1,777 research outputs found
Fast Cross-Polytope Locality-Sensitive Hashing
We provide a variant of cross-polytope locality sensitive hashing with
respect to angular distance which is provably optimal in asymptotic sensitivity
and enjoys hash computation time. Building on a recent
result (by Andoni, Indyk, Laarhoven, Razenshteyn, Schmidt, 2015), we show that
optimal asymptotic sensitivity for cross-polytope LSH is retained even when the
dense Gaussian matrix is replaced by a fast Johnson-Lindenstrauss transform
followed by discrete pseudo-rotation, reducing the hash computation time from
to . Moreover, our scheme achieves
the optimal rate of convergence for sensitivity. By incorporating a
low-randomness Johnson-Lindenstrauss transform, our scheme can be modified to
require only random bitsComment: 14 pages, 6 figure
Two-subspace Projection Method for Coherent Overdetermined Systems
We present a Projection onto Convex Sets (POCS) type algorithm for solving
systems of linear equations. POCS methods have found many applications ranging
from computer tomography to digital signal and image processing. The Kaczmarz
method is one of the most popular solvers for overdetermined systems of linear
equations due to its speed and simplicity. Here we introduce and analyze an
extension of the Kaczmarz method that iteratively projects the estimate onto a
solution space given by two randomly selected rows. We show that this
projection algorithm provides exponential convergence to the solution in
expectation. The convergence rate improves upon that of the standard randomized
Kaczmarz method when the system has correlated rows. Experimental results
confirm that in this case our method significantly outperforms the randomized
Kaczmarz method.Comment: arXiv admin note: substantial text overlap with arXiv:1204.027
Stable image reconstruction using total variation minimization
This article presents near-optimal guarantees for accurate and robust image
recovery from under-sampled noisy measurements using total variation
minimization. In particular, we show that from O(slog(N)) nonadaptive linear
measurements, an image can be reconstructed to within the best s-term
approximation of its gradient up to a logarithmic factor, and this factor can
be removed by taking slightly more measurements. Along the way, we prove a
strengthened Sobolev inequality for functions lying in the null space of
suitably incoherent matrices.Comment: 25 page
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