27 problems

In the double Fibonacci model, let the encoded subspace be the span of the modular orbit states $\{M\ket{w}\}_M$, where $M$ ranges over modular matrices and … with $\omega^6=1$. A…

Multipartite mutual-information conjecture. The EHLs can be associated with multipartite mutual informations.

Mutual-entropy ground-state area-law conjecture. The area law for ground states can be formulated in terms of mutual entropy instead of von Neumann entropy.

Let $A$ be a spherical region, with spatial complement $D_A$ and complementary domain $D_{\bar{A}}$. For a scaling dimension $\Delta$, let $G^{-1}_{A;\Delta}(\xi_1,w_1)$ be an inve…

Let $f(t,\mu)$ denote the limit of the re-scaled mutual information between the signal and the observation, parametrized by $(t,\mu)\in\mathbb{R}_{\geq 0}\times\mathcal{M}_+$. Domi…

Intrinsic-dimension conjecture. The optimal value of $k$ is related to the intrinsic dimension of the data distribution, even when the distribution is not strictly supported on a l…

Let the free theory be the Gaussian Euclidean field theory of a single massive scalar boson without interactions, and let the mutual information refer to the information shared bet…

Let $f$ denote the unique viscosity solution of the infinite-dimensional Hamilton–Jacobi equation governing the enriched free energy, and let $\mathbf{G}_N$ and $\sigma^*$ denote t…

Let $\overline{F}_N$ be the modified free energy, and let $f(t,\mu)$ solve the infinite-dimensional Hamilton–Jacobi equation specified by the model's enriched free energy. Free-ene…

Let $C(\sigma^2)$ denote the mutual information (MI) of a two-hop, double-scattering MIMO channel, and let $\overline{C}(\sigma^2)$ and $V(\sigma^2)$ be the asymptotic mean and var…

Mutual-information concavity conjecture. The mutual information $I(X_1,X_2;Y_1,Y_2,Y_3)$ is concave in $(T_1,T_2,T_3)$. The conjecture concerns the sensing-time allocation in the t…

Let $\mathcal H$ be a VC class of dimension $d$, let $\mathcal D$ be a realizable distribution, and let $\mathcal A_n$ be a learning algorithm; it may output hypotheses outside…

Let $\mathcal H$ be a VC class of dimension $d$, and let $\mathcal D$ be a realizable distribution. A proper learning algorithm is one whose output belongs to $\mathcal H$; write…

Let $\mathcal H \subseteq \mathcal X\to\mathcal Y$ be a VC class of dimension $d$, and let $\mathcal A$ be a proper learning algorithm. For a distribution $\mathcal D$, write…

Small-aspect-ratio channel equivalence conjecture. As $lpha\to0^+$, the asymptotic normalized mutual information of the original structured tensor estimation problem converges to t…

Conjecture. Condition … . The conjecture proposes that the stated continuity condition is not only sufficient but also necessary for convergence of the mutual information, independ…

Mutual-information subadditivity conjecture. The inequality

Let $\mathcal{X}$ be a finite alphabet with $|\mathcal{X}|>2$. For an information-distillation function $\mathrm{ID}_M(|\mathcal{X}|,\beta)$, where $M$ is the number of quantizatio…

Let $\mathbf{X}=[X_1\enskip\ldots\enskip X_n]$ be the input to a memoryless binary symmetric channel with error probability $0\leq p\leq\frac{1}{2}$, and let…

The channel input parameters determine the channel and its ability to convey information. Let $\varPsi$ denote the underlying nature of channel information, and let $\rm{I}$ be a c…

Vectorial replica-formula conjecture. Theorem 1-letter formula for $i^{\rm cs}$ and the measurement-MMSE corollary hold for any $B$.

Finite mutual-information conjecture. One should have

Let ${\mathcal I}$ be the achievable rate of the single-carrier ISI channel, and let $I_{\mathrm{SL}}$ denote the Shamai–Laroia approximation. Relaxed Shamai–Laroia conjecture. The…

The mutual-information bound. One has

Let $\bar{I}(\tau)$ and $\hat{I}(\tau)$ denote the two population time-delayed mutual information estimators, and define their difference by … A population is statistically homogen…