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2013

New bounds for circulant Johnson-Lindenstrauss embeddings

Zhang, Hui, Cheng, Lizhi

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This paper analyzes circulant Johnson-Lindenstrauss (JL) embeddings which, as an important class of structured random JL embeddings, are formed by randomizing the column signs of a circulant matrix generated by a random vector.

With the help of recent decoupling techniques and matrix-valued Bernstein inequalities, we obtain a new bound $k=O(\epsilon^{-2}\log^{(1+\delta)} (n))$ for Gaussian circulant JL embeddings.
Moreover, by using the Laplace transform technique (also called Bernstein's trick), we extend the result to subgaussian case.
The bounds in this paper offer a small improvement over the current best bounds for Gaussian circulant JL embeddings for certain parameter regimes and are derived using more direct methods.

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W. B. Johnson and J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, Contemp Math, 26, 189-206, 1984
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R. A. Horn, C. R. Johnson, Topics in Matrices Analysis, Posts and Telecom Press, 2007
2007
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