A:projection onto a uniform W.h.p,u,v∈V, random k-subspace 那甲w很≥“ 2 Step I: Reduce to unit vectors
W.h.p., ⇥u, v V , A: projection onto a uniform random k-subspace ⇥d k Au ⇥d k Av 2 ⇥ 3⇤u v⇤2 2 ⇥d k Au ⇥d k Av 2 ⇥ ⇤u v⇤2 2 Step I: Reduce to unit vectors
O(n2)pairs A:projection onto a uniform random k-subspace W.h.p.,u,v∈V, -4「s 2 2 d M-a 2d treat (u)as a vector, is a unit vector New goal: V unit vector u, with probability o(是) IlAull2> d or lAul<2d
W.h.p., ⇥u, v V , A: projection onto a uniform random k-subspace ⇥d k Au ⇥d k Av 2 ⇥ 3⇤u v⇤2 2 ⇥d k Au ⇥d k Av 2 ⇥ ⇤u v⇤2 2 A (u v) ⇤u v⇤ 2 ⇥ k 2d A (u v) ⇤u v⇤ 2 ⇥ 3k 2d unit vector u, Au2 > 3k 2d Au2 < k 2d with probability o( 1 n2 ) or New goal: O(n2) pairs treat (u v) as a vector, uv ⇥uv⇥ is a unit vector
A:projection onto a uniform random k-subspace V unit vector u, with probability o() IlAull2> d or 14ul2<2d Step II: Random projection of fixed unit vector 0 Fixed projection of random unit vector
unit vector u, Au2 > 3k 2d Au2 < k 2d with probability o( 1 n2 ) or Step II: Random projection of fixed unit vector Fixed projection of random unit vector A: projection onto a uniform random k-subspace ⇔
A:projection onto a uniform random k-subspace V unit vector u, with probability o(是) IlAull2> 2d or 14ul2<2a probabilistically random equivalent unit vector fixed fixed unit vector subspace random subspace "inner-products are invariant under rotations
unit vector u, Au2 > 3k 2d Au2 < k 2d with probability o( 1 n2 ) or A: projection onto a uniform random k-subspace fixed unit vector random subspace fixed subspace random unit vector probabilistically equivalent “inner-products are invariant under rotations
A:projection onto a uniform random k-subspace V unit vector u, with probability o() IAull2> d or 14ul2<2a uniform random unit vector: random unit vector Y=(Y,...,Yk,Y+1...Yd) Z=(Yi,.,Y) fixed subspace Z2> d or 112< k 2d
unit vector u, Au2 > 3k 2d Au2 < k 2d with probability o( 1 n2 ) or A: projection onto a uniform random k-subspace fixed subspace random unit vector Z = (Y1,...,Yk) Z2 > 3k 2d Z2 < k 2d or Y = (Y1,...,Yk, Yk+1,...,Yd) uniform random unit vector: