2,283 research outputs found
Counting faces of randomly-projected polytopes when the projection radically lowers dimension
This paper develops asymptotic methods to count faces of random
high-dimensional polytopes. Beyond its intrinsic interest, our conclusions have
surprising implications - in statistics, probability, information theory, and
signal processing - with potential impacts in practical subjects like medical
imaging and digital communications. Three such implications concern: convex
hulls of Gaussian point clouds, signal recovery from random projections, and
how many gross errors can be efficiently corrected from Gaussian error
correcting codes.Comment: 56 page
A Technique to Derive Improved Proper Motions for Kepler Objects of Interest
We outline an approach yielding proper motions with higher precision than
exists in present catalogs for a sample of stars in the Kepler field. To
increase proper motion precision we combine first moment centroids of Kepler
pixel data from a single Season with existing catalog positions and proper
motions. We use this astrometry to produce improved reduced proper motion
diagrams, analogous to a Hertzsprung-Russell diagram, for stars identified as
Kepler Objects of Interest. The more precise the relative proper motions, the
better the discrimination between stellar luminosity classes. With UCAC4 and
PPMXL epoch 2000 positions (and proper motions from those catalogs as
quasi-bayesian priors) astrometry for a single test Channel (21) and Season (0)
spanning two years yields proper motions with an average per-coordinate proper
motion error of 1.0 millisecond of arc per year, over a factor of three better
than existing catalogs. We apply a mapping between a reduced proper motion
diagram and an HR diagram, both constructed using HST parallaxes and proper
motions, to estimate Kepler Object of Interest K-band absolute magnitudes. The
techniques discussed apply to any future small-field astrometry as well as the
rest of the Kepler field.Comment: Accepted to The Astronomical Journal 15 August 201
A boundary integral formalism for stochastic ray tracing in billiards
Determining the flow of rays or non-interacting particles driven by a force or velocity field is fundamental to modelling many physical processes. These include particle flows arising in fluid mechanics and ray flows arising in the geometrical optics limit of linear wave equations. In many practical applications, the driving field is not known exactly and the dynamics are determined only up to a degree of uncertainty. This paper presents a boundary integral framework for propagating flows including uncertainties, which is shown to systematically interpolate between a deterministic and a completely random description of the trajectory propagation. A simple but efficient discretisation approach is applied to model uncertain billiard dynamics in an integrable rectangular domain
Observed Universality of Phase Transitions in High-Dimensional Geometry, with Implications for Modern Data Analysis and Signal Processing
We review connections between phase transitions in high-dimensional
combinatorial geometry and phase transitions occurring in modern
high-dimensional data analysis and signal processing. In data analysis, such
transitions arise as abrupt breakdown of linear model selection, robust data
fitting or compressed sensing reconstructions, when the complexity of the model
or the number of outliers increases beyond a threshold. In combinatorial
geometry these transitions appear as abrupt changes in the properties of face
counts of convex polytopes when the dimensions are varied. The thresholds in
these very different problems appear in the same critical locations after
appropriate calibration of variables.
These thresholds are important in each subject area: for linear modelling,
they place hard limits on the degree to which the now-ubiquitous
high-throughput data analysis can be successful; for robustness, they place
hard limits on the degree to which standard robust fitting methods can tolerate
outliers before breaking down; for compressed sensing, they define the sharp
boundary of the undersampling/sparsity tradeoff in undersampling theorems.
Existing derivations of phase transitions in combinatorial geometry assume
the underlying matrices have independent and identically distributed (iid)
Gaussian elements. In applications, however, it often seems that Gaussianity is
not required. We conducted an extensive computational experiment and formal
inferential analysis to test the hypothesis that these phase transitions are
{\it universal} across a range of underlying matrix ensembles. The experimental
results are consistent with an asymptotic large- universality across matrix
ensembles; finite-sample universality can be rejected.Comment: 47 pages, 24 figures, 10 table
Counting faces of randomly projected polytopes when the projection radically lowers dimension
1.1. Three surprises of high dimensions. This paper develops asymptotic methods to count faces of random high-dimensional polytopes; a seemingly dry and unpromising pursuit. Yet our conclusions have surprising implications- in statistics, probability, information theory, and signal processing- with potential impacts i
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