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Gromov-Monge quasi-metrics and distance distributions
Applications in data science, shape analysis and object classification
frequently require maps between metric spaces which preserve geometry as
faithfully as possible. In this paper, we combine the Monge formulation of
optimal transport with the Gromov-Hausdorff distance construction to define a
measure of the minimum amount of geometric distortion required to map one
metric measure space onto another. We show that the resulting quantity, called
Gromov-Monge distance, defines an extended quasi-metric on the space of
isomorphism classes of metric measure spaces and that it can be promoted to a
true metric on certain subclasses of mm-spaces. We also give precise
comparisons between Gromov-Monge distance and several other metrics which have
appeared previously, such as the Gromov-Wasserstein metric and the continuous
Procrustes metric of Lipman, Al-Aifari and Daubechies. Finally, we derive
polynomial-time computable lower bounds for Gromov-Monge distance. These lower
bounds are expressed in terms of distance distributions, which are classical
invariants of metric measure spaces summarizing the volume growth of metric
balls. In the second half of the paper, which may be of independent interest,
we study the discriminative power of these lower bounds for simple subclasses
of metric measure spaces. We first consider the case of planar curves, where we
give a counterexample to the Curve Histogram Conjecture of Brinkman and Olver.
Our results on plane curves are then generalized to higher dimensional
manifolds, where we prove some sphere characterization theorems for the
distance distribution invariant. Finally, we consider several inverse problems
on recovering a metric graph from a collection of localized versions of
distance distributions. Results are derived by establishing connections with
concepts from the fields of computational geometry and topological data
analysis.Comment: Version 2: Added many new results and improved expositio
Lecture 09: Hierarchically Low Rank and Kronecker Methods
Exploiting structures of matrices goes beyond identifying their non-zero patterns. In many cases, dense full-rank matrices have low-rank submatrices that can be exploited to construct fast approximate algorithms. In other cases, dense matrices can be decomposed into Kronecker factors that are much smaller than the original matrix. Sparsity is a consequence of the connectivity of the underlying geometry (mesh, graph, interaction list, etc.), whereas the rank-deficiency of submatrices is closely related to the distance within this underlying geometry. For high dimensional geometry encountered in data science applications, the curse of dimensionality poses a challenge for rank-structured approaches. On the other hand, models in data science that are formulated as a composition of functions, lead to a Kronecker product structure that yields a different kind of fast algorithm. In this lecture, we will look at some examples of when rank structure and Kronecker structure can be useful
Preliminary results for RR Lyrae stars and Classical Cepheids from the Vista Magellanic Cloud (VMC) Survey
The Vista Magellanic Cloud (VMC, PI M.R. Cioni) survey is collecting
-band time series photometry of the system formed by the two Magellanic
Clouds (MC) and the "bridge" that connects them. These data are used to build
-band light curves of the MC RR Lyrae stars and Classical Cepheids and
determine absolute distances and the 3D geometry of the whole system using the
-band period luminosity (), the period - luminosity - color ()
and the Wesenhiet relations applicable to these types of variables. As an
example of the survey potential we present results from the VMC observations of
two fields centered respectively on the South Ecliptic Pole and the 30 Doradus
star forming region of the Large Magellanic Cloud. The VMC -band light
curves of the RR Lyrae stars in these two regions have very good photometric
quality with typical errors for the individual data points in the range of
0.02 to 0.05 mag. The Cepheids have excellent light curves (typical
errors of 0.01 mag). The average magnitudes derived for both types
of variables were used to derive relations that are in general good
agreement within the errors with the literature data, and show a smaller
scatter than previous studies.Comment: 7 pages, 6 figure. Accepted for publication in Astrophysics and Space
Science. Following a presentation at the conference "The Fundamental Cosmic
Distance Scale: State of the Art and the Gaia Perspective", Naples, May 201
Physics-guided adversarial networks for artificial digital image correlation data generation
Digital image correlation (DIC) has become a valuable tool in the evaluation
of mechanical experiments, particularly fatigue crack growth experiments. The
evaluation requires accurate information of the crack path and crack tip
position, which is difficult to obtain due to inherent noise and artefacts.
Machine learning models have been extremely successful in recognizing this
relevant information given labelled DIC displacement data. For the training of
robust models, which generalize well, big data is needed. However, data is
typically scarce in the field of material science and engineering because
experiments are expensive and time-consuming. We present a method to generate
synthetic DIC displacement data using generative adversarial networks with a
physics-guided discriminator. To decide whether data samples are real or fake,
this discriminator additionally receives the derived von Mises equivalent
strain. We show that this physics-guided approach leads to improved results in
terms of visual quality of samples, sliced Wasserstein distance, and geometry
score
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