11 problems

Replica continuity conjecture. The analytic continuation of $f(\cdot)$ from $\mathbb{Z}^{+}$ to $\mathbb{R}^{+}$ equals the same function $f(m)$ for real $m$, and the limits with r…

Analytic continuation conjecture. The exact representation above, established for $n\in\mathbb Z^+$, also holds for $n$ away from the positive integers, so that the two-point corre…

The paper considers the asymptotic limit $N\to\infty$ for feature learning in imbalanced classification, using the mathematically non-rigorous replica method to derive results abou…

Consider the paper's Bayesian-optimal inference setting, equivalently statistical-mechanical models on the Nishimori line, in the asymptotic limit $N\to\infty$, and let the paper's…

The replica method gives a potential $\phi(\mathbf{q},\hat{\mathbf{q}},\mathbf{m},\hat{\mathbf{m}},\mathbf{q}_{12},\hat{\mathbf{q}}_{12})$, whose extrema determine self-consistency…

Consider a multilayer model with weight matrices whose aspect ratios are denoted by $\{\alpha_k,\tilde \alpha_k\}_{k=1}^\ell$. Assume that the weight matrices are drawn from an ens…

Consider a communication system with an independent and identically distributed random matrix and a postulated minimum-mean-square-error estimator. The decoupling conjecture. The p…

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.…

Consider a large-system MAP estimation problem whose analysis uses the replica method and satisfies the replica continuity assumption. Generic MAP-decoupling conjecture. Decoupling…

DFT replica conjecture. The replica result of Claim should also apply to a DFT sensing matrix.

Let $k$ and $m$ be parameters, let $\lambda$ be the Hermite-ensemble parameter, and let $s$ be complex. For positive integer $k,m$ and $\lambda>0$, the duality relation is … Here…