Separation of measures by Cartan–Hadamard sliced distances
Let $(M, g)$ be a finite-dimensional, complete, simply connected Riemannian manifold with nonpositive sectional curvature. Fix $o \in M$, let $S_o$ be the unit sphere in $T_o M$ and $\lambda_o$ its uniform probability measure. For $v \in S_o$ put $\gamma_v(t) = \exp_o(tv)$ and define two real-valued projections, $$G_v(x) = \operatorname*{argmin}_{t \in \mathbb{R}} d\bigl(x, \gamma_v(t)\bigr)^2, \qquad H_v(x) = \lim_{t \to \infty} \bigl( d(x, \gamma_v(t)) - t \bigr),$$ the coordinate of the nearest point on the geodesic and the Busemann function; both are well defined. For $1 \le p < \infty$ let $\mathcal{P}_p(M)$ be the Borel probability measures with finite $p$th moment of the distance, and for $F = G$ or $F = H$ let $$D_{F,p}(\mu,\nu)^p = \int_{S_o} W_p^p\bigl((F_v)_\# \mu, (F_v)_\# \nu\bigr) \, d\lambda_o(v),$$ where $W_p$ is the Wasserstein distance on $\mathbb{R}$. Bonet, Drumetz and Courty proved that both are finite pseudo-distances on $\mathcal{P}_p(M)$, that $D_{F,p} \le W_p$, and that they are distances when the metric of $M$ is the pullback of a Euclidean metric by a diffeomorphism, which includes the log-Euclidean metric on positive definite matrices; in that case $D_{F,p}$ coincides with the Euclidean sliced-Wasserstein distance of the pushforwards. $D_{F,p}(\mu,\nu) = 0$ means that $(F_v)_\# \mu = (F_v)_\# \nu$ for $\lambda_o$-almost every $v$, so separation is the injectivity on measures of a Radon transform over the level sets of the $F_v$: for $G$ the hypersurfaces $\exp_{\gamma_v(t)}(\gamma_v'(t)^\perp)$ orthogonal to $\gamma_v$ (totally geodesic when $M$ has constant curvature), horospheres for $H$.
Problem. For $F = G$ and for $F = H$, is it true that $D_{F,p}(\mu,\nu) = 0$ implies $\mu = \nu$, for every such manifold, every base point $o$, every $p \in [1,\infty)$ and every $\mu, \nu \in \mathcal{P}_p(M)$? Both projection families are to be decided, with directions integrated over the whole sphere $S_o$.
Source: Bonet, Drumetz and Courty (2025), §5.1, who conjecture that the Cartan–Hadamard sliced-Wasserstein distances separate measures; for hyperbolic spaces already Bonet, Chapel, Drumetz and Courty (2023), §3.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- DG, ST
- Keywords
- AIM Open Applied Problems, Cartan–Hadamard manifolds, Radon transforms, sliced Wasserstein distances
References
@article{bonet2025,
author = {Bonet, Cl{\'e}ment and Drumetz, Lucas and Courty, Nicolas},
title = {Sliced-{W}asserstein distances and flows on {C}artan--{H}adamard manifolds},
journal = {J. Mach. Learn. Res.},
volume = {26},
number = {32},
year = {2025},
pages = {1--76},
url = {https://www.jmlr.org/papers/v26/24-0359.html},
eprint = {2403.06560},
archivePrefix = {arXiv},
}
@inproceedings{bonet2023,
author = {Bonet, Cl{\'e}ment and Chapel, Laetitia and Drumetz, Lucas and Courty, Nicolas},
title = {Hyperbolic sliced-{W}asserstein via geodesic and horospherical projections},
booktitle = {Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML)},
series = {Proceedings of Machine Learning Research},
volume = {221},
publisher = {PMLR},
year = {2023},
pages = {334--370},
url = {https://proceedings.mlr.press/v221/bonet23a.html},
eprint = {2211.10066},
archivePrefix = {arXiv},
}
@misc{aim-cartan-hadamard-sliced-separation,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Separation of measures by Cartan–Hadamard sliced distances}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/548-cartan-hadamard-sliced-separation.md},
note = {Entry 548 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00283, "Separation of measures by Cartan–Hadamard sliced distances", clopen.solutions.