On Outer Bi-Lipschitz Extensions of Linear Johnson-Lindenstrauss Embeddings of Low-Dimensional Submanifolds of ℝN

Let M be a compact d-dimensional submanifold of ℝN with reach τ and volume Vmathcal M. Fix ε ∈ (0,1). In this paper we prove that a nonlinear function f: ℝN rightarrow ℝm exists with m ≤ C left(d / ε2 right) log left(frac√[d]Vmathcal Mτ right) such that (1 - ε) | bf x - bf y |2 ≤ left| f(bf x) - f(bf y) right|2 ≤ (1 + ε) | bf x - bf y |2 holds for all bf x ∈ M and bf y ∈ ℝN. In effect, f not only serves as a bi-Lipschitz function from M into ℝm with bi-Lipschitz constants close to one, but also approximately preserves all distances from points not in M to all points in M in its image. Furthermore, the proof is constructive and yields an algorithm which works well in practice. In particular, it is empirically demonstrated herein that such nonlinear functions allow for more accurate compressive nearest neighbor classification than standard linear Johnson-Lindenstrauss embeddings do in practice.