An improved connectionist activation function for energy minimization

Symmetric networks that are based on energy minimization, such as Boltzmann machines or Hopfield nets, are used extensively for optimization, constraint satisfaction, and approximation of NP-hard problems. Nevertheless, finding a global minimum for the energy function is not guaranteed, and even a local minimum may take an exponential number of steps. We propose an improvement to the standard activation function used for such networks. The improved algorithm guarantees that a global minimum is found in linear time for tree-like subnetworks. The algorithm is uniform and does not assume that the network is a tree. It performs no worse than the standard algorithms for any network topology. In the case where there are trees growing from a cyclic subnetwork, the new algorithm performs better than the standard algorithms by avoiding local minima along the trees and by optimizing the free energy of these trees in linear time. The algorithm is self-stabilizing for trees (cycle-free undirected graphs) and remains correct under various scheduling demons. However, no uniform protocol exists to optimize trees under a pure distributed demon and no such protocol exists for cyclic networks under central demon

Learned multi-stability in mechanical networks

We contrast the distinct frameworks of materials design and physical learning in creating elastic networks with desired stable states. In design, the desired states are specified in advance and material parameters can be optimized on a computer with this knowledge. In learning, the material physically experiences the desired stable states in sequence, changing the material so as to stabilize each additional state. We show that while designed states are stable in networks of linear Hookean springs, sequential learning requires specific non-linear elasticity. We find that such non-linearity stabilizes states in which strain is zero in some springs and large in others, thus playing the role of Bayesian priors used in sparse statistical regression. Our model shows how specific material properties allow continuous learning of new functions through deployment of the material itself

Inherent global stabilization of unstable local behavior in coupled map lattices

The behavior of two-dimensional coupled map lattices is studied with respect to the global stabilization of unstable local fixed points without external control. It is numerically shown under which circumstances such inherent global stabilization can be achieved for both synchronous and asynchronous updating. Two necessary conditions for inherent global stabilization are derived analytically.Comment: 17 pages, 10 figures, accepted for publication in Int.J.Bif.Chao