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

JL LEMMA: IMPROVEMENT ON LOWER BOUND
FOR k
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Dasgupta and Gupta (2003): mgf technique
For any 0 < ɛ < 1 and integer n, let k be such that
k ≥
24 ln n
3ɛ 2 − 2ɛ 3 . (3)
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,
[
P (1 − ɛ) ||u − v|| 2 ≤ ||f(u) − f(v)|| 2 ≤ (1 + ɛ) ||u − v|| 2] ≥ 1 − 2 n 2 .
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◮ f(x) = 1 √
k
xΓ, where x = u − v
◮ Gaussian random matrix:
k ||f(x)||2
||x|| 2
◮ best available lower bound for k
∼ χ 2 k
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