5,894 research outputs found
A Connection Between GRBF and MLP
Both multilayer perceptrons (MLP) and Generalized Radial Basis Functions (GRBF) have good approximation properties, theoretically and experimentally. Are they related? The main point of this paper is to show that for normalized inputs, multilayer perceptron networks are radial function networks (albeit with a non-standard radial function). This provides an interpretation of the weights w as centers t of the radial function network, and therefore as equivalent to templates. This insight may be useful for practical applications, including better initialization procedures for MLP. In the remainder of the paper, we discuss the relation between the radial functions that correspond to the sigmoid for normalized inputs and well-behaved radial basis functions, such as the Gaussian. In particular, we observe that the radial function associated with the sigmoid is an activation function that is good approximation to Gaussian basis functions for a range of values of the bias parameter. The implication is that a MLP network can always simulate a Gaussian GRBF network (with the same number of units but less parameters); the converse is true only for certain values of the bias parameter. Numerical experiments indicate that this constraint is not always satisfied in practice by MLP networks trained with backpropagation. Multiscale GRBF networks, on the other hand, can approximate MLP networks with a similar number of parameters
Relative Resolution: A Multipole Approximation at Appropriate Distances
Recently, we introduced Relative Resolution as a hybrid formalism for fluid
mixtures [1]. The essence of this approach is that it switches molecular
resolution in terms or relative separation: While nearest neighbors are
characterized by a detailed fine-grained model, other neighbors are
characterized by a simplified coarse-grained model. Once the two models are
analytically connected with each other via energy conservation, Relative
Resolution can capture the structural and thermal behavior of (nonpolar)
multi-component and multi-phase systems across state space. The current work is
a natural continuation of our original communication [1]. Most importantly, we
present the comprehensive mathematics of Relative Resolution, basically casting
it as a multipole approximation at appropriate distances; the current set of
equations importantly applies for all systems (e.g, polar molecules). Besides,
we continue examining the capability of our multiscale approach in molecular
simulations, importantly showing that we can successfully retrieve not just the
statics but also the dynamics of liquid systems. We finally conclude by
discussing how Relative Resolution is the inherent variant of the famous
"cell-multipole" approach for molecular simulations
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