Given a subset K of the unit Euclidean sphere, we estimate the minimal number m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x, y in K is nearly proportional to the Euclidean distance between x and y. Random hyperplanes prove to be almost ideal for this problem; they achieve the almost optimal bound m = O(w(K)^2) where w(K) is the Gaussian mean width of K. Using the map that sends x in K to the sign vector with respect to the hyperplanes, we conclude that every bounded subset K of R^n embeds into the Hamming cube -1, 1 ^m with a small distortion in the Gromov-Haussdorf metric. Since for many sets K one has m = m(K) << n, this yields a new discrete mechanism of dimension reduction for sets in Euclidean spaces.
Embeddings of subsets @math into normed spaces were studied in geometric functional analysis @cite_10 @cite_14 . In particular, Klartag and Mendelson @cite_10 were concerned with embeddings into @math . They showed that for @math there exists a linear map @math such that @math One can choose @math to be an @math random matrix with Gaussian entries as in Theorem , or with sub-gaussian entries. Schechtman @cite_15 gave a simpler argument for a Gaussian matrix, which also works for embeddings into general normed spaces @math . In the specific case of @math , Schechtman's result states that for @math one has @math This result also follows from Lemma below.
@cite_15 @cite_14 @cite_10
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Consider an @math matrix @math with the restricted isometry property of order k and level @math ; that is, the norm of any k-sparse vector in @math is preserved to within a multiplicative factor of @math under application of @math . We show that by randomizing the column signs of such a matrix @math , the resulting map with high probability embeds any fixed set of @math points in @math into @math without distorting the norm of any point in the set by more than a factor of @math . Consequently, matrices with the restricted isometry property and with randomized column signs provide optimal Johnson–Lindenstrauss embeddings up to logarithmic factors in N. In particular, our results improve the best known bounds on the necessary embedding dimension m for a wide class of structured random matrices; for partial Fourier and partial Hadamard matrices, we improve the recent bound @math given by Ailon and Liberty to $m... The JohnsonLindenstrauss lemma asserts that an n-point set in any Euclidean space can be mapped to a Euclidean space of dimension k = O(e-2 log n) so that all distances are preserved up to a multiplicative factor between 1 - e and 1 + e. 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 JohnsonLindenstrauss lemma that subsumes a basic versions by Indyk and Motwani and a version more suitable for efficient computations due to Achlioptas. (Another proof of this result, slightly different but in a similar spirit, was given independently by Indyk and Naor.) An even more general result was established by Klartag and Mendelson using considerably heavier machinery. Recently, Ailon and Chazelle showed, roughly speaking, that a good mapping can also be obtained by composing a suitable Fourier transform with a linear mapping that has a sparse random matrix M; a mapping of this form can be evaluated very fast. In their result, the nonzero entries of M are normally distributed. We show that the nonzero entries can be chosen as random ± 1, which further speeds up the computation. We also discuss the case of embeddings into Rk with the l1 norm. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008 Abstract This paper contains two results concerning linear embeddings of subsets of Euclidean space in low-dimensional normed spaces. The first is an improvement of the known dependence on ɛ in Dvoretzky's theorem from order of ɛ 2 to order of ɛ (except for log factors). The second is a joint generalization of (Milman's version of) Dvoretzky's theorem and (a recent generalization by Klartag and Mendelson of) the Johnson–Lindenstrauss Lemma. Given an @math matrix @math , its Schatten- @math norm, @math , is defined as @math , where @math is the @math -th largest singular value of @math . These norms have been studied in functional analysis in the context of non-commutative @math -spaces, and recently in data stream and linear sketching models of computation. Basic questions on the relations between these norms, such as their embeddability, are still open. Specifically, given a set of matrices @math , suppose we want to construct a linear map @math such that @math for each @math , where @math and @math , and further, @math for a given approximation factor @math and real number @math . Then how large do @math and @math need to be as a function of @math ? We nearly resolve this question for every @math , for the case where @math can be expressed as @math , where @math and @math are arbitrary matrices that are allowed to depend on @math , that is, @math can be implemented by left and right matrix multiplication. Namely, for every @math , we provide nearly matching upper and lower bounds on the size of @math and @math as a function of @math . Importantly, our upper bounds are oblivious , meaning that @math and @math do not depend on the @math , while our lower bounds hold even if @math and @math depend on the @math . As an application of our upper bounds, we answer a recent open question of about space-approximation trade-offs for the Schatten @math -norm, showing in a data stream it is possible to estimate the Schatten- @math norm up to a factor of @math using @math space.