Storage capacity of a Potts-perceptron

Abstract

We consider the properties of “Potts” neural networks where each neuron can be in Q different states. For a “Potts-perceptron” with N Q-states input neurons and one Q' states output neutron, we compute the maximal storage capacity for unbiased patterns. In the large N limit the maximal number of patterns that can be stored is found to be proportional to N(Q-1)f(Q'), where f(Q') is of order 1

Optimal capacity of the Blume-Emery-Griffiths perceptron

“…The factor N −1/2 is introduced to have the weights J j and K j of order unity. The aim is then to determine the maximal number of patterns, p, that can be stored in the perceptron, in other words to find the maximal value of the loading α = p/N for which couplings satisfying (10)- (11) can still be found. Following a Gardner-type analysis [19] the fundamental quantity that we want to calculate is then the volume fraction of weight space given by…”

“…The perceptron is by now a well-known and standard model in theoretical studies and practical applications in connection with learning and generalization [3], [4], [7] - [9]. Consequently, a number of extensions including many-state, graded response and colored perceptrons have been formulated in the literature [11]- [18].…”

Optimal colored perceptrons

“…The perceptron which was first analyzed with statistical mechanics techniques in the seminal paper of Gardner [1] is by now a well-known and standard model in theoretical studies and practical applications in connection with learning and generalization [2][3][4][5]. A number of extensions of the perceptron model have been formulated, including many-state and graded-response perceptrons (e.g., [6][7][8][9][10][11]). Here we present some new extensions allowing for so-called coloured or Ashkin-Teller type neurons, i.e., different types of binary neurons at each site possibly having different functions.…”

A new design method for the complex-valued multistate hopfield associative memory

“…In order to perform a search for a Hermitian coefficient matrix , such that the real-valued discrete quadratic form (5) attains a local minimum at each element of , we simply apply the definition of a strict local minimum, and impose a set of strict inequalities (6) to be satisfied for each . Here is the 1-neighborhood of and defined formally as 7By substituting (4) in (6), we express this condition as inequalities to be satisfied by the coefficient matrix (8) Incorporating now our initial design considerations and , condition (8) can be further expressed in terms of only the upper triangle entries of (9) for all…”

“…By replacing the activation functions of neurons in the conventional Hopfield network with this nonlinearity remarkable steps have been made toward the design of multistate associative memories [5]- [7]. It has also been shown in [8] that the maximum number of integral patterns that can be stored in such a network by any design procedure is proportional to , where is of order one. An alternative dynamical finite-state system operating on has been introduced in [9] as the complex-valued multistate Hopfield network.…”