? ? V n d f :

R^{d} → R^{k} u, v ∈

V

(1−ϵ)||u−v||^{2} ≤ ||f(u)−f(v)||^{2} ≤ (1+ϵ)||u−v||^{2},

0 < ϵ < 1 n, d

k ≥ 4(\frac{ϵ^{2}}{2} − \frac{ϵ^{3}}{3} )^{−1} ln n

d k k d V (1±

ϵ)

k O(1/n^{2})

(1± ϵ)

k k k N(0, 1)

X_{1}, …, X_{d}

d N(0, 1)

X = (X_{1}, X_{2}, …, X_{d})

Y = \frac{1}{||X||}X

Y d X_{i}

f(x_{i}) = \frac{1}{\sqrt{2π}} e^{−x_{i}^{2}/2} X Π_{i=1}^{d}[\frac{1}{\sqrt{2π}} e^{−x_{i}^{2}/2}] =

\frac{1}{2π^{d/2}} e^{− \frac{1}{2} ∑_{i=1}^{d} x_{i}^{2}}

X (X_{1}, …, X_{d})

Y ||X||

Y Y = (Y_{1}, …, Y_{d})

?? Z ∈

R^{K} Y k k L Z L = ||Z||^{2}

E[L] =

k/d.

E[L] = E[||Z^{2}||] = \frac{1}{||X||^{2}} E[X_{1}^{2}+⋯+X_{k}^{2}] = \frac{1}{||X||^{2}} ∑_{1=1}^{k} E[X_{i}^{2}] = k \frac{E[X_{1}^{2}]}{||X||^{2}}.

||Y ||^{2} = 1 = \frac{1}{||X||^{2}} ∑_{i=1}^{d} X_{i}^{2}.

d\frac{E[X_{1}^{2}]}{||X||^{2}} = 1.

E[L] = k \frac{E[X_{1}^{2}]}{||X||^{2}} = \frac{k}{d} .

?? k < d β < 1 P (L ≤

β \frac{k}{d}) ≤

e^{\frac{k}{2} (1−β+ln β)}. β > 1 P (L ≥

β \frac{k}{d}) ≤

e^{\frac{k}{2} (1−β+ln β)}. ?? k ≥ d k < d S k R^{d} k v ∈ V R^{k} v′ Y := \frac{v−w}{||v||−||w||}

Z := \frac{v′−w′}{||v||−||w||}

L =