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Audiovisual Mathematics Library

Audiovisual Mathematics Library

Documents 53C17 3 results

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Score matching and sub-Riemannian bridges - Grong, Erlend (Author of the conference) | CIRM H

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We discuss how to simulate bridge processes by conditioning a stochastic process on a manifold whose generator is a hypo-elliptic operator. This operator is, up to a drift-term, the sub-Laplacian of a bracketgenerating sub-Riemannian structure, meaning in particular that it has positive smooth density everywhere. The logarithmic gradient of this density is called the score, and we show that it is needed to describe the generator of the bridge process. We therefore discuss several methods for how we can estimate the score using a neural network, with examples. The results are from a joint work with Stefan Sommer (Copenhagen) and Karen Habermann (Warwick).[-]
We discuss how to simulate bridge processes by conditioning a stochastic process on a manifold whose generator is a hypo-elliptic operator. This operator is, up to a drift-term, the sub-Laplacian of a bracketgenerating sub-Riemannian structure, meaning in particular that it has positive smooth density everywhere. The logarithmic gradient of this density is called the score, and we show that it is needed to describe the generator of the bridge ...[+]

58J65 ; 53C17 ; 62R30

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Isoperimetry with density - Morgan, Frank (Author of the conference) | CIRM H

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In 2015 Chambers proved the Log-convex Density Conjecture, which says that for a radial density f on $R^n$, spheres about the origin are isoperimetric if and only if log f is convex (the stability condition). We discuss recent progress and open questions for other densities, unequal perimeter and volume densities, and other metrics.

49Q20 ; 53C17 ; 49N60

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Geometric control and sub-Riemannian geodesics - Part I - Rifford, Ludovic (Author of the conference) | CIRM H

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This will be an introduction to sub-Riemannian geometry from the point of view of control theory. We will define sub-Riemannian structures and prove the Chow Theorem. We will describe normal and abnormal geodesics and discuss the completeness of the Carnot-Carathéodory distance (Hopf-Rinow Theorem). Several examples will be given (Heisenberg group, Martinet distribution, Grusin plane).

53C17 ; 49Jxx

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