20 problems

Let $\Sigma_1$ and $\Sigma_2$ be the covariance matrices, and let $H$ be their joint limiting spectral distribution. For any $n_1,n_2\in\mathbb{N}$ and every selection of non-negat…

Let $f$ be a quadratic function, and consider the covariance matrix maintained by CMA-ES. Hansen's covariance convergence conjecture. The covariance matrix of CMA-ES should converg…

Covariance-structure conjecture. This effect is related to the structure of the covariance matrix $V$ and its positive definiteness.

Assume conditions (A1)–(A4), and suppose $\lambda$ and $\alpha$ satisfy either $\alpha\in[0,1)$, or $\alpha=1$ and $\lambda=\sigma_1^2(A)/(\sigma_1^2(A)+c)$ for a positive constant…

Let $T_3(Y)$ be the test statistic and let $\phi_3(Y)$ denote its test, under the same conditions used for the consistency result for $\phi_1(Y)$. Consistency conjecture for…

Let $T_2(Y)$ be the test statistic and let $phi_2(Y)$ denote its test, under the same regularity conditions used for the consistency result for $phi_1(Y)$. Consistency conjecture f…

Let an eigenspectrum determine a set of quantum covariance matrices. Problem 1 asks for the eigenspectra for which the corresponding quantum covariance matrices form a single orbit…

Let $d$ be the ambient dimension, let $r$ be the rank parameter, and let $C_{d-r,1}$ and $C_{d-r,2}$ be the constants appearing in the covariance eigenvalue bounds, with $d-r$ the…

Exponential joint error-rate conjecture. The joint error probability should satisfy

Pairwise-max positive-semidefiniteness conjecture. The matrix $R(\overrightarrow{r}_{pm})$ is always positive semidefinite; equivalently, $\overrightarrow{r}_{pm}$ is a valid choic…

Let the samples be drawn independently from a Gaussian distribution with covariance matrix $\Sigma(r)$, and let $\sigma$ and $\alpha$ denote the noise and signal parameters, respec…

Let $\lambda \geq 0$ and $a_{\lambda} \geq 0$ be prespecified constants. Suppose $(\bar{\mathbf{x}}_T-\bar{\mathbf{x}}_C)\mid\mathbf{x}\sim\mathcal{N}(0,\mathbf{\Sigma})$, with…

Let $Z\in\mathcal{G}(n,p)$, let $\lambda_k$ denote its ordered eigenvalues, let $q_k$ be the corresponding classical locations, and write $\tilde{k}=k\wedge(n+1-k)$. Assume…

Matheron's product-form normal conjecture. Every normal vector of $\mathscr{U}_{N}^{*}$ is of the form

Consistency conjecture. The estimators of $H(t)$ and the corresponding functions, such as $\frac{1}{n}\operatorname{tr}\mathbf{T}_n^2$, obtained by the above method, are also consi…

Let $X\sim\mathcal{N}_{v}(0,\Lambda)$, where $\Lambda=\operatorname{diag}(\lambda)$, and let … be the Euclidean ball of radius $\rho$. Consider the covariance matrix of the vector…

Let $C$ be a positive semidefinite matrix and let … Consider high-dimensional empirical covariance matrices $YY'$ and rank-$r$ projections onto their leading singular spaces. Wisha…

Let $\mathit{BCD}(p)$ denote Efron's biased coin design, and let $\bolds{\Sigma}_{\mathbf{T}_n}$ be the covariance matrix associated with the treatment assignment vector for a samp…

Large-system consistency conjecture. In this joint limit, $\widehat{k}$ is a consistent estimator of $k_{\mathrm{eff}}(c\mid {\bf R})$.

Classical consistency conjecture. With $n$ fixed and $m\to\infty$, $\widehat{k}$ is a consistent estimator of the true number of signals $k$.