1,158 research outputs found
A variant of the Johnson-Lindenstrauss lemma for circulant matrices
We continue our study of the Johnson-Lindenstrauss lemma and its connection
to circulant matrices started in \cite{HV}. We reduce the bound on from
proven there to . Our
technique differs essentially from the one used in \cite{HV}. We employ the
discrete Fourier transform and singular value decomposition to deal with the
dependency caused by the circulant structure
A Simple proof of Johnson-Lindenstrauss extension theorem
Johnson and Lindenstrauss proved that any Lipschitz mapping from an -point
subset of a metric space into Hilbert space can be extended to the whole space,
while increasing the Lipschitz constant by a factor of . We
present a simplification of their argument that avoids dimension reduction and
the Kirszbraun theorem.Comment: 3 pages. Incorporation of reviewers' suggestion
Almost Optimal Unrestricted Fast Johnson-Lindenstrauss Transform
The problems of random projections and sparse reconstruction have much in
common and individually received much attention. Surprisingly, until now they
progressed in parallel and remained mostly separate. Here, we employ new tools
from probability in Banach spaces that were successfully used in the context of
sparse reconstruction to advance on an open problem in random pojection. In
particular, we generalize and use an intricate result by Rudelson and Vershynin
for sparse reconstruction which uses Dudley's theorem for bounding Gaussian
processes. Our main result states that any set of real
vectors in dimensional space can be linearly mapped to a space of dimension
k=O(\log N\polylog(n)), while (1) preserving the pairwise distances among the
vectors to within any constant distortion and (2) being able to apply the
transformation in time on each vector. This improves on the best
known achieved by Ailon and Liberty and by Ailon and Chazelle.
The dependence in the distortion constant however is believed to be
suboptimal and subject to further investigation. For constant distortion, this
settles the open question posed by these authors up to a \polylog(n) factor
while considerably simplifying their constructions
Acceleration of Randomized Kaczmarz Method via the Johnson-Lindenstrauss Lemma
The Kaczmarz method is an algorithm for finding the solution to an
overdetermined consistent system of linear equations Ax=b by iteratively
projecting onto the solution spaces. The randomized version put forth by
Strohmer and Vershynin yields provably exponential convergence in expectation,
which for highly overdetermined systems even outperforms the conjugate gradient
method. In this article we present a modified version of the randomized
Kaczmarz method which at each iteration selects the optimal projection from a
randomly chosen set, which in most cases significantly improves the convergence
rate. We utilize a Johnson-Lindenstrauss dimension reduction technique to keep
the runtime on the same order as the original randomized version, adding only
extra preprocessing time. We present a series of empirical studies which
demonstrate the remarkable acceleration in convergence to the solution using
this modified approach
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