39 problems

Let $d$ and $k$ be positive integers, and consider a general mixture of $k$ Gaussian densities in $\mathbb{R}^d$. A mode is a local maximum of the mixture density. Sturmfels's conj…

Let $f$ be a mixture of three Gaussian densities in ambient dimension $d=2$, with all three component covariance matrices equal; a mode is a local maximum of $f$. Four-mode conject…

Finite Gaussian mixtures with unknown locations and unknown covariance matrices form the location-scale setting. In this setting, covariance perturbations create higher-order inter…

Cantero–Miranda–Vitale conjecture. Half-spaces bounded by vertical hyperplanes are extremal sets for the isoperimetric problem of the mixture.

In the normal mean model, let the prior be either compactly supported or subgaussian, and consider the minimax regret over the corresponding class of priors. The previously establi…

Let $d\geq K$, let $\Delta$ be the separation parameter, and suppose the mixture means are sampled according to the prior distribution considered in the paper. Let $G^{*,sp}$ be th…

Let $n$, $d$, and $K$ be positive integers with $d\geq K$, and let $\Delta_*$ denote the separation parameter for the Gaussian-mixture clustering model. Write $a\gtrsim_{\log} b$ w…

Let GF denote the Gaussian gradient flow method and CBO the consensus-based optimization method for approximating Gaussian mixture target distributions, with multiple particles use…

In the Gaussian mixture clustering model, let $n$ be the number of observations, $p$ the ambient dimension, and $K$ the number of clusters. The informational threshold concerns sta…

Consider a Gaussian mixture model with $n$ observations in $\mathbb{R}^p$, a fixed number $K$ of clusters, and signal separation $\mathit{\Delta}^2$. Assume the asymptotic regime…

Galois-width conjecture.

Consider the numerical evaluation of the Polyfit approximation to Gaussian mixture entropy in the case $q=4$ and $n=8$, where the approximation diverges as $C$ grows larger. The co…

Let $\bar{h}^{\mathrm{Polyfit}}_{C}({\mathbf X})$ denote the Polyfit approximation to the differential entropy $h({\mathbf X})$ of a Gaussian mixture, with approximation parameter…

Let the sampling problem have a target density whose complexity may be measured, for example, by its number of modes, and let the problem dimension be the number of variables in th…

Aden-Aden-Arieli et al.'s sample-complexity conjecture. Only

Let $s\geq 0$ and $p=b/n$. For independent random variables $V,W,W',Z$, with $V\sim\operatorname{Ber}(p)$, $\lVert W\rVert,\lVert W'\rVert\leq s$, and…

Gaussian degree-4 secant defect conjecture. The (tangential) defect of $\sigma_m\operatorname{GM}_{4}(\mathbb{C}^n)$ is precisely $\binom{m}{2}$, unless…

Gaussian moment identifiability conjecture. For all $m,n,d\in\mathbb{N}$ with $n\geq 2$ and $d\geq 5$, $\operatorname{GM}_{d}(\mathbb{C}^n)$ is $m$-identifiable whenever

Let $\bar{f}$ be the population objective and $\bar{f}_{n}$ its empirical counterpart for the location parameter, and consider the empirical iterates of the ELU updates. The first…

Spurious fixed-point conjecture. There are only two spurious fixed points of $f(\theta)$ in $\mathbb{R}_{-}$. Moreover, there exists $\delta_{\mathrm{cr}}\in(0,1/2)$ such that the…

A $k$-mixture of spherical Gaussians is a probability distribution formed by combining $k$ Gaussian components in which each covariance matrix is a scalar multiple of the identity.…

Let $\a>0$, let $x_m$ be mixture locations, let $w_m$ be mixture weights, and let $\phi$ denote the Gaussian density. Gaussian-mixture derivative conjecture. Certain mixtures posse…

Consider the nonparametric maximum likelihood estimator for a multivariate Gaussian mixture based on a subgaussian sample of size $n$ in dimension $d$. A mixing distribution is $k$…

Let the mixing distribution of a Gaussian location mixture be supported on the compact interval $[-1,1]$, and let $K_n$ denote its statistical degree. Statistical-degree conjecture…

Given samples from an equal mixture of $\mathcal N(\mu_1,I_d)$ and $\mathcal N(\mu_2,I_d)$, suppose that $\mu_1$ and $\mu_2$ have bounded $\ell_2$ norms and that $\mu_1-\mu_2$ is…