# Epsilon-entropy

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

of a set in a metric space

The logarithm to the base 2 of the smallest number of points in an -net for this set. In other words, the -entropy of a set lying in a metric space is where and is a ball of radius with centre at . The definition of was given by A.N. Kolmogorov in , motivated by ideas and definitions of information theory (cf. Information theory). is also called the relative -entropy; it depends on the space in which the set is situated, that is, on the metric extension of . The quantity where the infimum is taken over all metric extensions of , is called the absolute -entropy of . It may also be defined directly (Kolmogorov, ): for a metric space , is the logarithm to the base 2 of the cardinality of the most economic (by the number of sets) -covering of . Here a system of sets is called a -covering of if , and the diameter of each set does not exceed . It has been shown  that the absolute -entropy is the minimum relative -entropy. The quantities and are inverse to the widths (cf. Width) and . This characterizes the fact that it is possible to recover the elements of from tables of elements and to approximate by -point sets.

The investigation of the asymptotic behaviour of the -entropy of various function classes is a special branch of approximation theory.

How to Cite This Entry:
Epsilon-entropy. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Epsilon-entropy&oldid=12568
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article