25 problems

Quadratic-rank conjecture. The linear system suffices for tensor decompositions of rank up to $O(n^2)$, with a leading coefficient improved over existing results.

Determinantal radical conjecture. These relations generate the radical of the variety defined by the $|Y|\times |Y|$ minors of $\mathbf{A}$.

Very efficient format conjecture. 1. The format $(n,R_c(n))$ is very efficient if $c$ is such that $|E_1|\geq |Y|$. 2. The format $(n,r)$ is efficient if there exists $c$ with…

Let $C^*$ be a ground-truth piecewise-linear curve with vertices $c^*_0,\dots,c^*_M$, let $\widehat{m}_3$ be the estimated third moment tensor, and let the tensor power method with…

Positive minimal tensor-rank conjecture. If the decomposition is positive and minimal, meaning that $\beta_i>0$ for every $i$ and $R$ is the lowest natural number possible, then ea…

Consider decompositions of sums of powers of quadratic forms, and let the rank be the number of summands. The number of variables is $n$. Linear-variable threshold conjecture. Effi…

Expected generic rank conjecture. In general, the generic rank equals this expected value unless $d=2$. This conjecture concerns the dimension and generic rank of secant varieties…

The tensor decomposition model uses complex-valued tensors, while the cited AMP results are derived for real-valued tensors. Complex-valued AMP conjecture. The asymptotic behavior…

Grammians conjecture. There exists a constant $\kappa_n>0$ and an increasing function $\gamma_n:(0,\kappa_n)\to(0,1)$, both depending only on $n$, such that $\gamma_n(\kappa)\to1$…

Defining-ideal conjecture. The Zariski closure of $SOT_{2,n}$ in $\mathbb{C}^{n^4}$ is cut out by the vanishing of $\mathcal P_n$, $\mathcal Q_n$, and $h_n$. The ideal defined by t…

Consider the optimization landscape for the symmetric tensor decomposition problem with rank $R$ and component-vector length $L$. A local maximum is called bad if it is not a globa…

Let $\{a_i\}_{i=1}^R$ be $R$ independent standard Gaussian vectors in $\mathbb{R}^L$, let $c_n<\frac{1}{n!}$ be a constant, and consider the power method iterations given by the pa…

Let four general forms in three variables have degrees $3,3,3,2$, respectively, and consider their minimal simultaneous Waring decompositions. Conjecture on quadruple decomposition…

Consequence of Conjecture. Under these hypotheses,

Generalized Kruskal conjecture. Under these hypotheses, a nontrivial zero-sum subcollection must exist. This relaxes general position to lower bounds on the dimensions of the subsy…

Uniqueness conjecture. For the dictionaries considered, the solution $X$ is unique among real matrices when

Let $\{\mathbf{A}^k\}$ be the sequence generated by Algorithm, where $\mathbf{A}^k$ is the solution $(\mathbf{A}^{(1)},\mathbf{A}^{(2)},\ldots,\mathbf{A}^{(N)})$ in the $k$-th iter…

Randomization conjecture. Randomization prevents getting stuck in a local minimum caused by overfitting the noise.

Let $V_1,\ldots,V_d$ be real inner product spaces, and let $T\in V_1\otimes\cdots\otimes V_d$ have an orthogonal decomposition … For $l\in\{1,\ldots,d\}$, define $T *_l T$ by contr…

Let $G(k,n)$ denote the Grassmannian of $k$-planes in an $n$-dimensional vector space, and let $\sigma_r(G(k,n))$ denote its $r$-th secant variety. A secant variety is defective wh…

Let $\mathcal{I}$ be the identity tensor of order $m$ and dimension $n$, defined by $\mathcal{I}_{i_1\cdots i_m}=1$ if $i_1=\cdots=i_m$ and $\mathcal{I}_{i_1\cdots i_m}=0$ otherwis…

Let a tensor have format $(n_1,\dots,n_d)$, and let … Call the format perfect when $R(n_1,\dots,n_d)$ is an integer, and call a format generically identifiable when a general tenso…

Let $S^d(\mathbb C^n)$ be the space of symmetric tensors, and let the odeco variety be the Zariski closure of the orthogonally decomposable tensors. For a tensor…

Let $k\ge 3$ and let $n_i$ denote the dimensions of the factors, for $1\le i\le k$. Write $\mathbf{n}=(n_1,\ldots,n_k)$ and define … A generic tensor of order $k$ and rank $r$ is a…

Let $k>1$, let $\boldsymbol{E}=S^{\delta_{1}}(E_{1})\otimes \cdots \otimes S^{\delta_{k}}(E_{k})$, where at least one $\delta_i>1$, and let $\Xi_r(\boldsymbol{E})$ be the $r$-secan…