935 research outputs found
Optimally fast incremental Manhattan plane embedding and planar tight span construction
We describe a data structure, a rectangular complex, that can be used to
represent hyperconvex metric spaces that have the same topology (although not
necessarily the same distance function) as subsets of the plane. We show how to
use this data structure to construct the tight span of a metric space given as
an n x n distance matrix, when the tight span is homeomorphic to a subset of
the plane, in time O(n^2), and to add a single point to a planar tight span in
time O(n). As an application of this construction, we show how to test whether
a given finite metric space embeds isometrically into the Manhattan plane in
time O(n^2), and add a single point to the space and re-test whether it has
such an embedding in time O(n).Comment: 39 pages, 15 figure
DIMAL: Deep Isometric Manifold Learning Using Sparse Geodesic Sampling
This paper explores a fully unsupervised deep learning approach for computing
distance-preserving maps that generate low-dimensional embeddings for a certain
class of manifolds. We use the Siamese configuration to train a neural network
to solve the problem of least squares multidimensional scaling for generating
maps that approximately preserve geodesic distances. By training with only a
few landmarks, we show a significantly improved local and nonlocal
generalization of the isometric mapping as compared to analogous non-parametric
counterparts. Importantly, the combination of a deep-learning framework with a
multidimensional scaling objective enables a numerical analysis of network
architectures to aid in understanding their representation power. This provides
a geometric perspective to the generalizability of deep learning.Comment: 10 pages, 11 Figure
Diffusion Variational Autoencoders
A standard Variational Autoencoder, with a Euclidean latent space, is
structurally incapable of capturing topological properties of certain datasets.
To remove topological obstructions, we introduce Diffusion Variational
Autoencoders with arbitrary manifolds as a latent space. A Diffusion
Variational Autoencoder uses transition kernels of Brownian motion on the
manifold. In particular, it uses properties of the Brownian motion to implement
the reparametrization trick and fast approximations to the KL divergence. We
show that the Diffusion Variational Autoencoder is capable of capturing
topological properties of synthetic datasets. Additionally, we train MNIST on
spheres, tori, projective spaces, SO(3), and a torus embedded in R3. Although a
natural dataset like MNIST does not have latent variables with a clear-cut
topological structure, training it on a manifold can still highlight
topological and geometrical properties.Comment: 10 pages, 8 figures Added an appendix with derivation of asymptotic
expansion of KL divergence for heat kernel on arbitrary Riemannian manifolds,
and an appendix with new experiments on binarized MNIST. Added a previously
missing factor in the asymptotic expansion of the heat kernel and corrected a
coefficient in asymptotic expansion KL divergence; further minor edit
On the moduli space of positive Ricci curvature metrics on homotopy spheres
We show that the moduli space of Ricci positive metrics on certain homotopy
spheres has infinitely many connected components.Comment: 28 pages, 11 figures. The text has been substantially re-written to
improve the expositio
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