Topic
Smooth structure
About: Smooth structure is a research topic. Over the lifetime, 486 publications have been published within this topic receiving 6511 citations.
Papers published on a yearly basis
1 / 2
Papers
TL;DR: This paper focuses on the geodesic-minimal-spanning-tree (GMST) method, which uses the overall lengths of the MSTs to simultaneously estimate manifold dimension and entropy.
Abstract: In the manifold learning problem, one seeks to discover a smooth low dimensional surface, i.e., a manifold embedded in a higher dimensional linear vector space, based on a set of measured sample points on the surface. In this paper, we consider the closely related problem of estimating the manifold's intrinsic dimension and the intrinsic entropy of the sample points. Specifically, we view the sample points as realizations of an unknown multivariate density supported on an unknown smooth manifold. We introduce a novel geometric approach based on entropic graph methods. Although the theory presented applies to this general class of graphs, we focus on the geodesic-minimal-spanning-tree (GMST) to obtaining asymptotically consistent estimates of the manifold dimension and the Re/spl acute/nyi /spl alpha/-entropy of the sample density on the manifold. The GMST approach is striking in its simplicity and does not require reconstruction of the manifold or estimation of the multivariate density of the samples. The GMST method simply constructs a minimal spanning tree (MST) sequence using a geodesic edge matrix and uses the overall lengths of the MSTs to simultaneously estimate manifold dimension and entropy. We illustrate the GMST approach on standard synthetic manifolds as well as on real data sets consisting of images of faces.
330 citations
TL;DR: It is shown under a natural regularity condition that critical points of partly smooth functions are stable: small perturbations to the function cause small movements of the critical point on the active manifold.
Abstract: Nonsmoothness pervades optimization, but the way it typically arises is highly structured. Nonsmooth behavior of an objective function is usually associated, locally, with an active manifold: on this manifold the function is smooth, whereas in normal directions it is "vee-shaped." Active set ideas in optimization depend heavily on this structure. Important examples of such functions include the pointwise maximum of some smooth functions and the maximum eigenvalue of a parametrized symmetric matrix. Among possible foundations for practical nonsmooth optimization, this broad class of "partly smooth" functions seems a promising candidate, enjoying a powerful calculus and sensitivity theory. In particular, we show under a natural regularity condition that critical points of partly smooth functions are stable: small perturbations to the function cause small movements of the critical point on the active manifold.
235 citations
1 May 1987
TL;DR: Views of objects composed of smooth surface patches whose intersections form smooth space curves are classified by mappings and diagrams of mappings from the line to the plane or from the plane to the planes.
Abstract: Views of objects composed of smooth surface patches whose intersections form smooth space curves are classified. These views may he described algebraically by mappings and diagrams of mappings from the line to the plane (for the crease) or from the plane to the plane (for the apparent contour). It is possible to derive a finite catalogue of generic views and their transitions, so that every view is either (up to smooth coordinate changes in source and target) equivalent to one of those in the catalogue or is of sufficiently high codimension.
75 citations
TL;DR: In this paper, an operadic description of the obstructions to deforming smooth immersions into smooth embeddings is given, based on the Goodwillie-Klein-Weiss manifold calculus.
Abstract: In the homotopical study of spaces of smooth embeddings, the functor calculus method (Goodwillie-Klein-Weiss manifold calculus) has opened up important connections to operad theory. Using this and a few simplifying observations, we arrive at an operadic description of the obstructions to deforming smooth immersions into smooth embeddings. We give an application which in some respects improves on recent results of Arone-Turchin and Dwyer-Hess concerning high-dimensional variants of spaces of long knots.
75 citations
TL;DR: In this article, the moduli space of all infinitesimal special Lagrangian deformations of L in a symplectic manifold with non-integrable almost complex structure is also a smooth manifold of dimension b1(L), the first Betti number of L.
Abstract: In [7], R. C. McLean showed that the moduli space of nearby submanifolds of a smooth, compact, orientable special Lagrangian submanifold L in a Calabi-Yau manifold X is a smooth manifold and its dimension is equal to the dimension of ℋ1(L), the space of harmonic 1-forms on L. In this paper, we will show that the moduli space of all infinitesimal special Lagrangian deformations of L in a symplectic manifold with non-integrable almost complex structure is also a smooth manifold of dimension b1(L), the first Betti number of L.
62 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 9 |
| 2020 | 9 |
| 2019 | 10 |
| 2018 | 11 |
| 2017 | 26 |
| 2016 | 23 |