Nonconvergence of a sum-of-squares hierarchy for global polynomial optimization based on push-forward measures

Let $\mathbf{X} \subseteq \mathbb{R}^n$ be a closed set, and consider the problem of computing the minimum $f_{\min}$ of a polynomial $f$ on $\mathbf{X}$. Given a measure $\mu$ supported on $\mathbf{X}$, Lasserre (SIAM J. Optim. 21(3), 2011) proposes a decreasing sequence of upper bounds on $f_{\min}$, each of which may be computed by solving a semidefinite program. When $\mathbf{X}$ is compact, these bounds converge to $f_{\min}$ under minor assumptions on $\mu$. Later, Lasserre (Math. Program. 190, 2020) introduces a related, but far more economical sequence of upper bounds which rely on the push-forward measure of $\mu$ by $f$. While these new bounds are weaker a priori, they actually achieve similar asymptotic convergence rates on compact sets. In this work, we show that no such free lunch exists in the non-compact setting. While convergence of the standard bounds to $f_{\min}$ is guaranteed when $\mathbf{X} = \mathbb{R}^n$ and $\mu$ is a Gaussian distribution, we prove that the bounds relying on the push-forward measure fail to converge to $f_{\min}$ in that setting already for polynomials of degree $6$.