A note on random projections for preserving paths on a manifold

Random projections are typically used to study low distortion linear embeddings that approximately preserve Euclidean distances between pairs of points in a set S ⊂ R. Of particular interest is when the set S is a low-dimensional submanifold of R. Recent results by Baraniuk and Wakin [2007] and Clarkson [2007] shed light on how to pick the projection dimension to achieve low distortion of Euclidean distances between points on a manifold. While preserving ambient Euclidean distances on a manifold does imply preserving intrinsic path-lengths between pairs of points on a manifold, here we investigate how one can reason directly about preserving path-lengths without having to appeal to the ambient Euclidean distances between points. In doing so, we can improve upon Baraniuk and Wakin’s result by removing the dependence on the ambient dimension D, and simplify Clarkson’s result by using a single covering quantity and giving explicit dependence on constants.