A note on random projections for preserving paths on a manifold

Nakul Verma
December 22, 2011

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^D . Of particular interest is when the set S is a low-dimensional submanifold of R^D . 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.

How to view this document

The authors of these documents have submitted their reports to this technical report series for the purpose of non-commercial dissemination of scientific work. The reports are copyrighted by the authors, and their existence in electronic format does not imply that the authors have relinquished any rights. You may copy a report for scholarly, non-commercial purposes, such as research or instruction, provided that you agree to respect the author's copyright. For information concerning the use of this document for other than research or instructional purposes, contact the authors. Other information concerning this technical report series can be obtained from the Computer Science and Engineering Department at the University of California at San Diego, techreports@cs.ucsd.edu.

[ Search ]

This server operates at UCSD Computer Science and Engineering.
Send email to webmaster@cs.ucsd.edu