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

IMPROVEMENT ON DASGUPTA AND
GUPTA BOUND
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Right-tail prob.
∑d−1
P[||y|| 2 ≥ α 1 ] = e −λ λ y 1 1
y!
y=0
( ) ( )
1 + ɛ λ d−1
1
≤
e −λ 1 (13)
ɛ (d − 1)!
Left-tail prob.
∞∑
P[||y|| 2 ≤ α 2 ] = e −λ λ y 2 2
y!
y=d
( ) ( )
1 − ɛ λ d−1
2
≤
ɛ (d − 1)!
( ) ( )
1 + ɛ λ d−1
1
≤
ɛ (d − 1)!
e −λ 2
e −λ 1
◮ lower bound for k:
P[||y|| 2 ≥ α 1 ] + P[||y|| 2 ≤ α 2 ] ≤ 2e −λ 1
( ) ( )
1+ɛ λ d−1
1
≤ 2/n ɛ (d−1)!
2+β 28 / 86