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Dense fast random projections and lean Walsh transforms

By Edo Liberty, Nir Ailon and Amit Singer


Abstract. Random projection methods give distributions over k × d matrices such that if a matrix Ψ (chosen according to the distribution) is applied to a vector x ∈ R d the norm of the resulting vector, Ψx ∈ R k, is up to distortion ɛ equal to the norm of x w.p. at least 1−δ. The Johnson Lindenstrauss lemma shows that such distributions exist over dense matrices for k (the target dimension) in O(log(1/δ)/ε 2). Ailon and Chazelle and later Matousek showed that there exist entry-wise i.i.d. distributions over sparse matrices Ψ which give the same guaranties for vectors whose ℓ ∞ is bounded away from their ℓ2 norm. This allows to accelerate the mapping x ↦ → Ψx. We claim that setting Ψ as any column normalized deterministic dense matrix composed with random ±1 diagonal matrix also exhibits this property for vectors whose ℓp (for any p> 2) is bounded away from their ℓ2 norm. We also describe a specific tensor product matrix which we term lean Walsh. It is applicable to any vector in R d in O(d) operations and requires a weaker ℓ ∞ bound on x then the best current result, under comparable running times, using sparse matrices due to Matousek

Topics: Key words, Random Projections, Lean Walsh Transforms, Johnson Lindenstrauss, Dimension reduction
Publisher: Springer
Year: 2008
OAI identifier: oai:CiteSeerX.psu:
Provided by: CiteSeerX
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