Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Using Toeplitz and Circulant Matrices for Johnson-Lindenstrauss
  Transforms   The Johnson-Lindenstrauss lemma is one of the corner stone results in
dimensionality reduction. It says that given $N$, for any set of $N$ vectors $X
\subset \mathbb{R}^n$, there exists a mapping $f : X \to \mathbb{R}^m$ such
that $f(X)$ preserves all pairwise distances between vectors in $X$ to within
$(1 \pm \varepsilon)$ if $m = O(\varepsilon^{-2} \lg N)$. Much effort has gone
into developing fast embedding algorithms, with the Fast Johnson-Lindenstrauss
transform of Ailon and Chazelle being one of the most well-known techniques.
The current fastest algorithm that yields the optimal $m =
O(\varepsilon^{-2}\lg N)$ dimensions has an embedding time of $O(n \lg n +
\varepsilon^{-2} \lg^3 N)$. An exciting approach towards improving this, due to
Hinrichs and Vyb\'iral, is to use a random $m \times n$ Toeplitz matrix for the
embedding. Using Fast Fourier Transform, the embedding of a vector can then be
computed in $O(n \lg m)$ time. The big question is of course whether $m =
O(\varepsilon^{-2} \lg N)$ dimensions suffice for this technique. If so, this
would end a decades long quest to obtain faster and faster
Johnson-Lindenstrauss transforms. The current best analysis of the embedding of
Hinrichs and Vyb\'iral shows that $m = O(\varepsilon^{-2}\lg^2 N)$ dimensions
suffices. The main result of this paper, is a proof that this analysis
unfortunately cannot be tightened any further, i.e., there exists a set of $N$
vectors requiring $m = \Omega(\varepsilon^{-2} \lg^2 N)$ for the Toeplitz
approach to work.

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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: quant-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ex
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The categories of this paper are: cond-mat
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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an