Covering Numbers for Semicontinuous Functions
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Authors
Royset, Johannes O.
Subjects
covering numbers
metric entropy numbers
semicontinuous functions
epi-distance
Attouch-Wets topology
epi-convergence
epi-spline
approximation theory
metric entropy numbers
semicontinuous functions
epi-distance
Attouch-Wets topology
epi-convergence
epi-spline
approximation theory
Advisors
Date of Issue
2016-04-29
Date
Publisher
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Abstract
Considering the metric space of extended real-valued lower semicontinuous functions under the epi-distance, the paper gives an upper bound on the covering numbers of bounded subsets of such functions. No assumptions about continuity, smoothness, variation, and even finiteness of the functions are needed. The bound is shown to be nearly sharp through the construction of a set of functions with covering numbers deviating from the upper bound only by a logarithmic factor. The analogy between lower and upper semicontinuous functions implies that identical covering numbers hold for bounded sets of the latter class of functions as well, but now under the hypo-distance metric.
Type
Preprint
Description
This paper is in review.
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Identifiers
NPS Report Number
Sponsors
DARPA under grant HR0011-14-1-0060
Funder
DARPA under grant HR0011-14-1-0060
Format
11 p.
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Rights
This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.