7 problems

Let $f_X(x)=e^{-\phi(x)}$ be a symmetric log-concave distribution, where $\phi$ is upper bounded by some fixed polynomial. Consider one-bit non-adaptive and adaptive estimators of…

For the generalized Gaussian family with shape parameter $\beta>0$ and unit-variance density … where the distribution is strictly log-concave for $\beta>1$, the non-adaptive lower-…

Fixed-sample concentration conjecture. For a given sample size $n$, a concentration inequality of the same type should hold without the $\log(n)$ term.

Finite-covariance conjecture. Whenever $p$ has finite covariance, a version of the paper’s main theorem holds for some $\epsilon_{n,\delta}(p)$ that approaches the high-dimensional…

High-dimensional sub-Gaussian lower-bound conjecture. Mean estimators cannot outperform this sub-Gaussian rate in the asymptotic regime as $n\to\infty$.

Tightness conjecture. The upper bound above is tight for all $d$; equivalently, a matching lower bound of the same order should hold, including when $\delta>0$ and $d\gg1$. The con…

Let $\theta$ be estimated from sequential one-bit measurements with Gaussian noise of variance $\sigma^2$, using the adaptive encoding and estimation scheme defined by the posterio…