7 problems

MMSE conjecture. If

Let $Y\sim\operatorname{Unif}(\{\pm1\})$, $X=Y$, and let $\mathcal{S}_k$ be the family of functions used in the numerical experiment, with $k\ll n$. Small empirical square-loss con…

Let $B$ denote the section size in the compressed-sensing model, and let Theorem and Corollary be the replica formula and its associated MMSE and Y-MMSE conclusions for the model.…

Discrete optimal-input conjecture. The optimal input for $\mathcal{C}_1(\mathsf{snr},\mathsf{snr}_0,\beta)$ is discrete.

Let $D(R,\beta)$ denote the asymptotic MMSE at measurement rate $R$ and inverse noise variance (SNR) parameter $\beta$, and let $m_a$ be the a-priori magnetization. In the low-nois…

Let $X$ be a finite-variance random variable in the scalar case $n=1$, and consider the optimization problem of maximizing mutual information subject to a constraint on the MMSE. D…

Let $X$ and $Z$ be zero-mean random variables, and for $\rho\in[0,\infty)$ let ${\mathsf{mmse}}(X,\rho)$ denote the minimum mean-square error in estimating $X$ from a Gaussian-nois…