Space Manifold dynamics

Intrinsic scale space on manifold

Scale space obtained by diffusion processes suffers from notorious volume shrinkage and unnatural smoothing result. We find that these drawbacks come from the mis-match of vertices between adjacent scales, which is caused by the incomplete catch of vertex migration during diffusion. We use generalized optical flow tracking technique, borrowed from computer vision, to correct such mis-match. Our...

On images of continuous functions from a compact manifold to Euclidean space

Quotient Manifold Projections and Hierarchical Dynamics

In this paper we explore the mathematical structure of hierarchical organization in smooth dynamical systems. We start by making precise what we mean by a level in a hierarchy, and how the higher levels need to respect the dynamics on the lower levels. We derive a mathematical construction for identifying distinct levels in a hierarchical dynamics. The construction is expressed through a quotie...

Manifold Stochastic Dynamics for Bayesian Learning

We propose a new Markov Chain Monte Carlo algorithm, which is a generalization of the stochastic dynamics method. The algorithm performs exploration of the state-space using its intrinsic geometric structure, which facilitates efficient sampling of complex distributions. Applied to Bayesian learning in neural networks, our algorithm was found to produce results comparable to the best state-of-t...

Soft manifold dynamics behind negative thermal expansion.

Minimal models are developed to examine the origin of large negative thermal expansion in underconstrained systems. The dynamics of these models reveals how underconstraint can organize a thermodynamically extensive manifold of low-energy modes which not only drives negative thermal expansion but extends across the Brillioun zone. Mixing of twist and translation in the eigenvectors of these mod...