The Johnson-Lindenstrauss lemma and the sphericity of some graphs
نویسندگان
چکیده
منابع مشابه
The Johnson-Lindenstrauss lemma and the sphericity of some graphs
A simple short proof of the Johnson-Lindenstrauss lemma (concerning nearly isometric embeddings of finite point sets in lower-dimensional spaces) is given. This result is applied to show that if G is a graph on n vertices and with smallest eigenvalue i then its sphericity sph(G) is less than cA2 log n. It is also proved that if G or its complement is a forest then sph(G) < c log n holds. Q 19%8...
متن کاملThe Johnson-Lindenstrauss Lemma
Definition 1.1 Let N(0, 1) denote the one dimensional normal distribution. This distribution has density n(x) = e−x 2/2/ √ 2π. LetNd(0, 1) denote the d-dimensional Gaussian distribution, induced by picking each coordinate independently from the standard normal distribution N(0, 1). Let Exp(λ) denote the exponential distribution, with parameter λ. The density function of the exponential distribu...
متن کامل10 : The Johnson - Lindenstrauss Lemma ∗
In this chapter, we will prove that given a set P of n points in IR, one can reduce the dimension of the points to k = O(ε−2 log n) and distances are 1 ± ε reserved. Surprisingly, this reduction is done by randomly picking a subspace of k dimensions and projecting the points into this random subspace. One way of thinking about this result is that we are “compressing” the input of size nd (i.e.,...
متن کاملOn variants of the Johnson-Lindenstrauss lemma
The Johnson–Lindenstrauss lemma asserts that an n-point set in any Euclidean space can be mapped to a Euclidean space of dimension k = O(ε−2 log n) so that all distances are preserved up to a multiplicative factor between 1 − ε and 1 + ε. Known proofs obtain such a mapping as a linear map Rn → Rk with a suitable random matrix. We give a simple and self-contained proof of a version of the Johnso...
متن کاملThe Johnson-Lindenstrauss Lemma Meets Compressed Sensing
We show how two fundamental results in analysis related to n-widths and Compressed Sensing are intimately related to the Johnson-Lindenstrauss lemma. Our elementary approach is based on the same concentration inequalities for random inner products that have recently provided simple proofs of the Johnson-Lindenstrauss lemma. We show how these ideas lead to simple proofs of Kashin’s theorems on w...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series B
سال: 1988
ISSN: 0095-8956
DOI: 10.1016/0095-8956(88)90043-3