Prob 6 - ans 3

Prob 6 - ans 3 - University of Washington Lecturer: James...

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University of Washington Lecturer: James R. Lee CSE 599I: Geometric embeddings and high-dimensional pheneomena February 1, 2007 Scribe: Dang-Trinh Huynh-Ngoc Lecture 6 Our goal in this lecture is to prove the following theorem. Theorem 1. For arbitrary large n N , there exists n -point subsets X 1 such that for any D -embedding of X into d 1 , d n Ω(1 /D 2 ) . Note that the best known upper bound for dimension reduction in 1 is due to Talagrand. It says that every n -dimensional subspace of 1 embeds into O ( n log n ) 1 with O (1) distortion. In particular, this holds for every n -point subset of 1 . 1 Uniform convexity Our proof of Theorem 1 relies on the following uniform convexity inequality for points in L p , for p (1 , 2]. Lemma 2. Fix 1 < p 2 and a,b L p . Then, k a + b k 2 p + ( p - 1) k a - b k 2 p 2( k a k 2 p + k b k 2 p ) . Proof. See course web page. Note that when p = 2, Lemma 2 is the well-known Parallelogram law for Euclidean spaces. The inequality expresses a property of the L p spaces (for p (1 , 2]) called uniform convexity. The geometry of every norm space is reflected in its unit ball and we know that the unit ball of every norm space is convex. For some spaces the ball is even strictly convex, in the following sense. If one fixes X and Y on the boundary of the unit ball, then the midpoint M not only lies inside the ball, but at some non-zero distance inside the boundary. As we can see from Figure 1, the unit ball of L 1 is not strictly convex, while for p (1 , 2], the unit ball of L p is strictly convex. Furthermore, the L p balls for p (1 , 2] are not just strictly convex, they are uniformly convex. In other words, if the length of the segment XY is ² , then there is a certain distance δ
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Prob 6 - ans 3 - University of Washington Lecturer: James...

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