Circles - then d 1 ~x ~ y is just the Hamming distance...

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What shape is a circle? Complex Systems Summer School, Santa Fe Institute Tom Carter http://astarte.csustan.edu/˜ tom/SFI-CSSS June 24, 2006 1
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How do we define a circle? We usually define a circle as C = { x | || x || = 1 } (i.e., the set of all vectors x of length 1). Of course, we need to make sense of || x || , the length of a vector. Most people start with the definition || x || = i x 2 i 1 2 , but since we are mathematicians, and don’t like our definitions to be too special, we can generalize: || x || p = i | x i | p 1 p . for p > 0. 2
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We then have the more general definition of the p circle (in dimension n ): C n p = { x | || x || p = 1 } . What does C n p look like? Let’s work in two dimensions, and leave out the dimension label. C 2 is the familiar circle: 3
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C 1 is a diamond: Note that if our vector space is over { 0 , 1 } , then a vector is just a string of zeros and ones, and || x || 1 is just the number of ones in the string. We can convert our length measures into a distance measures: d p ( x, y ) = || x - y || p = i | x i - y i | p 1 p . 4
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In particular, if our vector space is over { 0 ,
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Unformatted text preview: } , then d 1 ( ~x, ~ y ) is just the Hamming distance between the two vectors (i.e., the number of places in which the two strings differ). We can also define || ~x || ∞ = lim p →∞ ( || ~x || p ) . Going through the math, we have that || ~x || ∞ = max i ( | x i | ) . We then have that C ∞ is a square: 5 This means that for a mathematician, a circle is a circle, is a diamond, is a square (which may explain why I always had trouble with those “shape matching” tests . . . :-) In general, we have the following sort of picture of various circles: 6 Homework exercises: What happens for 0 < p < 1? What happens if we take the limit as p goes to 0? Show that in the limit as p goes to 0, the corresponding distance d ( ~x, ~ y ) = || ~x-~ y || is a generalized Hamming distance that counts the number of coordinates that are different from each other . . . 7...
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