EECS 281 B / STAT 241 B : Advanced Topics in Statistical Learning Spring 2009 Lecture 16 — March 13 Lecturer :
نویسنده
چکیده
16.1 Recap Last time (and also in Homework #3), we introduced the notion of covering and packing numbers. Definition 1. Given a set S and a metric ρ and tolerance ǫ, the ǫ-covering number is given by N(ǫ;S, ρ) = min { n | S ⊆ ∪i=1Bǫ(y), for some y(i) ∈ S} } . Here Bǫ(y) = {x | ρ(x, y) ≤ ǫ} is the ball of radius ǫ centered at y. Thus, the ǫ-covering number is the minimum number of balls of radius ǫ required to cover the set S. Of interest to us is the growth rate of the covering number as ǫ → 0, which we measure in terms of the metric entropy H(ǫ;S, ρ) := logN(ǫ;S, ρ). This metric entropy provides some notion of the richness of the set S. These notions go back to the seminal work of Kolmogorov and Tikhomirov (1961), and are studied extensively in the approximation theory literature. A closely related quantity is the packing number: Definition 2. The ǫ-packing number of a set S is the maximal number M = M(ǫ;S, ρ) of points {y(1), . . . , y(M)} with ρ(y(i), y(j)) > ǫ for all i 6= j. Note: This definition of packing number is slightly different than the version M̃ from HW #3; they are related by M̃(ǫ) = M(2ǫ). In HW #3, we proved the following relationships (translated appropriately for our new definition M of packing): M(2ǫ;S, ρ) ≤ N(ǫ;S, ρ) ≤ M(ǫ;S, ρ). (16.1)
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