OP-00686 Solution claimed

Variance ordering for Gaussian alpha-divergence approximations

Let $d \ge 2$, $\mu \in \mathbb{R}^d$, and let $\Sigma$ be a real symmetric positive-definite $d \times d$ matrix that is not diagonal; write $p$ for the density of $N(\mu, \Sigma)$. Let $\mathcal{Q}$ be the family of Gaussian densities $N(\nu, \Psi)$ with $\nu \in \mathbb{R}^d$ and $\Psi$ diagonal with positive entries. For $\alpha > 1$ let $$D_\alpha(p \,\Vert\, q) = \frac{1}{\alpha(\alpha - 1)} \left( \int_{\mathbb{R}^d} p(x)^\alpha q(x)^{1-\alpha} \, dx - 1 \right),$$ with value $+\infty$ when the integral diverges. Over $\mathcal{Q}$ it has a unique minimizer $q_\alpha = N(\mu, \Psi(\alpha))$. Margossian, Pillaud-Vivien and Saul proved that the optimal diagonal variances increase with $\alpha$ on $(0, 1)$: for $0 < \alpha_1 < \alpha_2 < 1$, $\Psi_{ii}(\alpha_1) \le \Psi_{ii}(\alpha_2)$ for every $i$, with strict inequality for some $i$. For $\alpha > 1$ they proved that the optimal variances dominate, in the same sense, those of the forward Kullback–Leibler approximation, which are the marginal variances $\Sigma_{ii}$.

Problem. Is it true that, for every such $\mu$ and $\Sigma$ and every $1 < \alpha_1 < \alpha_2$, $$\Psi_{ii}(\alpha_1) \le \Psi_{ii}(\alpha_2) \qquad (i = 1, \dots, d),$$ with strict inequality for at least one $i$?

Source: Margossian, Pillaud-Vivien and Saul (2025), Remark 12, in the ordering of their Definition 2.

Imported from github.com/MColbrook/AIM.

Classification

Primary: PR — Probability
Secondary: ST
Keywords: AIM Open Applied Problems, factorized Gaussian approximations, variational inference, α-divergences

References

@article{margossian2025,
  author        = {Margossian, Charles C. and Pillaud-Vivien, Loucas and Saul, Lawrence K.},
  title         = {Variational inference for uncertainty quantification: an analysis of trade-offs},
  journal       = {J. Mach. Learn. Res.},
  volume        = {26},
  number        = {202},
  year          = {2025},
  pages         = {1--41},
  url           = {https://www.jmlr.org/papers/v26/24-0878.html},
  eprint        = {2403.13748},
  archivePrefix = {arXiv},
}
@misc{aim-gaussian-variational-variance-ordering,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Variance ordering for Gaussian alpha-divergence approximations}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/research/resolved/662-gaussian-variational-variance-ordering.md},
  note         = {Entry 662 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

\textbf{Gaussian $\alpha$-projections are not coordinatewise variance-monotone} An exact counterexample to AIM problem 558, with a determinant-monotonicity theorem v1 GitHub ↗

Siavash Sadeghi

For a fixed correlated Gaussian target, consider its optimal diagonal Gaussian approximation under $D_\alpha(p\Vert q)$, with $\alpha>1$. We give a rational three-dimensional target whose optimal covariance is the identity at $\alpha=10$, but whose second and third variances are strictly smaller than one for every $\alpha>10$. The first variance is strictly larger than one, so the covariances are incomparable in the …

Counterexample · not verified · fidelity not established · posted on GitHub 1 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: AIM @ a153f46, siavash-sadeghi-457-558/aim_558_counterexample.pdf.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00686, "Variance ordering for Gaussian alpha-divergence approximations", clopen.solutions.