30 problems

Let $D$ be the tensor rank, let ${\bm{\sigma}}=(\sigma_1,\ldots,\sigma_D)\in S_n^D$ be a trace-invariant, and let $K_\mathrm{p}({\bm{\sigma}},\eta)$ denote the number of pure conne…

Irreducible-representation melonic conjecture. Any irreducible tensor representation of $\mathrm{O}(N)$ or $\mathrm{Sp}(N)$ can support a melonic large $N$ limit.

Let $T$involve Bernoulli random tensor entries, and let $d_{i_{1},\dots,i_{k-m}}$ denote the degree of a tuple $(i_{1},\dots,i_{k-m})$, namely … Assume…

Let $B_1,\ldots,B_n$ be melonic, $D$-coloured graphs, and write $p_i=\#V(B_i)/2$. Define … For positive integers $p_1,\ldots,p_n$, let … Here $(x)_m$ is the $m$-th Pochhammer symbo…

Let $M_1,\ldots,M_q$ be melonic, connected $3$-coloured graphs, with $q\in\mathbb{Z}_{>0}$. Define … For each $i$, let $p_i$ satisfy $2p_i=\#V(M_i)$. Strong conjecture. … The right…

The paper considers real eigenpairs of a real antisymmetric tensor of order $p$ and dimension $N$, consisting of a real eigenvalue and $p$ orthonormal $N$-dimensional real eigenvec…

The probe and reference systems are parametrized by inverse temperatures $b$ and $b_{\rm ref}$, an overlap parameter $c$, and solution branches labeled solution 1 and solution 3. T…

Large $N$ factorization conjecture. The large $N$ factorization property should hold for the simplest case of a Gaussian random tensor.

Isotropy conjecture. The tensor $H'$ is isotropic.

Let $s<r$ and consider the setting of Theorem, including Assumption, except that the orthogonality condition between \hat\pmb v_i and \hat\pmb v_j for $1\leq i\leq s<j\leq r$ i…

Let (\hat\gamma_1\hat\pmb v_1,\dots,\hat\gamma_r\hat\pmb v_r) be critical points of the likelihood, with summary statistics consisting of…

Let $T\in(\mathbb{R}^d)^{\otimes r}$ be a symmetric jointly Gaussian tensor with mean zero, and let $\sigma_{(1,\mathcal{I}_2)}$ denote the first parameter in the paper's variance-…

Let $\mathcal{C}_{r,p}(d)$ be the Type-$2$ constant for $2\leq p<\infty$, and let $E_{r,p}$ depend only on $r$ and $p$. Conjectured type-2 bound. One has … Moreover, the stronger i…

Let ${\bm{\tau}}\in S_n^D$, and let $\nabla({\bm{\tau}};\eta)$ be the nonnegative quantity whose vanishing characterizes the relevant compatible configurations. For a permutation o…

Let $h(y)$ be the large-$N$ exponent governing the size density of eigenvalues, with its roots displayed in Fig. 1 for the relevant values of the rescaled noise parameter. Largest-…

Gaussian MPS convergence conjecture. As the bond dimension $q$ increases, the normalized injective norm of the Gaussian MPS converges to that of the corresponding Gaussian tensor:

Real-complex asymptotic equivalence conjecture. The asymptotic limits of the injective norms of corresponding real and complex Gaussian tensors are identical.

Inverse-square-root correction conjecture. The first correction term to the asymptotic injective norm is of order $\frac{1}{\sqrt{d}}$.

Let $B$ be a state in the entangled regime with $\beta=0$ and $\epsilon=1/(D-1)$. Assume that tensors satisfying $\beta+\epsilon(D-1)=1$ saturate the number of degrees of freedom,…

Consider an asymmetric spiked tensor maximum-likelihood problem with signal-to-noise ratio $\beta$. Let $(\lambda_*,{\bm{u}}_*^{(1)},\ldots,{\bm{u}}_*^{(d)})$ be the tuple associat…

Consider an asymmetric spiked tensor maximum-likelihood problem with signal-to-noise ratio $\beta$. Let $(\lambda_*,{\bm{x}}^{(1)}_*,\ldots,{\bm{x}}^{(d)}_*)$ denote the tuple asso…

Let ${\bm{\mathsfit{Y}}}$ be an asymmetric spiked tensor with signal-to-noise ratio $\beta$, and let the maximum-likelihood problem be the optimization problem in Eq. (objective).…

Let $\widehat\Sigma_N=\frac{1}{N}\sum_{k=1}^N\mathbf{x}_{pk}\mathbf{x}_{pk}^{\top}$ be the sample covariance matrix of independent copies of a random vector following the random te…

Let $A\in(\mathbb{R}^N)^{\otimes p}$ have independent entries distributed as $\mathcal{N}(0,N^{-(p-1)})$, and let $\mathcal{B}_N\subset\mathcal{H}_N$ denote the binary and Hilbert-…

Hilbert-cube OGP conjecture. For every even $p\geq4$, there exist $\mu>0$ and $0<\nu_1<\nu_2<1$ such that $\mathcal{A}$ satisfies the OGP with domain $\mathcal{S}_N=\mathcal{H}_N$,…