7 problems

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…

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…

Error-order conjecture. The expected coordinatewise error satisfies

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…

Let $\hat L^{k_1,k_2}$ be a block of the matrix $\hat L$ arising in the covariance-based cryo-electron microscopy algorithm, with $k_1,k_2$ its associated frequency indices. Let…