Convex Optimization · Problem 4.46

Duality — Boyd & Vandenberghe

Mathematics — Optimization · multidimensional-moments

Problem

Multidimensional moments. The moments of a random variable

t

\mathbf{R}^2

are defined as

\mu_{ij} = \mathbf{E}\, t_1^i t_2^j

, where

i, j

are nonnegative integers. In this problem we derive necessary conditions for a set of numbers

\mu_{ij}

0 \leq i, j \leq 2k

i + j \leq 2k

, to be the moments of a distribution on

\mathbf{R}^2

. Let

p : \mathbf{R}^2 \to \mathbf{R}

be a polynomial of degree

k

with coefficients

c_{ij}

p(t) = \sum_{i=0}^k \sum_{j=0}^{k-i} c_{ij} t_1^i t_2^j,

and let

t

be a random variable with moments

\mu_{ij}

. Suppose

c \in \mathbf{R}^{(k+1)(k+2)/2}

contains the coefficients

c_{ij}

in some specific order, and

\mu \in \mathbf{R}^{(k+1)(2k+1)}

contains the moments

\mu_{ij}

in the same order. Show that

\mathbf{E}\, p(t)^2

can be expressed as a quadratic form in

c

\mathbf{E}\, p(t)^2 = c^T H(\mu) c,

where

H : \mathbf{R}^{(k+1)(2k+1)} \to \mathbf{S}^{(k+1)(k+2)/2}

is a linear function of

\mu

. From this, conclude that

\mu

must satisfy the LMI

H(\mu) \succeq 0

. Remark: For random variables on

\mathbf{R}

, the matrix

H

can be taken as the Hankel matrix defined in (4.52). In this case,

H(\mu) \succeq 0

is a necessary and sufficient condition for

\mu

to be the moments of a distribution, or the limit of a sequence of moments. On

\mathbf{R}^2

, however, the LMI is only a necessary condition.