Dimension Reduction Methods with Application to ... - Rice University
Dimension Reduction Methods with Application to ... - Rice University
Dimension Reduction Methods with Application to ... - Rice University
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L 2 -L 2 RP: IMPROVEMENT ON ACHLIOPTAS<br />
BOUND<br />
Rojo and Nguyen (2009b): Method 3<br />
◮ Pinelis inequality<br />
———————————————————————<br />
Let U i ’s be independent rademacher random variables. Let<br />
d 1 , . . . , d m be real numbers such that ∑ m<br />
i=1 d2 i = 1. Let<br />
S m = ∑ m<br />
i=1 d iU i , then for any t > 0,<br />
( )<br />
1<br />
P[|S m | ≥ t] ≤ min<br />
t , 2(1 − Φ(t − 1.495/t)) 2<br />
———————————————————————<br />
◮ Method 3: lower bound for k: 15% improvement (ɛ = 0.1)<br />
k ≥ 2(p − 1)a2 n<br />
pɛ 2 , (24)<br />
where a n = Qn+√ Q 2 n +4(1.495)<br />
2<br />
, and Q n = Φ −1 ( 1 − 1<br />
n 2+β )<br />
.<br />
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