31 problems

Let $\bm{S}$ be the sample scatter matrix and let $\widehat{\bm{\Sigma}}^U=n^{-1}\bm{S}$ be the unbiased estimator of the covariance matrix $\bm{\Sigma}$. For an orthogonally equiv…

Let $\mathcal C(\sigma_{\min},\sigma_{\max})$ be the bounded-spectrum correlation class, and let $\widehat{\bm{\Sigma}}$ estimate the latent correlation matrix $\bm{\Sigma}$. Sharp…

Let $\widehat{\bm{S}}_n$ be the normal-scores covariance, and let $m_n$ and $m_F$ denote its empirical Stieltjes transform and the limiting generalized Marčenko--Pastur Stieltjes t…

The covariance matrix … , but its privacy term incurs a logarithmic-factor loss. Unavoidability conjecture. This logarithmic loss is unavoidable for adaptive estimation under diffe…

Let $\Sigma$ denote the asymptotic covariance matrix in the pseudo-observation regression setting, and let … be the Huber--White-type estimator. Overgaard's conjecture. This estima…

In pseudo-observation regression, let the usual Huber--White-type sandwich estimator refer to the variance estimator used for estimating-regression parameters from pseudo-observati…

Nonnegative covariance-constrained optimum conjecture. If the right-hand optimization is constrained so that every element of $X_L^\top X_L$ is nonnegative, then the optimizer sati…

Consider the proportional-asymptotic Gaussian-design setting in which the covariate covariance matrix $\bm{\Sigma}$ is unknown and is estimated from the data. No-sample-splitting c…

Covariance operator estimation concerns recovering a covariance operator from observations, while kernel density estimation is studied using Fourier analysis techniques. Fourier-an…

Optimal low-rank covariance approximation conjecture. For almost any positive definite matrix $A$, for every iteration $m\leq n$, there exist $d(m)\ll n$ and $r(d)\leq d(m)$ such t…

Dithering-scale conjecture. The observed numerical degradation stems from the factor $\sqrt{\log i}$ in $\lambda_i$: although this factor may facilitate theoretical analysis, it ca…

Error-order conjecture. The expected coordinatewise error satisfies

Let $\epsilon$ denote the averaged discrepancy between the Ledoit–Wolf shrunken eigenvalues and the corresponding population quantities over an interval $I\subset[0,\infty)$, and l…

Let $\mathbf{R}$ be the population covariance matrix, with eigenvectors $\mathbf{u}_i$, and let $\hat{d}_j$ denote the shrunken eigenvalues of the Ledoit–Wolf estimator. For some…

Let $X_i$ be observations from a Gaussian mixture model with noise covariance matrix $\boldsymbol\varepsilon_i\sim N(0,\Sigma)$, where the positive-definite covariance matrix…

Preliminary-estimation conjecture. Replacing the unknown idiosyncratic components by $\widehat{\mathbf e}_{nt}^*$ in the computation of $\widehat{\bm\Gamma}_n^{e*}$ should not affe…

The problem concerns estimating a positive semidefinite covariance matrix $T\in\mathbb{R}^{d\times d}$ from query access to entries of independent samples $x^{(1)},\ldots,x^{(n)}$…

Consider an $\epsilon$-contamination model in which $(1-\epsilon)n$ samples come from an elliptical distribution with shape matrix $\bm S_{in}$ and the remaining $\epsilon n$ sampl…

Covariance spectral-approximation sample-size conjecture. The improvement obtainable from accelerating stochastic gradient descent, relative to averaged stochastic gradient descent…

Let $Z(\boldsymbol{s}_0)$ be the process value at an unobserved location, let $\widehat{Z}_{n}(\mu,\kappa,\beta_1)$ be the predictor based on $n$ observations, and let…

Genericity conjecture. It should be possible to prove, similarly to the genericity result for the negative result and the generic resolution result for the adjustment procedure, th…

Let Tyler's M-estimator be formed from high-dimensional data samples, with the empirical spectral density given by the empirical distribution of its eigenvalues. Tyler's M-estimato…

Improvement conjecture. These two aspects can be improved via other proof techniques.

For frequency indices $k_1$ and $k_2$, let $\hat L^{k_1,k_2}$ be the corresponding block of $\hat L$, and write $\kappa(\hat L^{k_1,k_2})$ for its condition number. Condition-numbe…

For frequency indices $k_1$ and $k_2$, let $\hat L^{k_1,k_2}$ denote the corresponding block of the matrix $\hat L$, and let $\lambda_{\max}(\hat L^{k_1,k_2})$ be its maximal eigen…