Curvature Bias in Statistics on Quotient Spaces

Definitions and mathematical statement

Let $X$ be a space with an equivalence relation $\sim$. Its quotient is $Q=X/{\sim}=\{[x]:x\in X\}$, where $[x]=\{y\in X:y\sim x\}$. The source does not specify $X$, the equivalence relation, a metric, a statistical model, or an estimator class.

For clarity, if a metric $d$ on $Q$ and a Borel probability measure $P$ satisfying $\int_Q d(q_0,y)^2\,P({\,\mathrm{d}} y)<\infty$ for some $q_0\in Q$ are specified, the Fréchet functional and its mean set are \[ F_P(q)=\int_Q d(q,y)^2\,P({\,\mathrm{d}} y),\qquad \mathcal M(P)=\operatorname*{arg\,min}_{q\in Q}F_P(q). \] This conditional definition explains the reference to a mean shape; it does not supply a metric or model omitted by the source. Neither existence nor uniqueness of a Fréchet mean is assumed. The source also does not specify whether its mean shape is a population mean, an empirical mean, or another estimator.

Problem as supplied. For statistical estimators on quotient spaces, determine:

1. the geometric and statistical conditions under which curvature-induced bias occurs;
2. whether this bias is confined to estimation of the mean shape or also affects other statistical analyses;
3. its magnitude and practical importance in applications; and
4. when correction is warranted and how such correction should be performed.

Unresolved formulation requirements. This is a research programme rather than a single fully quantified proposition. A precise instance must specify the quotient geometry and relevant notion of curvature, the data-generating model, the estimators and their targets, and the meaning of bias. In a general quotient space, subtraction and hence a vector expression such as $\mathbb E[\widehat\theta]-\theta$ need not be defined. The source does not choose an alternative definition, an asymptotic regime, a class of analyses beyond the mean, application-specific error scales, or a criterion for deciding whether correction is beneficial. No universal quantification over all quotient spaces or all estimators is asserted here.

Short English statement

The problem asks when quotient-space geometry biases statistical estimators, whether the effect extends beyond mean shapes, how much it matters in applications, and when and how to correct it. The supplied record labels the problem partially solved, but leaves the geometry, statistical model, definition of bias, and correction criterion unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003714$, title Curvature Bias in Statistics on Quotient Spaces, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. The record gives proposed year 2018 and names no individual proposer. This draft uses the supplied record without independently inspecting registry files or external sources.
• Supplied report citation: Statistics for Data with Geometric Structure (2019), Oberwolfach Reports, pp. 125--186, DOI 10.4171/owr/2018/3, https://doi.org/10.4171/owr/2018/3. The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports. The supplied literature list labels this same reference Oberwolfach Report 3/2018. The differing year designations are retained without reconciliation.
• Supplied literature reference: Miolane--Holmes--Pennec, "asymptotic bias," https://doi.org/10.1137/16M1084493. The supplied record gives no fuller bibliographic details.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem partially_solved and states that asymptotic bias of Fréchet means on quotient-shaped spaces has been quantified using Taylor expansions and bootstrap or correction procedures, while the scope across other statistical analyses and practical necessity remain unsettled. This is supplied dated curation derived from the OWR curation, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This supplied editorial label does not establish mathematical correctness or independent source verification of this draft.