The Bregman distance, approximate compactness and convexity of Chebyshev sets in Banach spaces
نویسندگان
چکیده
منابع مشابه
The Bregman distance, approximate compactness and convexity of Chebyshev sets in Banach spaces
We present some sufficient conditions ensuring the upper semicontinuity and the continuity of the Bregman projection operator Π g C and the relative projection operator P g C in terms of the D-approximate (weak) compactness for a nonempty closed set C in a Banach space X . We next present certain sufficient conditions as well as equivalent conditions for the convexity of a Chebyshev subset of a...
متن کاملBregman distance, approximate compactness and convexity of Chebyshev sets in Banach spaces
We present some sufficient conditions ensuring the upper semicontinuity and the continuity of the Bregman projection operator ΠgC and the relative projection operator P g C in terms of the D-approximate (weak) compactness for a nonempty closed set C in a Banach space X. We next present certain sufficient conditions as well as equivalent conditions for the convexity of a Chebyshev subset of a Ba...
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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 Analys...
متن کاملBregman Distances and Klee Sets in Banach Spaces
In this paper, we first present some sufficient conditions for the upper semicontinuity and/or the continuity of the Bregman farthest-point map QgC and the relative farthest-point map S g C for a nonempty D-maximally approximately compact subset C of a Banach space X. We next present certain sufficient conditions as well as equivalent conditions for a Klee set to be singleton in a Banach space ...
متن کاملBregman 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 th...
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ژورنال
عنوان ژورنال: Journal of Approximation Theory
سال: 2010
ISSN: 0021-9045
DOI: 10.1016/j.jat.2009.12.006