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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JL LEMMA: DASGUPTA AND GUPTA<br />
Sketch of Proof<br />
◮ Minimize e −s(1+ɛ) (1 − 2s) −1/2 wrt s, <strong>with</strong> s ∗ = 1 2<br />
]<br />
P<br />
[||y|| 2 ≥ α 1 ≤ exp<br />
(− k )<br />
2 (ɛ − ln(1 + ɛ)) (<br />
≤ exp − k<br />
)<br />
12 (3ɛ2 − 2ɛ 3 )<br />
( ɛ<br />
1+ɛ)<br />
,<br />
(7)<br />
◮ if k ≥<br />
and<br />
24 ln n<br />
, then<br />
3ɛ 2 −2ɛ 3<br />
P<br />
[<br />
||f(x)|| 2 ≥ (1 + ɛ) ||x|| 2] ≤ 1/n 2<br />
P<br />
[<br />
||f(x)|| 2 ≤ (1 − ɛ) ||x|| 2] ≤ 1/n 2 22 / 86
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