Random Projections

We consider the distribution of matrices R such that each R(i, j) is drawn independently from a normal distribution with mean zero and variance 1, R(i, j) ∼ N (0, 1). We show that for this distribution Equation 1 holds for some k ∈ O(log(n)/ε). First consider the random variable z = ∑d i=1 r(i)x(i) where r(i) ∼ N (0, 1). To understand how the variable z distributes we recall the two-stability of the normal distribution. Namely, if z3 = z2 + z1 and z1 ∼ N (μ1, σ1) and z2 ∼ N (μ2, σ2) then, z3 ∼ N (μ1 + μ2, √