5 problems

Let $(M,g)$ be a closed Riemannian manifold with $n=\dim M$, let $\xi^*\in H^1(M,\mathbb{Z}_2)$ be a flat bundle class selected by the variational principle, and let…

Let $\mathcal{K}$ be the space of kernels, let $\mathcal{S}^*(k)$ be the optimized path entropy for a kernel held fixed at $k$, and let a self-consistent kernel $k^*$ be a fixed po…

Let $\mathcal{X} \subseteq \mathbb{R}^d$, let $\nu$ have a density, and let $P$ and $Q$ be measures on $\mathcal{X}$ with densities bounded away from zero, satisfying…

Let $\tilde{\boldsymbol X}$ and $\tilde{\boldsymbol y}$ be new training data and labels independent of $\boldsymbol W_1$. Assume the activation $\sigma$ is odd, Assumption 1 holds,…

Huang–Jebara Bethe permanent kernel conjecture. The function $\kappa_{\operatorname{perm}_{\mathrm{B}}}$ is a positive definite kernel on $\mathcal{X}^n\times\mathcal{X}^n$. The cl…