15 research outputs found
Learning Probability Measures with respect to Optimal Transport Metrics
We study the problem of estimating, in the sense of optimal transport
metrics, a measure which is assumed supported on a manifold embedded in a
Hilbert space. By establishing a precise connection between optimal transport
metrics, optimal quantization, and learning theory, we derive new probabilistic
bounds for the performance of a classic algorithm in unsupervised learning
(k-means), when used to produce a probability measure derived from the data. In
the course of the analysis, we arrive at new lower bounds, as well as
probabilistic upper bounds on the convergence rate of the empirical law of
large numbers, which, unlike existing bounds, are applicable to a wide class of
measures.Comment: 13 pages, 2 figures. Advances in Neural Information Processing
Systems, NIPS 201
Orphan-Free Anisotropic Voronoi Diagrams
We describe conditions under which an appropriately-defined anisotropic
Voronoi diagram of a set of sites in Euclidean space is guaranteed to be
composed of connected cells in any number of dimensions. These conditions are
natural for problems in optimization and approximation, and algorithms already
exist to produce sets of sites that satisfy them.Comment: 17 pages, 6 figure
The Fourteenth Data Release of the Sloan Digital Sky Survey: First Spectroscopic Data from the extended Baryon Oscillation Spectroscopic Survey and from the second phase of the Apache Point Observatory Galactic Evolution Experiment
The fourth generation of the Sloan Digital Sky Survey (SDSS-IV) has been in
operation since July 2014. This paper describes the second data release from
this phase, and the fourteenth from SDSS overall (making this, Data Release
Fourteen or DR14). This release makes public data taken by SDSS-IV in its first
two years of operation (July 2014-2016). Like all previous SDSS releases, DR14
is cumulative, including the most recent reductions and calibrations of all
data taken by SDSS since the first phase began operations in 2000. New in DR14
is the first public release of data from the extended Baryon Oscillation
Spectroscopic Survey (eBOSS); the first data from the second phase of the
Apache Point Observatory (APO) Galactic Evolution Experiment (APOGEE-2),
including stellar parameter estimates from an innovative data driven machine
learning algorithm known as "The Cannon"; and almost twice as many data cubes
from the Mapping Nearby Galaxies at APO (MaNGA) survey as were in the previous
release (N = 2812 in total). This paper describes the location and format of
the publicly available data from SDSS-IV surveys. We provide references to the
important technical papers describing how these data have been taken (both
targeting and observation details) and processed for scientific use. The SDSS
website (www.sdss.org) has been updated for this release, and provides links to
data downloads, as well as tutorials and examples of data use. SDSS-IV is
planning to continue to collect astronomical data until 2020, and will be
followed by SDSS-V.Comment: SDSS-IV collaboration alphabetical author data release paper. DR14
happened on 31st July 2017. 19 pages, 5 figures. Accepted by ApJS on 28th Nov
2017 (this is the "post-print" and "post-proofs" version; minor corrections
only from v1, and most of errors found in proofs corrected
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Unique specular shape from two specular flows
When a curved mirror-like surface moves relative to its environment, it induces a motion fieldâor specular flowâon the image plane that observes it. This specular flow is related to the mirrorâs shape through a non-linear partial differential equation, and there is interest in understanding when and how this equation can be solved for surface shape. Existing analyses of this âshape from specular flow equationâ have focused on closed-form solutions, and while they have yielded insight, their critical reliance on externally-provided initial conditions and/or specific motions makes them difficult to apply in practice. This paper resolves these issues. We show that a suitable reparameterization leads to a linear formulation of the shape from specular flow equation. This formulation radically simplifies the reconstruction process and allows, for example, both motion and shape to be recovered from as few as two specular flows even when no externally-provided initial conditions are available. The result of our analysis is a practical method for recovering shape from specular flow that operates under arbitrary, unknown motions in unknown illumination environments and does not require additional shape information from other sources.Engineering and Applied Science