Distribution (differential geometry)

About: Distribution (differential geometry) is a research topic. Over the lifetime, 911 publications have been published within this topic receiving 10149 citations.

Слоения на распределениях с финслеровой метрикой

Abstract: A distribution D with a admissible Finsler metric is defined on a smooth manifold X. Let F be a foliation on X. On the distribution of D as on a smooth manifold foliation F corresponds to the foliation T F. Using this foliation and connection over distribution we define analog exterior derivative. Exterior differential forms is applied to a special form.

Nonequilibrium equivalence principle, dynamical phase transitions and coarse graining

Abstract: We introduce the Fisher information in the basis of decay modes of Markovian dynamics, arguing that it encodes important information about the behavior of nonequilibrium systems. In particular we generalize an orthonormality relation between decay eigenmodes of detailed balanced systems to normal generators that commute with their time-reversal. Viewing such modes as tangent vectors to the manifold of statistical distributions, we relate the result to the choice of a coordinate patch that makes the Fisher- Rao metric Euclidean at the steady distribution, realizing a sort of statistical equivalence principle. We then classify nonequilibrium systems according to their spectrum, showing that a degenerate Fisher matrix is the signature of the insurgence of a class of dynamical phase transitions between nonequilibrium regimes, characterized by level crossing and power-law decay in time of suitable order parameters. An important consequence is that normal systems cannot manifest critical behavior. Finally, we study the Fisher matrix of systems with time-scale separation.

Riemannian metrics and Laplacians for generalized smooth distributions

Abstract: We show that any generalized smooth distribution on a smooth manifold, possibly of non-constant rank, admits a Riemannian metric. Using such a metric, we attach a Laplace operator to any smooth dis...

The use of tangent vector fields

Abstract: Recently, no-waiting stations with spatial arrival process and ar- bitrary service time distribution have been analyzed (2), (3). Such stations represent adequate models of, for example, mobile communication networks based on code division multiple access (CDMA) techniques. Also, the inclusion of customer movements in space in such models has been performed. In (4), (5), movements were represented as group operations, which mapped a region R of interest into itself. In this paper we present a method to construct these group operations by means of velocity tangent vector fields. Field curves are to be interpreted as curves along which customers move. We show, how the varying behaviour of customers in space (e.g. calls in progress in a mobile com- munication network model) in a system fed by a spatial arrival process can be adequately described. Former results allow to compute the transient as well as equilibrium distribution of numbers of customers in any Borel subset of R.

Zero-Shot Leaning with Manifold Embedding

Abstract: Zero-Shot Learning (ZSL) has gained its popularity recently owing to its promising characteristic that requires no training data to recognize new visual classes. One key technique is to transfer knowledge from the seen classes to the new unseen classes in an intermediate embedding space for both visual and textual modalities. Therefore, the construction of the embedding space is extremely important. Manifold embedding is able to well capture the intrinsic structure of the embedding space. To this end, with the assumption that the distribution of the semantic categories in the word vector space has an intrinsic manifold structure, this paper proposes a Manifold Embedding based ZSL (ME-ZSL) approach by formulating the manifold structure for the visual to textual embedding with the intra-class compactness, the inter-class separability, and the locality preservation. The linear, closed-form solution makes ME-ZSL efficient to compute. Extensive experiments on the popular AwA and CUB datasets validate the effectiveness of ME-ZSL.