50 research outputs found
A general theorem of existence of quasi absolutely minimal Lipschitz extensions
In this paper we consider a wide class of generalized Lipschitz extension
problems and the corresponding problem of finding absolutely minimal Lipschitz
extensions. We prove that if a minimal Lipschitz extension exists, then under
certain other mild conditions, a quasi absolutely minimal Lipschitz extension
must exist as well. Here we use the qualifier "quasi" to indicate that the
extending function in question nearly satisfies the conditions of being an
absolutely minimal Lipschitz extension, up to several factors that can be made
arbitrarily small.Comment: 33 pages. v3: Correction to Example 2.4.3. Specifically,
alpha-H\"older continuous functions, for alpha strictly less than one, do not
satisfy (P3). Thus one cannot conclude that quasi-AMLEs exist in this case.
Please note that the error remains in the published version of the paper in
Mathematische Annalen. v2: Several minor corrections and edits, a new
appendix (Appendix A
Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds
The Euclidean scattering transform was introduced nearly a decade ago to
improve the mathematical understanding of convolutional neural networks.
Inspired by recent interest in geometric deep learning, which aims to
generalize convolutional neural networks to manifold and graph-structured
domains, we define a geometric scattering transform on manifolds. Similar to
the Euclidean scattering transform, the geometric scattering transform is based
on a cascade of wavelet filters and pointwise nonlinearities. It is invariant
to local isometries and stable to certain types of diffeomorphisms. Empirical
results demonstrate its utility on several geometric learning tasks. Our
results generalize the deformation stability and local translation invariance
of Euclidean scattering, and demonstrate the importance of linking the used
filter structures to the underlying geometry of the data.Comment: 35 pages; 3 figures; 2 tables; v3: Revisions based on reviewer
comment