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A Sparse Johnson-Lindenstrauss Transform using Fast Hashing · arXivDesk

2305.03110May 4, 202334 pages

A Sparse Johnson-Lindenstrauss Transform using Fast Hashing

Jakob Bæk Tejs Houen, Mikkel Thorup

Abstract

The Sparse Johnson-Lindenstrauss Transform of Kane and Nelson (SODA 2012) provides a linear dimensionality-reducing map $A \in \mathbb{R}^{m \times u}$ in $\ell_2$ that preserves distances up to distortion of $1 + \varepsilon$

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The Sparse Johnson-Lindenstrauss Transform of Kane and Nelson (SODA 2012) provides a linear dimensionality-reducing map A ∈ R^m × u in ℓ₂ that preserves distances up to distortion of 1 + ε with probability 1 - δ, where m = O(ε⁻² 1/δ) and each column of A has O(ε m) non-zero entries. The previous analyses of the Sparse Johnson-Lindenstrauss Transform all assumed access to a Ω( 1/δ)-wise independent hash function.

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The paper addresses: A Sparse Johnson-Lindenstrauss Transform using Fast Hashing.
The Sparse Johnson-Lindenstrauss Transform of Kane and Nelson (SODA 2012) provides a linear dimensionality-reducing map A ∈ R^m × u in ℓ₂ that preserves distances up to distortion of 1 + ε with probability 1 - δ, where m = O(ε⁻² 1/δ) and each column of A has O(ε m) non-zero entries.