Dimension Reduction Methods with Application to ... - Rice University

RP: JOHNSON-LINDENSTRAUSS (JL)
LEMMA (1984)
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Johnson-Lindenstrauss Lemma
For any 0 < ɛ < 1 and integer n, let k = O(ln n/ɛ 2 ). For any set
V of n points in R p , there is a linear map f : R p → R k such that
for any u, v ∈ V,
(1 − ɛ) ||u − v|| 2 ≤ ||f(u) − f(v)|| 2 ≤ (1 + ɛ) ||u − v|| 2 .
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◮ n points in p−dimensional space can be projected onto a
k−dimensional space such that the pairwise distance bet. any 2
points is preserved within (1 ± ɛ).
◮ Euclidean distance in both original and projected spaces
(L 2 -L 2 projection)
◮ f is linear, but not specified
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