Singular Curvature and Stickiness of Fréchet Means

Definitions and mathematical statement

For a metric space $(X,d)$ and a Borel probability measure $\mu$ satisfying \[ \int_X d(p,x)^2\,\mathrm d\mu(x)<\infty \] for some $p\in X$, define the Fréchet functional and its set of minimizers by \[ F_\mu(y)=\int_X d(y,x)^2\,\mathrm d\mu(x),\qquad \operatorname{Mean}(\mu)=\operatorname*{arg\,min}_{y\in X}F_\mu(y). \] An element of this set is a Fréchet mean; the set need not be nonempty or a singleton under these assumptions alone. This provides notation for the question, rather than hypotheses supplied by the source. In the Riemannian setting, the relevant distance is the Riemannian distance wherever it is finite. A tangent-space covariance additionally requires a specified mean, a suitable logarithm map or other convention, and appropriate integrability assumptions; none is specified in the record.

Source conjecture, with unresolved formulation. Curvature-dependent corrections to Fréchet means and covariances on Riemannian manifolds are conjectured to be precursors of stickiness of the mean in limiting cases where curvature becomes singular.

Here stickiness informally concerns persistence of the mean on a distinguished singular subset under perturbations of the probability distribution. The source record supplies no precise definition of stickiness, perturbation class, or distinguished subset, so this explanation does not fix the intended mathematical property.

A quantified conjecture cannot be recovered from the supplied record: it specifies neither the family of Riemannian manifolds and probability measures, nor the correction terms, limiting parameter, notion of convergence, meaning of singular curvature, or mathematical relation expressed by "precursors." It also gives no existence or uniqueness hypotheses for the means or covariance conventions. The background assessment additionally mentions a global flag-based unexplained-variance criterion, but supplies no definitions of the flags, variance functional, or criterion, and no relationship between that criterion and the stated conjecture. These qualifications remain unresolved.

Short English statement

The conjecture asks whether curvature corrections to Fréchet means and covariances on smooth Riemannian manifolds anticipate stickiness of the mean as curvature becomes singular. The supplied record lacks the limiting framework and definitions needed for a precise quantified assertion. Its additional reference to a global flag-based unexplained-variance criterion is also unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003715$, title Singular Curvature and Stickiness of Fréchet Means, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2018$ and no named proposer.
• Supplied source citation: Statistics for Data with Geometric Structure (2019), Oberwolfach Reports, pp. 125--186, DOI 10.4171/owr/2018/3. Supplied source URL: https://doi.org/10.4171/owr/2018/3. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Oberwolfach Report 3/2018, https://doi.org/10.4171/owr/2018/3. Literature assessment dated exactly 2026-08-21: the proposed link between singular curvature and stickiness of the Fréchet mean, together with the global flag-based unexplained-variance criterion, is described as a conjectural research direction rather than a proved criterion. This is supplied dated curation derived from the supplied OWR curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry supplies status open and statement status corrected_verified. Its verification note says that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. These are supplied registry labels and editorial claims, not independent mathematical verification of this draft.