Linear and logarithmic capacities in associative neural networks

A model of associate memory incorporating global linearity and pointwise nonlinearities in a state space of n-dimensional binary vectors is considered. Attention is focused on the ability to store a prescribed set of state vectors as attractors within the model. Within the framework of such associative nets, a specific strategy for information storage that utilizes the spectrum of a linear operator is considered in some detail. Comparisons are made between this spectral strategy and a prior scheme that utilizes the sum of Kronecker outer products of the prescribed set of state vectors, which are to function nominally as memories. The storage capacity of the spectral strategy is linear in n (the dimension of the state space under consideration), whereas an asymptotic result of n/4 log n holds for the storage capacity of the outer product scheme. Computer-simulated results show that the spectral strategy stores information more efficiently. The preprocessing costs incurred in the two algorithms are estimated, and recursive strategies are developed for their computation

How much information can one bit of memory retain about a Bernoulli sequence?

The maximin problem of the maximization of the minimum amount of information that a single bit of memory retains about the entire past is investigated. The problem is to estimate the supremum over all possible sequences of update rules of the minimum information that the bit of memory at epoch (n+1) retains about the previous n inputs. Using only elementary techniques, it is shown that the maximin covariance between the memory at epoch (n+1) and past inputs is Θ(1/n), the maximum average covariance is Θ(1/n), and the maximin mutual information is Ω(1/n^2). In a consideration of related issues, the authors also provide an exact count of the number of Boolean functions of n variables that can be obtained recursively from Boolean functions of two variables, discuss extensions and applications of the original problem, and indicate links with issues in neural computation

On edge-colored interior planar graphs on a circle and the expected number of RNA secondary structures

AbstractUsing a mathematical model for an RNA molecule as a family of disjoint edge-colored interior planar graphs on a circle, we determine the expected number of secondary RNA structures that can form under various assumptions on the type and number of ribonucleotide bonds

Sensor Network Devolution and Breakdown in Survivor Connectivity

As batteries fail in wireless sensor networks there is an inevitable devolution of the network characterised by a breakdown in connectivity between the surviving nodes of the network. A sharp limit theorem characterising the time at which this phenomena makes an appearance is derived

On the Finite Sample Performance of the Nearest Neighbor Classifier

The finite sample performance of a nearest neighbor classifier is analyzed for a two-class pattern recognition problem. An exact integral expression is derived for the m-sample risk R_m given that a reference m-sample of labeled points, drawn independently from Euclidean n-space according to a fixed probability distribution, is available to the classifier. For a family of smooth distributions characterized by asymptotic expansions in general form, it is shown that the m-sample risk R_m has a complete asymptotic series expansion R_m ~ R_∞ + Σ^∞_(k=1) c_km^(-k/n) (m → ∞) where R_∞ denotes the nearest neighbor risk in the infinite-sample limit. Improvements in convergence rate are shown under stronger smoothness assumptions, and in particular, R_m = R_∞ + O(m^(-2/n)) if the class-conditional probability densities have uniformly bounded third derivatives on their probability one support. This analysis thus provides further analytic validation of Bellman's curse of dimensionality. Numerical simulations corroborating the formal results are included, and extensions of the theory discussed. The analysis also contains a novel application of Laplace's asymptotic method of integration to a multidimensional integral where the integrand attains its maximum on a continuum of points