Efficient Clustering on Riemannian Manifolds: A Kernelised Random Projection Approach

Reformulating computer vision problems over Riemannian manifolds has demonstrated superior performance in various computer vision applications. This is because visual data often forms a special structure lying on a lower dimensional space embedded in a higher dimensional space. However, since these manifolds belong to non-Euclidean topological spaces, exploiting their structures is computationally expensive, especially when one considers the clustering analysis of massive amounts of data. To this end, we propose an efficient framework to address the clustering problem on Riemannian manifolds. This framework implements random projections for manifold points via kernel space, which can preserve the geometric structure of the original space, but is computationally efficient. Here, we introduce three methods that follow our framework. We then validate our framework on several computer vision applications by comparing against popular clustering methods on Riemannian manifolds. Experimental results demonstrate that our framework maintains the performance of the clustering whilst massively reducing computational complexity by over two orders of magnitude in some cases

Conditional and Unconditional Statistical Independence

Conditional independence almost everywhere in the space of the conditioning variates does not imply unconditional independence, although it may well imply unconditional independence of certain functions of the variables. An example that is important in linear regression theory is discussed in detail. This involves orthogonal projections on random linear manifolds, which are conditionally independent but not unconditionally independent under normality. Necessary and sufficient conditions are obtained under which conditional independence does imply unconditional independence

Masking Strategies for Image Manifolds

We consider the problem of selecting an optimal mask for an image manifold, i.e., choosing a subset of the pixels of the image that preserves the manifold's geometric structure present in the original data. Such masking implements a form of compressive sensing through emerging imaging sensor platforms for which the power expense grows with the number of pixels acquired. Our goal is for the manifold learned from masked images to resemble its full image counterpart as closely as possible. More precisely, we show that one can indeed accurately learn an image manifold without having to consider a large majority of the image pixels. In doing so, we consider two masking methods that preserve the local and global geometric structure of the manifold, respectively. In each case, the process of finding the optimal masking pattern can be cast as a binary integer program, which is computationally expensive but can be approximated by a fast greedy algorithm. Numerical experiments show that the relevant manifold structure is preserved through the data-dependent masking process, even for modest mask sizes

A dynamical approximation for stochastic partial differential equations

Random invariant manifolds often provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the random invariant manifold is almost surely asymptotically complete. The asymptotic dynamical behavior is thus described by a stochastic ordinary differential system on the random invariant manifold, under suitable conditions. As an application, stationary states (invariant measures) is considered for one example of stochastic partial differential equations.Comment: 28 pages, no figure

Polynomial Roots and Calabi-Yau Geometries

The examination of roots of constrained polynomials dates back at least to Waring and to Littlewood. However, such delicate structures as fractals and holes have only recently been found. We study the space of roots to certain integer polynomials arising naturally in the context of Calabi-Yau spaces, notably Poincare and Newton polynomials, and observe various salient features and geometrical patterns.Comment: 22 pages, 13 Figure

A Motivating Exploration on Lunar Craters and Low-Energy Dynamics in the Earth -- Moon System

It is known that most of the craters on the surface of the Moon were created by the collision of minor bodies of the Solar System. Main Belt Asteroids, which can approach the terrestrial planets as a consequence of different types of resonance, are actually the main responsible for this phenomenon. Our aim is to investigate the impact distributions on the lunar surface that low-energy dynamics can provide. As a first approximation, we exploit the hyberbolic invariant manifolds associated with the central invariant manifold around the equilibrium point L_2 of the Earth - Moon system within the framework of the Circular Restricted Three - Body Problem. Taking transit trajectories at several energy levels, we look for orbits intersecting the surface of the Moon and we attempt to define a relationship between longitude and latitude of arrival and lunar craters density. Then, we add the gravitational effect of the Sun by considering the Bicircular Restricted Four - Body Problem. As further exploration, we assume an uniform density of impact on the lunar surface, looking for the regions in the Earth - Moon neighbourhood these colliding trajectories have to come from. It turns out that low-energy ejecta originated from high-energy impacts are also responsible of the phenomenon we are considering.Comment: The paper is being published in Celestial Mechanics and Dynamical Astronomy, vol. 107 (2010