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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.


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TL;DR: In this paper, the authors considered the problem of finding a ruled submanifold with constant tangent space along the rulings of a smooth regular curve, such that its tangent bundle along the regular curve coincides with the tangent bundles along a regular curve.
Abstract: Given a smooth distribution $\mathscr{D}$ of $m$-dimensional planes along a smooth regular curve $\gamma$ in $\mathbb{R}^{m+n}$, we consider the following problem: To find an $m$-dimensional developable submanifold of $\mathbb{R}^{m+n}$, that is, a ruled submanifold with constant tangent space along the rulings, such that its tangent bundle along $\gamma$ coincides with $\mathscr{D}$. In particular, we give sufficient conditions for the local well-posedness of the problem, together with a parametric description of the solution.
Journal ArticleDOI
TL;DR: The notion of a parallelizable distribution has been introduced and investigated in this article, and a non-integrable parallelizable distributed distribution carries a natural sub-Riemannian structure.
Abstract: The notion of a parallelizable distribution has been introduced and investigated. A non-integrable parallelizable distribution carries a natural sub-Riemannian structure. The geometry of this struc...
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TL;DR: Persistent Homology for Wasserstein Auto-Encoders, called PHom-WAE, is a new methodology to assess and measure the data distribution of a generative model and uses persistent homology, the study of the topological features of a space at different spatial resolutions, to compare the nature of the latent manifold and the reconstructed distribution.
Abstract: Auto-encoders are among the most popular neural network architecture for dimension reduction. They are composed of two parts: the encoder which maps the model distribution to a latent manifold and the decoder which maps the latent manifold to a reconstructed distribution. However, auto-encoders are known to provoke chaotically scattered data distribution in the latent manifold resulting in an incomplete reconstructed distribution. Current distance measures fail to detect this problem because they are not able to acknowledge the shape of the data manifolds, i.e. their topological features, and the scale at which the manifolds should be analyzed. We propose Persistent Homology for Wasserstein Auto-Encoders, called PHom-WAE, a new methodology to assess and measure the data distribution of a generative model. PHom-WAE minimizes the Wasserstein distance between the true distribution and the reconstructed distribution and uses persistent homology, the study of the topological features of a space at different spatial resolutions, to compare the nature of the latent manifold and the reconstructed distribution. Our experiments underline the potential of persistent homology for Wasserstein Auto-Encoders in comparison to Variational Auto-Encoders, another type of generative model. The experiments are conducted on a real-world data set particularly challenging for traditional distance measures and auto-encoders. PHom-WAE is the first methodology to propose a topological distance measure, the bottleneck distance, for Wasserstein Auto-Encoders used to compare decoded samples of high quality in the context of credit card transactions.
Journal ArticleDOI
01 Mar 1970
TL;DR: In this article, the authors define the parameter of distribution and the line of striction in relation to the enveloping space and show that they have the usual properties in the hyperbolic case.
Abstract: Bonnet's theorem on ruled surfaces [2, p. 449] deals solely with intrinsic properties if an intrinsic definition of the line of striction is adopted. Contrasting this, our aim is to define the parameter of distribution and the line of striction in relation to the enveloping space and to show that they have the usual properties. Attention will be called occasionally to the changes required to treat the hyperbolic case. The two theorems hold in hyperbolic space with minor variations and follow by formally analogous proofs. In fact, in the first one K S -1 must be assumed and in the second the hyperbolic tangent of the distance has to be used. Notation and terminology will largely be taken from [1]. Let "D be the standard connection on the Euclidean 4-space such that "DvW= (VWi)ei where ei, i= 1, 2, 3, 4 constitutes the natural frame field. Our elliptic 3-space is represented by the unit hypersphere (x, x) = 1, so the position vector x serves as unit normal and we have IDvx = V. If now 'D stands for the induced connection on the elliptic space we find for vectors tangent to it
Book ChapterDOI
01 Jan 1993
TL;DR: Theorem 7.3.5 as discussed by the authors gives an exact functor from RH(D X ) into the category of regular holonomic modules on the conjugate complex manifold defined by
Abstract: The first section treats analytic D-module theory on real analytic manifolds and some basic results concerned with extendible distributions is presented in section 2 as a preparation to section 3. There we prove that every regular holonomic D X -module on a complex manifold is locally a cyclic module generated by a distribution on the underlying real manifold. The main result is Theorem 7.3.5 which gives an exact functor from RH(D X ) into the category of regular holonomic modules on the conjugate complex manifold defined by

Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20221
202159
202067
201953
201843
201733