Benign non-convexity conjecture for dimensional logarithmic Sobolev optimization

Let $U_r\in\mathbb{R}^{d\times r}$ be an orthonormal matrix, and let $\mathrm{Gr}(d,r)$ denote the Grassmann manifold of $r$-dimensional subspaces of $\mathbb{R}^d$. Define

$$\bar{\mathcal{J}}_\mathrm{KL}^\downarrow:\mathrm{Gr}(d,r)\rightarrow\mathbb{R},\qquad \bar{\mathcal{J}}_\mathrm{KL}^\downarrow(\mathrm{range}\{U_r\})=\mathcal{J}_\mathrm{KL}^\downarrow(U_r),$$

as in the paper's definition of $\mathcal{J}_\mathrm{KL}^\downarrow$. Benign non-convexity conjecture. The function $\bar{\mathcal{J}}_\mathrm{KL}^\downarrow$ is benignly non-convex on the Grassmann manifold $\mathrm{Gr}(d,r)$. The conjecture is motivated by numerical experiments in which Riemannian gradient descent appears to converge to the same solution from all initial conditions, despite the absence of global geodesic convexity on the compact Grassmann manifold; the precise meaning of “benignly non-convex” and a proof of the claimed landscape property remain to be established.