58 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 $x$ be an object with assembly index $a(x)$, let $k_x$ be the minimal kernel distinguishing $x$ from its chemical precursors, and let $\mathcal{H}$ be the reproducing-kernel Hi…

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…

Weaker-singularity convergence conjecture. Kernels with weaker singularities should converge more rapidly.

Let $(\mathcal{H}_{k_\varepsilon})_\vartheta$ denote the power space associated with the kernel $k_\varepsilon$, where $\vartheta>1$, and let $\tau$ be the Sobolev smoothness param…

Kernel discrepancy subset-selection complexity conjecture. The subset selection problem is NP-hard.

Let KRAS denote the kernel adversarial estimator, and consider the source conditions imposed in the paper. An iterated Tikhonov KRAS estimator is obtained by applying Tikhonov regu…

Let an adaptive iterated KRAS estimator be an iterated Tikhonov KRAS estimator whose regularization or stopping choice is selected adaptively, as in the adaptive LRAS procedure dis…

Let $t$ be a positive integer, and let a $t$-iterated Tikhonov KRAS estimator mean the KRAS estimator obtained by applying Tikhonov regularization for $t$ iterations. Let Assumptio…

Let KRAS denote the kernel adversarial estimator, and consider the finite-sample setting and source-condition assumptions described in the paper. The estimator may be penalized usi…

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 $\mathtt{RFM}(\alpha)$ be the modified Recursive Feature Machines algorithm that draws fresh samples and reweights the data using an average of the Average Gradient Outer Produ…

Let $p\in\mathbb{N}$, and consider Gaussian data in the regime $n=d^{p+\delta}$ with $\delta\in(0,1/2)$. Let $\hat f$ be the kernel ridge regression prediction function with regula…

Kernel-basis approximation conjecture. The choice of kernel functions as basis functions can be very useful for the $n$-dimensional space of Gibbs distributions to accurately appro…

Let $s>2$ denote the source smoothness parameter, let $\beta$ be the eigenvalue-decay parameter, and let $n$ be the sample size. Kernel ridge regression (KRR) is tuned through its…

Zero-bandwidth limit conjecture. In this limit, the kernelized Wasserstein gradient-flow equation

Let Proposition denote the paper's linearization result for finite-degree polynomial kernels with positive Taylor coefficients. Smooth-kernel extension conjecture. Methods similar…

Let $d$ be the input dimension and $m$ the layer width, as in the stated feature-learning bound, whose approximation term contains the factor $\sqrt{d+m}$. Bietti et al.'s feature-…

KODE's kernel sufficiency conjecture. This problem is difficult for KODE because the MMD between the reference and target samples is very small and the chosen kernel $K$ is not suf…

In a filtering experiment, KME-Dynamics is compared with the Ensemble Kalman Filter and a Kalman-adjusted KME-Dynamics method. The KME-Dynamics inference step involves a linear inv…

The KME-Dynamics method is applied to transport probability distributions, including a Gaussian-to-mixture example in which splitting probability mass presents challenges. A noise…

Laplacian and Matérn kernel conjecture. The assumptions of part (i) are also true for the Laplacian and Matérn kernels when $C=1$.

Let $K_\Phi$ be the kernel matrix associated with a Matérn-type kernel $\Phi$, and let $\sigma_{\operatorname{min}}(K_\Phi)$ denote its smallest singular value. Let $q_n$ be the re…

The time-varying bandit setting concerns sequential optimization problems in which the objective or reward function may change over time. Time-varying bandit extension conjecture.…

For inputs of order one, let $\gamma$ denote the bandwidth parameter of the Gaussian spike, and consider the inaccuracy of the sine approximation to the Gaussian neural tangent ker…