Near-Optimal Sample Complexity Bounds for Circulant Binary Embedding

Near-Optimal Sample Complexity Bounds for Circulant Binary Embedding by Samet Oymak

Binary embedding is the problem of mapping points from a high-dimensional space to a Hamming cube in lower dimension while preserving pairwise distances. An efficient way to accomplish this is to make use of fast embedding techniques involving Fourier transform e.g.~circulant matrices. While binary embedding has been studied extensively, theoretical results on fast binary embedding are rather limited. In this work, we build upon the recent literature to obtain significantly better dependencies on the problem parameters. A set of N points in Rn can be properly embedded into the Hamming cube {±1}k with δ distortion, by using k∼δ−3logN samples which is optimal in the number of points N and compares well with the optimal distortion dependency δ−2. Our optimal embedding result applies in the regime logN≲n1/3. Furthermore, if the looser condition logN≲n√ holds, we show that all but an arbitrarily small fraction of the points can be optimally embedded. We believe our techniques can be useful to obtain improved guarantees for other nonlinear embedding problems.

 
 

Join the CompressiveSensing subreddit or the Google+ Community or the Facebook page and post there !
Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin.