Tractable characterization of the multivariate truncated moment problem

Determine a tractable necessary-and-sufficient condition for deciding whether a matrix of the form \(\operatorname{mat}(\boldsymbol{\Phi}\lambda)\) is a moment matrix associated with a nonnegative measure in the truncated multivariate polynomial moment problem.

Background

For polynomial features, boundedness of the data-driven linear program requires matching the matrix $\operatorname{mat}(\boldsymbol{\Phi}\lambda)$ with a moment matrix $\int_Z p(z)p(z)^\top c(dz)$ generated by a nonnegative measure. Positive semidefiniteness is necessary, but for polynomial degrees greater than two it is not generally sufficient. The paper identifies the absence of a tractable necessary-and-sufficient test for this truncated, multivariate moment problem as an unresolved issue; the quadratic case is the special case in which positive semidefiniteness suffices.