Foundations of Data Science

Using the substitution 2z = y 2 ,
|E(x s i )| = 1 + √ 1 ∫ ∞
2 s z s−(1/2) e −z dz
π
≤ 2 s s!.
The last inequality is from the Gamma integral.
Since E(x i ) = 0, V ar(x i ) = E(x 2 i ) ≤ 2 2 2 = 8. Unfortunately, we do not have
|E(x s i )| ≤ 8s! as required in Theorem 12.5. To fix this problem, perform one more change
of variables, using w i = x i /2. Then, V ar(w i ) ≤ 2 and |E(wi s )| ≤ 2s!, and our goal is
now to bound the probability that |w 1 + . . . + w d | ≥ β√ d
. Applying Theorem 12.5 where
2
σ 2 = 2 and n = d, this occurs with probability less than or equal to 3e − β2
96 .
In the next sections we will see several uses of the Gaussian Annulus Theorem.
2.7 Random Projection and Johnson-Lindenstrauss Lemma
One of the most frequently used subroutines in tasks involving high dimensional data
is nearest neighbor search. In nearest neighbor search we are given a database of n points
in R d where n and d are usually large. The database can be preprocessed and stored in
an efficient data structure. Thereafter, we are presented “query” points in R d and are to
find the nearest or approximately nearest database point to the query point. Since the
number of queries is often large, query time (time to answer a single query) should be
very small (ideally a small function of log n and log d), whereas preprocessing time could
be larger (a polynomial function of n and d). For this and other problems, dimension
reduction, where one projects the database points to a k dimensional space with k ≪ d
(usually dependent on log d) can be very useful so long as the relative distances between
points are approximately preserved. We will see using the Gaussian Annulus Theorem
that such a projection indeed exists and is simple.
The projection f : R d → R k that we will examine (in fact, many related projections are
known to work as well) is the following. Pick k vectors u 1 , u 2 , . . . , u k , independently from
1
the Gaussian distribution exp(−|x| 2 /2). For any vector v, define the projection
(2π) d/2
f(v) by:
f(v) = (u 1 · v, u 2 · v, . . . , u k · v).
The projection f(v) is the vector of dot products of v with the u i . We will show that
with high probability, |f(v)| ≈ √ k|v|. For any two vectors v 1 and v 2 , f(v 1 − v 2 ) =
f(v 1 ) − f(v 2 ). Thus, to estimate the distance |v 1 − v 2 | between two vectors v 1 and v 2 in
R d , it suffices to compute |f(v 1 ) − f(v 2 )| = |f(v 1 − v 2 )| in the k dimensional space since
the factor of √ k is known and one can divide by it. The reason distances increase when
we project to a lower dimensional space is that the vectors u i are not unit length. Also
notice that the vectors u i are not orthogonal. If we had required them to be orthogonal,
we would have lost statistical independence.
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