1,287 research outputs found
Approximate Near Neighbors for General Symmetric Norms
We show that every symmetric normed space admits an efficient nearest
neighbor search data structure with doubly-logarithmic approximation.
Specifically, for every , , and every -dimensional
symmetric norm , there exists a data structure for
-approximate nearest neighbor search over
for -point datasets achieving query time and
space. The main technical ingredient of the algorithm is a
low-distortion embedding of a symmetric norm into a low-dimensional iterated
product of top- norms.
We also show that our techniques cannot be extended to general norms.Comment: 27 pages, 1 figur
Practical sketching algorithms for low-rank matrix approximation
This paper describes a suite of algorithms for constructing low-rank
approximations of an input matrix from a random linear image of the matrix,
called a sketch. These methods can preserve structural properties of the input
matrix, such as positive-semidefiniteness, and they can produce approximations
with a user-specified rank. The algorithms are simple, accurate, numerically
stable, and provably correct. Moreover, each method is accompanied by an
informative error bound that allows users to select parameters a priori to
achieve a given approximation quality. These claims are supported by numerical
experiments with real and synthetic data
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