Injectivity of Procrustes Means of Covariance Operators

Definitions and mathematical statement

The supplied record asks whether injectivity is preserved by a Procrustes Fréchet mean of finitely many covariance operators. The following gives a precise conditional formulation using the standard positive trace-class interpretation of covariance operators and an equally weighted empirical mean; these conventions are not explicitly specified in the source record.

Let $H$ be a Hilbert space and let \[ \mathcal C(H)=\{A:H\to H: A\text{ is positive, self-adjoint, and trace-class}\}. \] The source does not specify whether $H$ is real or complex, or whether separability is assumed; the intended scope of these choices requires confirmation. An operator $A$ is injective when $\ker A=\{0\}$. For $A,B\in\mathcal C(H)$, use the Procrustes distance \[ d_{\mathrm P}(A,B)^2 =\operatorname{tr}(A)+\operatorname{tr}(B) -2\operatorname{tr}\!\left((A^{1/2}BA^{1/2})^{1/2}\right). \] For an integer $k\geq 1$ and $\Sigma_1,\ldots,\Sigma_k\in\mathcal C(H)$, define the set of equally weighted Fréchet means by \[ \mathcal M(\Sigma_1,\ldots,\Sigma_k) =\operatorname*{arg\,min}_{A\in\mathcal C(H)} \frac1k\sum_{i=1}^{k}d_{\mathrm P}(A,\Sigma_i)^2. \] A precise universal version of the supplied question is: assuming this set of minimizers is nonempty, does \[ \left(\ker\Sigma_i=\{0\}\text{ for every }i=1,\ldots,k\right) \quad\Longrightarrow\quad \left(\ker\overline\Sigma=\{0\}\text{ for every } \overline\Sigma\in\mathcal M(\Sigma_1,\ldots,\Sigma_k)\right) \] hold?

The source's singular phrase "their Fréchet mean" does not specify existence, uniqueness, or the intended quantifier if several minimizers exist. Thus the displayed universal version requires confirmation; existence and uniqueness are not asserted here. No commutativity hypothesis is imposed in the supplied general question.

Short English statement

If finitely many covariance operators have trivial kernels, must their Procrustes Fréchet mean also have trivial kernel? The displayed formulation uses positive trace-class operators, equal weights, and every existing minimizer; the source leaves these conventions and the precise Hilbert-space setting insufficiently explicit.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30003716$, title "Injectivity of Procrustes Means of Covariance Operators," exact source code catalogue entry, collection "Oberwolfach Reports Open Problems." The proposed year is recorded as 2018; no individual proposer is supplied.
• 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 record also labels this reference "Oberwolfach Report 3/2018" and describes the reports as documenting mathematical workshops and their open-problem discussions. The differing year labels are retained without reconciliation.
• Supplied literature reference: Masarotto--Panaretos--Zemel, Procrustes covariance metrics. https://arxiv.org/abs/1801.01990.
• Supplied dated literature curation, checked exactly 2026-08-21: the record labels the problem open and states that injectivity is proved in restricted commuting settings but remains unresolved for general operators. The listed sources checked are the two references above. This assessment is supplied dated curation derived from the supplied OWR curation, not independently verified; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and explains that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. These are supplied editorial labels, not independent mathematical verification of this draft.