32 problems

Let $r$ be the activation power, let $L$ be the depth, and fix input and output widths $d_0$ and $d_L$. For an architecture $\mathbf{d}=(d_0,d_1,\dots,d_L)$, call it minimal fillin…

Two-orthogonal uniqueness and stabilization conjecture. A generic two-orthogonal tensor in $\mathbb{R}^{n_1}\otimes\cdots\otimes\mathbb{R}^{n_d}$ has a unique two-orthogonal decomp…

Fix a tuple of graphs $(\mathcal{G}_1,\dots,\mathcal{G}_d)$ on vertex set $[r]$, and let $W \subset (\mathbb{R}^2)^{\otimes d}$ be the set described by this tuple. Assume that the…

Dimension conjecture for binary tensors. For binary tensors, equality holds:

Let $W_r$ denote the set of real tensors admitting a two-orthogonal decomposition with at most $r$ summands, and let $r_g$ be the generic rank in…

HyperCube regularizer minimizer conjecture. The unitary group representation $A_g=B_g=C_g^\intercal=\varrho(g)$ describes the unique minimizer of $\mathcal{H}$ up to unitary basis…

Descent-based recovery conjecture. Descent-based algorithms, such as those used for tensor decompositions, together with the construction procedure for TSS matrices, may prove to b…

Bounded-dimensional support conjecture. There exists a positive integer $C(d,k)$ such that, for every order-$d$ tensor $T$ over an arbitrary field $\mathbb{F}$ with slice rank $k$,…

Let $\mathcal{S}$ be the Segre variety of rank-$1$ tensors in $\mathbb{K}^{n_1\times\dots\times n_d}$, and let $\sigma_r(\mathcal{S})$ be the Zariski closure of the set of tensors…

Let $\mathcal E$ denote the class of existentially closed II$_1$ factors. A McDuff factor $P$ admits only the canonical McDuff decomposition if, for every tensor decomposition … wh…

Let $\mathcal E$ denote the class of existentially closed II$_1$ factors. Tensor indecomposability conjecture. For every $P\in\mathcal E$, there do not exist II$_1$ factors $A$ and…

Let $G_1,\dots,G_n$ be connected non-compact simple real Lie groups with finite center, and let $\Gamma$ be an icc irreducible lattice in $G_1\times\dots\times G_n$. Ioana's primen…

Let $P_{abc}$ be a unital tensor that admits a covering system of partial algebras. A pointwise decomposition is a positive tensor rank decomposition … such that every factor $p^i$…

Let $P_{abc}$ be a unital symmetric real tensor, meaning that its indices are symmetric and its unit conditions are $P_{1ab}=P_{a1b}=P_{ab1}=\delta_{ab}$. A positive tensor rank de…

Definiteness conjecture. Then $\mathcal{A}$ is neither a positive semidefinite tensor nor a negative semidefinite tensor.

Let $\mathcal{A}\in\mathbb{R}_{cps}^{n^4}\backslash\mathbb{R}_s^{n^4}$ be a real fourth-order conjugate partial-symmetric tensor that is not symmetric. A CPS rank decomposition is…

Condition-number equality conjecture.

Let $q$ be the ALS fixed-point map for canonical tensor decomposition, and let $x^*$ be a fixed point. Write $q'$ for the Jacobian evaluated at $x^*$, and let $\rho_{q'}$ denote it…

Sharp invariant rank bound conjecture. One has

Finite expected angular condition number conjecture.

Infinite expected regular condition number conjecture.

Let $C_1^0$ be the open condition on the parameters $\mathbf{h}$ requiring $\det \mathbb{H}_{\Lambda}^B\neq 0$, and let $C_2$ be the closed condition requiring the matrices … to co…

Let $n_1,n_2,n_3$ satisfy $n_1 \ge n_2 \ge n_3 \ge 3$, and let random rank-$1$ tuples of length $r$ in $\mathbb{R}^{n_1 \times n_2 \times n_3}$ have their expected condition number…

Let $f$ be the tensor-decomposition objective considered in the paper, let $d$ and $n$ denote its dimension and number of components, respectively, and let $\zeta=O(\sqrt{d/n})$. A…

DCPD distributional convergence conjecture. There exists at least one absolutely continuous probability measure $\mu$ on tensors $\boldsymbol{\mathcal{T}}$ in $\mathscr{T}^{(R)}$ s…