On outer bi-Lipschitz extensions of linear Johnson-Lindenstrauss embeddings of subsets of RN

The celebrated Johnson-Lindenstrauss lemma states that for all epsilon is an element of (0, 1) and finite sets X subset of R-N with n > 1 elements, there exists a matrix Phi is an element of R-mxN with m = O(epsilon(-2) log n) such that (1 - epsilon)parallel to x - y parallel to(2) <= parallel to Phi x - Phi y parallel to(2) <= (1 + epsilon)parallel to x - y parallel to(2) for all x, y is an element of X. Herein we consider so-called "terminal embedding" results which have recently been introduced in the computer science literature as stronger extensions of the Johnson-Lindenstrauss lemma for finite sets. After a short survey of this relatively recent line of work, we extend the theory of terminal embeddings to hold for arbitrary (e.g., infinite) subsets X subset of R-N, and then specialize our generalized results to the case where X is a low-dimensional compact submanifold of R-N. In particular, we prove the following generalization of the Johnson-Lindenstrauss lemma: For all epsilon is an element of (0, 1) and X subset of R-N, there exists a terminal embedding f : R-N -> R-m such that (1 - epsilon)parallel to x - y parallel to(2) <= parallel to f(x) - f(y)parallel to(2) <= (1 + epsilon)parallel to x - y parallel to(2) for all x is an element of X and for all y is an element of R-N. Crucially, we show that the dimension m of the range of f above is optimal up to multiplicative constants, satisfying m = O(epsilon(-2)omega(2)(S-X)), where omega(S-X) is the Gaussian width of the set of unit secants of X, S-X = <({(x - y)/parallel to x - y parallel to(2) : x not equal y is an element of X})over bar>. Furthermore, our proofs are constructive and yield algorithms for computing a general class of terminal embeddings f, an instance of which is demonstrated herein to allow for more accurate compressive nearest neighbor classification than standard linear Johnson-Lindenstrauss embeddings do in practice.