Inequalities and Positive-Definite Functions Arising from a Problem in Multidimensional Scaling

Abstract

We solve the following variational problem: Find the maximum of E|X - Y| subject to E|X|2 ≤ 1, where X and Y are i.i.d. random n-vectors, and |·| is the usual Euclidean norm on Rn. This problem arose from an investigation into multidimensional scaling, a data analytic method for visualizing proximity data. We show that the optimal X is unique and is (1) uniform on the surface of the unit sphere, for dimensions n ≥ 3, (2) circularly symmetric with a scaled version of the radial density ρ/(1 - ρ2)1/2, 0 ≤ ρ ≤ 1, for n = 2, and (3) uniform on an interval centered at the origin, for n = 1 (Plackett's theorem). By proving spherical symmetry of the solution, a reduction to a radial problem is achieved. The solution is then found using the Wiener-Hopf technique for (real) $n 3$. The results are reminiscent of classical potential theory, but they cannot be reduced to it. Along the way, we obtain results of independent interest: for any i.i.d. random n-vectors X and Y, E|X - Y| ≤ E|X + Y|. Further, the kernel Kp,β(x, y) = |x + y|β p - |x - y|β p, x, y ∈ Rn and |x|p = (∑|xi|p)1/p, is positive-definite, that is, it is the covariance of a random field, Kp,β(x, y) = E[ Z(x)Z(y)] for some real-valued random process Z(x), for 1 ≤ p ≤ 2 and $0 \textbackslashbeta \textbackslashleq p \textbackslashleq 2$ (but not for $\textbackslashbeta p$ or $p 2$ in general). Although this is an easy consequence of known results, it appears to be new in a strict sense. In the radial problem, the average distance D(r1, r2) between two spheres of radii r1 and r2 is used as a kernel. We derive properties of D(r1, r2), including nonnegative definiteness on signed measures of zero integral.