109 problems

Let $(T,d)$ be a metric space and let $(G_t)_{t\in T}$ be a centered Gaussian process. A majorizing measure is the system of weights introduced by Fernique to improve the covering-…

Let the points $x_1,dots,x_N$ be iid drawn from a bounded probability density function $\rho$ with support in $D=[0,1]^d$. Let the Matérn kernel with parameters $\nu$ and $\ell$ be…

Let $\widehat{\sigma}_{1,n}^2$ be the maximum likelihood estimator under a misspecified decay parameter, and let $C_{\nu,\alpha_i}$ denote the Matérn normalization constants for pa…

Weak variance conjecture. The expected spectral norm satisfies

Let $X=(X_1,\dots,X_n)$ be a centered Gaussian random vector with $E[X_i^2]=1$ for $1\le i\le n$, and let $n\ge 4$. Consider the covariance matrix of $X$. Gaussian random vector ma…

Gaussian limit conjecture. The Gaussian limit results should continue to hold for

True-rate conjecture. For $m\geq3$, this upper rate is also the true rate of $e_{N,r}(FBS)$. The source notes that the true rate is known for $m=1$ and $m=2$, while the higher-dime…

Let $X$ be a Gaussian process for which the preceding upper bound gives a rate for the product quantization error $e^{(\mathrm{prod})}_{N,r}(X)$ as $N\to\infty$. True-rate conjectu…

Let $h$ be a function admitting a Fourier-like expansion in the Chebyshev basis, represented by an infinite vector $\vec{h}$, and let $D_{\infty}$ denote the diagonal matrix arisin…

Let $M$ be a compact smooth manifold of dimension $d$ without boundary, and let $\rho_j(u)$ denote the Gaussian kinematic densities … where $H_j$ is the $j$th Hermite polynomial. L…

Let $X$ be a centered Gaussian random vector in $\mathbbm{R}^n$ such that … let $S$ be the centered Gaussian random vector associated with the regular simplex, and let $H_X$ and…

Let $X$ be a centered Gaussian random vector in $\mathbbm{R}^n$ such that … and let $S$ be a centered Gaussian random vector whose covariance matrix is defined by the regular-simpl…

Joint-pair Gaussianity conjecture. If $(X_1,X_2)$ is a centered multivariate Gaussian, then $X$ is a Gaussian process.

Let $f$ be a smooth stationary Gaussian process, let $K(r)$ denote its correlation function, and let the Poisson approximation refer to the approximation for the locations and size…

Cosine covariance conjecture. For every $n\geq 1$, every correlation matrix $\Sigma$, and every $t\geq 0$,

Litvak's conjecture. For every $n\times n$ correlation matrix $\Sigma$ and $p>0$,

Gaussian correlation conjecture. One has

Information rates quantify time-directed information flows in stochastic processes; here, Gaussian processes are the processes for which the proposed estimators would be developed.…

Bounded-activation extension conjecture. The main quantitative convergence result remains valid even without the bounded-activation Assumption.

Let $S\subset\mathbb{S}^{n-1}$ and let $\delta>0$. A matrix $G\in\mathbb{R}^{m\times n}$ induces a $\delta$-uniform tessellation of $S$ if, for all $x,y\in S$, … where…

Let a Gaussian process be used to model an error, cost, or risk function in machine learning, and consider covariance kernels that are isotropic but non-stationary. Benning et al.'…

Consider the SPDE approach with coefficient regularity parameter $\beta$ and the associated posterior contraction rate for the unknown function under the conditions described above…

Let A1 and A5 denote the assumptions introduced for the model, and let Theorem … should hold even without assumption A5. The authors state that they do not know how to prove this.…

Let $cmathsf{h}$ denote the spatial resolution parameter in the estimator, and consider the upper bound in Theorem 2.2 for estimating the block Cholesky factor, which contains an a…

The paper considers homogeneously scattered observation points and a matching technique based on Hall's marriage theorem that relates them to points arranged on a regular lattice.…