21,153 research outputs found
Klee sets and Chebyshev centers for the right Bregman distance
We systematically investigate the farthest distance function, farthest
points, Klee sets, and Chebyshev centers, with respect to Bregman distances
induced by Legendre functions. These objects are of considerable interest in
Information Geometry and Machine Learning; when the Legendre function is
specialized to the energy, one obtains classical notions from Approximation
Theory and Convex Analysis.
The contribution of this paper is twofold. First, we provide an affirmative
answer to a recently-posed question on whether or not every Klee set with
respect to the right Bregman distance is a singleton. Second, we prove
uniqueness of the Chebyshev center and we present a characterization that
relates to previous works by Garkavi, by Klee, and by Nielsen and Nock.Comment: 23 pages, 2 figures, 14 image
Bregman distances and Chebyshev sets
A closed set of a Euclidean space is said to be Chebyshev if every point in
the space has one and only one closest point in the set. Although the situation
is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that
in Euclidean spaces a closed set is Chebyshev if and only if the set is convex.
In this paper, from the more general perspective of Bregman distances, we show
that if every point in the space has a unique nearest point in a closed set,
then the set is convex. We provide two approaches: one is by nonsmooth
analysis; the other by maximal monotone operator theory. Subdifferentiability
properties of Bregman nearest distance functions are also given
Cosmographic analysis with Chebyshev polynomials
The limits of standard cosmography are here revised addressing the problem of
error propagation during statistical analyses. To do so, we propose the use of
Chebyshev polynomials to parameterize cosmic distances. In particular, we
demonstrate that building up rational Chebyshev polynomials significantly
reduces error propagations with respect to standard Taylor series. This
technique provides unbiased estimations of the cosmographic parameters and
performs significatively better than previous numerical approximations. To
figure this out, we compare rational Chebyshev polynomials with Pad\'e series.
In addition, we theoretically evaluate the convergence radius of (1,1)
Chebyshev rational polynomial and we compare it with the convergence radii of
Taylor and Pad\'e approximations. We thus focus on regions in which convergence
of Chebyshev rational functions is better than standard approaches. With this
recipe, as high-redshift data are employed, rational Chebyshev polynomials
remain highly stable and enable one to derive highly accurate analytical
approximations of Hubble's rate in terms of the cosmographic series. Finally,
we check our theoretical predictions by setting bounds on cosmographic
parameters through Monte Carlo integration techniques, based on the
Metropolis-Hastings algorithm. We apply our technique to high-redshift cosmic
data, using the JLA supernovae sample and the most recent versions of Hubble
parameter and baryon acoustic oscillation measurements. We find that
cosmography with Taylor series fails to be predictive with the aforementioned
data sets, while turns out to be much more stable using the Chebyshev approach.Comment: 17 pages, 6 figures, 5 table
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