2,371 research outputs found
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
A Topological Distance between Multi-fields based on Multi-Dimensional Persistence Diagrams
The problem of computing topological distance between two scalar fields based
on Reeb graphs or contour trees has been studied and applied successfully to
various problems in topological shape matching, data analysis, and
visualization. However, generalizing such results for computing distance
measures between two multi-fields based on their Reeb spaces is still in its
infancy. Towards this, in the current paper we propose a technique to compute
an effective distance measure between two multi-fields by computing a novel
\emph{multi-dimensional persistence diagram} (MDPD) corresponding to each of
the (quantized) Reeb spaces. First, we construct a multi-dimensional Reeb graph
(MDRG), which is a hierarchical decomposition of the Reeb space into a
collection of Reeb graphs. The MDPD corresponding to each MDRG is then computed
based on the persistence diagrams of the component Reeb graphs of the MDRG. Our
distance measure extends the Wasserstein distance between two persistence
diagrams of Reeb graphs to MDPDs of MDRGs. We prove that the proposed measure
is a pseudo-metric and satisfies a stability property. Effectiveness of the
proposed distance measure has been demonstrated in (i) shape retrieval contest
data - SHREC and (ii) Pt-CO bond detection data from computational
chemistry. Experimental results show that the proposed distance measure based
on the Reeb spaces has more discriminating power in clustering the shapes and
detecting the formation of a stable Pt-CO bond as compared to the similar
measures between Reeb graphs.Comment: Acepted in the IEEE Transactions on Visualization and Computer
Graphic
Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph
Data-sensitive metrics adapt distances locally based the density of data
points with the goal of aligning distances and some notion of similarity. In
this paper, we give the first exact algorithm for computing a data-sensitive
metric called the nearest neighbor metric. In fact, we prove the surprising
result that a previously published -approximation is an exact algorithm.
The nearest neighbor metric can be viewed as a special case of a
density-based distance used in machine learning, or it can be seen as an
example of a manifold metric. Previous computational research on such metrics
despaired of computing exact distances on account of the apparent difficulty of
minimizing over all continuous paths between a pair of points. We leverage the
exact computation of the nearest neighbor metric to compute sparse spanners and
persistent homology. We also explore the behavior of the metric built from
point sets drawn from an underlying distribution and consider the more general
case of inputs that are finite collections of path-connected compact sets.
The main results connect several classical theories such as the conformal
change of Riemannian metrics, the theory of positive definite functions of
Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop
novel proof techniques based on the combination of screw functions and
Lipschitz extensions that may be of independent interest.Comment: 15 page
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