OP-00677 Solution claimed

Monotone variance in the Gaussian approximation to DrMMD flow

Fix $s > 2$, let $\pi = N(0, s)$ on $\mathbb{R}$, let $k(x, y) = \exp(-(x - y)^2/2)$ be the Gaussian kernel and let $$T_\pi f(y) = \int_{\mathbb{R}} k(x, y) f(x) \, d\pi(x)$$ be its integral operator on $L^2(\pi)$. For $0 < v < s$ put $\nu_v = N(0, v)$ and $r_v = d\nu_v/d\pi - 1 \in L^2(\pi)$, and for a fixed regularization parameter $\lambda > 0$ let $$h_{v,\lambda} = T_\pi (T_\pi + \lambda I)^{-1} r_v$$ be the normalized de-regularized MMD (DrMMD) witness, taken as its differentiable Gaussian-kernel representative. In the centred Gaussian moment approximation to the DrMMD gradient flow of Chen, Mustafi, Glaser, Korba, Gretton and Sriperumbudur, the variance evolves by $$\dot v(t) = -2(1 + \lambda) \int_{\mathbb{R}} y \, h_{v(t),\lambda}'(y) \, d\nu_{v(t)}(y),$$ and $v = s$ is an equilibrium. They proved that the variance increases in the two limiting regimes $\lambda = 0$ (the $\chi^2$ flow) and $\lambda = \infty$ (the MMD flow).

Problem. Is $$\int_{\mathbb{R}} y \, h_{v,\lambda}'(y) \, d\nu_v(y) < 0$$ for every $s > 2$, every $\lambda > 0$ and every $0 < v < s$?

Source: Chen, Mustafi, Glaser, Korba, Gretton and Sriperumbudur (2025), Appendix C, who conjecture that the variance of the Gaussian approximation increases for every λ > 0. Stated by AIM — Open Applied Problems as the underlying sign inequality, for every 0 < v < s.

Imported from github.com/MColbrook/AIM.

Classification

Primary: PR — Probability
Secondary: ST
Keywords: AIM Open Applied Problems, Gaussian approximation, kernel gradient flows, maximum mean discrepancy

References

@article{chen2025drmmd,
  author        = {Chen, Zonghao and Mustafi, Aratrika and Glaser, Pierre and Korba, Anna and Gretton, Arthur and Sriperumbudur, Bharath K.},
  title         = {({D}e)-regularized maximum mean discrepancy gradient flow},
  journal       = {J. Mach. Learn. Res.},
  volume        = {26},
  number        = {235},
  year          = {2025},
  pages         = {1--77},
  url           = {https://www.jmlr.org/papers/v26/24-1574.html},
  eprint        = {2409.14980},
  archivePrefix = {arXiv},
}
@misc{aim-drmmd-gaussian-variance-monotonicity,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Monotone variance in the Gaussian approximation to DrMMD flow}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/research/resolved/663-drmmd-gaussian-variance-monotonicity.md},
  note         = {Entry 663 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Gaussian variance monotonicity for de-regularized MMD v1 GitHub ↗

Siavash Sadeghi

For a centered one-dimensional Gaussian target and the Gaussian kernel, we prove the strict sign inequality governing the variance in the Gaussian moment approximation to de-regularized maximum mean discrepancy (DrMMD) flow. The result holds for every target variance $s>0$, every finite regularization parameter $\lambda>0$, and every current variance $0<v<s$, removing the target statement's restriction $s>2$. The proof reduces the Gaussian …

Full resolution · 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-560/aim560_proof.pdf.

Citing this problem

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

OP-00677, "Monotone variance in the Gaussian approximation to DrMMD flow", clopen.solutions.