37 problems

Noncommutative alpha-geometry conjecture. Suppose $n\geq2$. If $p\in(1,2)$, then $\operatorname{Scal}_p$ is strictly Schur-decreasing; if $p\in(2,+\infty]$, then…

Commutative alpha-geometry conjecture. Suppose $n>2$. If $p\in(1,2)$, then $\operatorname{Scal}_p$ is strictly Schur-decreasing; if $p\in(2,+\infty]$, then $\operatorname{Scal}_p$…

Petz's conjecture. The scalar curvature of the BKM metric is a Schur-increasing function; equivalently,

The statistical geometrodynamics model describes spatial geometry through a Riemannian metric and includes an arbitrary scale factor $\sigma(x)$, whose choice represents a gauge fr…

Let $\Theta$ be the neural tangent kernel of a diffusion model and let $k_H$ denote the Hellinger kernel of its data distribution. A Hellinger kernel is understood here as the uniq…

Let $\beta_1=|E|-|V|+c$ be the cycle rank (first Betti number) of the undirected skeleton of a directed acyclic graph, where $c$ is the number of connected components, and let…

Let a bitnet be a binary Bayesian network equipped with its Fisher information metric, and let $\langle R\rangle$ denote its volume-averaged Ricci scalar. Rodríguez's conjecture. F…

Incommensurability conjecture. For $n\ge 3$, these two constants generally possess no algebraic consistency; that is, they are conjectured to be incommensurable.

Let $\Delta$ be the $\ell^2$-simplex equipped with the Fisher–Rao metric, and let the gradient flow lines of $F_{(c_n)}$ be given by the integral curves in Proposition. In finite-d…

Let the density manifold be equipped with the Otto metric, and let the exponential connection be defined as the connection dual to the mixture connection with respect to this metri…

Sensitivity vanishing and sign-change conjecture. The sensitivity $\alpha_4$ vanishes for $c(x)=\beta^{-1}$ and becomes negative for $c(x)>\beta^{-1}$, irrespective of the value of…

Practical reference-function conjecture. In practical scenarios, one can choose a specific reference function $\phi$ with inherent properties that contribute to rapid convergence.

Let $M$ be a manifold and let $\operatorname{Prob}(M)$ denote the space of probability densities on $M$. For $p\in(1,\infty)$, consider the geodesic equation of the $L^p$-Fisher--R…

Let $(\rho_t^{\mathrm{WFR}})_{t\geq 0}$ denote the Wasserstein–Fisher–Rao gradient flow of the Kullback–Leibler divergence, and let the relevant expansion refer to the explicit asy…

Skewness-tensor splitting conjecture. The skewness tensor describing this dual structure should split into a purely classical part related to the geometry of the Fisher–Rao metric…

Let $\theta=\pi(\overline{\theta})$ be a parameter in $\mathcal{M}_{p,k}$, let $\operatorname{err}^{\mathcal{M}_{p,k}}_{\theta}(\widehat{\theta})$ denote the proposed error measure…

Symmetry conjecture for the canonical divergence. The condition above together with $\nabla\,\mathcal{R}\equiv 0$ is sufficient for

Let $({\rm g},\nabla,\nabla^*)$ be a dualistic structure, with canonical divergences $\mathcal{D}(p,q)$ and $\mathcal{D}^*(p,q)$ and pseudo-squared-distance $r(p,q)$. Canonical-div…

Four-dimensional support conjecture. The support of an optimal information distribution is almost everywhere a finite-dimensional submanifold with three spacelike and one timelike…

Boundary information inequality conjecture.

Ruppeiner's conjecture. The scalar curvature is related to the correlation volume by

Let $D_\gamma$ be the family of quantum distances with $\gamma\in[-1,2]$ on $\N^+_\star$. A quantum Brègman distance is a distance representable in the form specified by the source…

Consider a head-on collision between two Gaussian wave packets, and let the resulting quantum entanglement be described in terms of the underlying microscopic statistical structure…

Noncommutative Musielak–Orlicz manifold conjecture. A similar construction can be carried out in the noncommutative case.

Jenčová's extension conjecture. Jenčová's smooth manifold construction can be extended from $\N^+_{\star01}$ to $\N^+_{\star0}$.