Chapter 5 On Complexity , Sampling , and ε - Nets and ε - Samples
نویسنده
چکیده
x " I've never touched the hard stuff, only smoked grass a few times with the boys to be polite, and that's all, though ten is the age when the big guys come around teaching you all sorts to things. But happiness doesn't mean much to me, I still think life is better. Happiness is a mean son of a bitch and needs to be put in his place. Him and me aren't on the same team, and I'm cutting him dead. I've never gone in for politics, because somebody always stand to gain by it, but happiness is an even crummier racket, and their ought to be laws to put it out of business. " – Momo, Emile Ajar. In this chapter we will try to quantify the notion of geometric complexity. It is intuitively clear that a a (i.e., disk) is a simpler shape than an c (i.e., ellipse), which is in turn simpler than a (i.e., smiley). This becomes even more important when we consider several such shapes and how they interact with each other. As these examples might demonstrate, this notion of complexity is somewhat elusive. Next, we show that one can capture the structure of a distribution/point set by a small subset. The size here would depend on the complexity of the shapes/ranges we care about, but it would be independent of the size of the point set. 5.1 VC Dimension Definition 5.1.1 A range space S is a pair (X, R), where X is a ground set (finite or infinite) and R is a (finite or infinite) family of subsets of X. The elements of X are points and the elements of R are ranges. Our interest is in the size/weight of the ranges in the range space. For technical reasons, it will be easier to consider a finite subset x as the underlining ground set. x This work is licensed under the Creative Commons Attribution-Noncommercial 3.0 License. To view a copy of this license, visit
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