Alianna J. Maren

Alianna J. Maren

Despite being invented in 1951 by R. Kikuchi, the 2-D Cluster Variation Method (CVM), has not yet received attention. Nevertheless, this method can usefully characterize 2-D topographies using just two parameters; the activation enthalpy and the interaction enthalpy. This Technical Report presents 2-D CVM details, including the dependence of the various configuration variables on the enthalpy parameters, as well as illustrations of various topographies (ranging from scale-free-like to rich club-like) that result from different parameter selection. The complete derivation for the analytic solution, originally presented simply as a result in Kikuchi and Brush (1967) is given here, along with careful comparison of the analytically-predicted configuration variables versus those obtained when performing computational free energy minimization on a 2-D grid. The 2-D CVM can potentially function as a secondary free energy minimization within the hidden layer of a neural network, providing a basis for extending node activations over time and allowing temporal correlation of patterns.

Alianna J. Maren

The derivation of key equations for the variational Bayes approach is well-known in certain circles. However, translating the fundamental derivations (e.g., as found in Beal's work) to Friston's notation is somewhat delicate. Further, the notion of using variational Bayes in the context of a system with a Markov blanket requires special attention. This Technical Report presents the derivation in detail. It further illustrates how the variational Bayes method provides a framework for a new computational engine, incorporating the 2-D cluster variation method (CVM), which provides a necessary free energy equation that can be minimized across both the external and representational systems' states, respectively.

Alianna J. Maren

A new approach for general artificial intelligence (GAI), building on neural network deep learning architectures, can make use of one or more hidden layers that have the ability to continuously reach a free energy minimum even after input stimulus is removed, allowing for a variety of possible behaviors. One reason that this approach has not been developed until now has been the lack of a suitable free energy equation. The Cluster Variation Method (CVM) offers a means for characterizing 2-D local pattern distributions, or configuration variables, and provides a free energy formalism in terms of these configuration variables. The equilibrium distribution of these configuration variables is defined in terms of a single interaction enthalpy parameter, h, for the case of equiprobable distribution of bistate units. For non-equiprobable distributions, the equilibrium distribution can be characterized by providing a fixed value for the fraction of units in the active state (x1), corresponding to the influence of a per-unit activation enthalpy, together with the pairwise interaction enthalpy parameter h. This paper provides verification and validation (V&V) for code that computes the configuration variable and thermodynamic values for 2-D CVM grids characterized by different interaction enthalpy parameters, or h-values. This work provides a foundation for experimenting with a 2-D CVM-based hidden layer that can, as an alternative to responding strictly to inputs, also now independently come to its own free energy minimum and also return to a free energy-minimized state after perturbations, which will enable a range of input-independent behaviors. A further use of this 2-D CVM grid is that by characterizing local patterns in terms of their corresponding h-values (together with their x1 values), we have a means for quantitatively characterizing different kinds of neural topographies.