], modelling of neural be- haviour of the brain [11,21], and research on applications, e.g. pattern recognition [10,11,18,20,22] and combinatorial optimization [3,15]. Especially the research on recognition and synthesis of speech with a Boltzmann machine has recently shown a number of interesting achievements [ls,20,22]. In this paper we discuss a formal model and two different applications of the Boltzmann ma- chine. In §2 we present a graph-theoretical model of the structure of the Boltzmann machine and