18 problems

Let the data points lie in a symmetrically perpendicular CAT($\kappa$) space, and consider an iteration formed from compositions and convex combinations of proximal mappings of the…

Let the underlying space be a Hadamard space, let the data points be points in that space, and consider an iteration formed from compositions and convex combinations of proximal ma…

Let $X$ be a random variable supported on a set $A\subset \mathbb{S}^m$ whose convex closure has nonzero volume and which has a unique Fréchet mean $\mu$. A point $\mu$ is Type I f…

Let $R_1,\ldots,R_m$ be reference points in the Riemannian manifold under consideration. The exponential barycentric subspace is … where $\mathcal{P}_m^*$ is the set of…

Let the setting be that of the preceding result, with transformed empirical Fréchet means and a steep variational inequality of exponent $\beta$. The steep variational inequality i…

Converse tail-condition conjecture. The result of Theorem 2 should hold for any non-compact symmetric space: convergence $\mu_n\overset{\mathbb{P}}{\to}\mu$ should imply the corres…

Let $(M,d)$ be the metric space under consideration, and let $P\in\mathcal{P}^{1}(M)$ have a unique Fréchet mean. Unique-means extension conjecture. The results of the paper on sti…

In Scenario $3$, let $\mu\in M$ be prismatic, let $P\in\mathcal{Q}^{1}_{x,\kappa}$, and let $\Sigma_\mu$ denote the space of directions at $\mu$. Write $F_P$ for the Fréchet functi…

Let $X$ be a random variable supported on a set $A \subset \mathbb{S}^m$ whose convex closure has nonzero volume and which has a unique mean $\mu$. A point $\mu$ is Type I finite s…

Let $X$ be a compact metric space, and consider random variables taking values in $X$ whose Fréchet means and sample Fréchet means are defined. A distribution is smeary when its Fr…

Let $X$ be a random variable on a metric space. A distribution is smeary when its Fréchet mean has a limiting convergence rate slower than the classical $n^{-1/2}$ rate, and it has…

Let the $q$-skeleton be considered as a subspace of $p$-Wasserstein space, and let its “medial axis” denote the singular set relevant to the distance from that skeleton. A medial-a…

Let $X$ be a random variable on $\mathbb{S}^m$ whose local Fréchet mean is smeary, as in the curse of dimensionality theorem, and let the corresponding local Fréchet means be those…

Let $m\geq 2$, and let $F$ and $G$ denote the Fréchet functions associated with the intrinsic and comparison distance settings, respectively, near the origin $0$ in the real projec…

Stickiness conjecture. This phenomenon is related to the phenomenon of stickiness of the Fréchet mean in stratified spaces.

Let $\mathcal H$ be the underlying Hilbert space, and let $\Sigma_1,\dots,\Sigma_n$ be covariance operators on $\mathcal H$. Assume that $\Sigma_1$ is injective, and let…

Let $(M,g)$ be a compact orientable Riemannian manifold, and let $Q$ be a probability measure on $M$ having a twice continuously differentiable density. The Fréchet function is def…

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