43 problems

Drew–Johnson–Loewy conjecture. The CP-rank of any completely positive matrix over $\mathbb{R}$ is at most

Salce–Zanardo conjecture. If $R$ satisfies $\operatorname{ID}_2$, then $R$ must be a Bézout domain.

Consider the belief propagation equations for the matrix factorization model and their fixed points. Bayes-optimal inference refers to inference under the true generative distribut…

Let $A\in\mathbb{R}_{\geq 0}^{m\times r}$ and consider the cone toolkit, in which rows of $A$ determine facets of the associated cone and its extreme rays provide candidate vectors…

Let $n\geq 2$, and let $\mathcal{T}_n$ be the affine variety of $n\times n$ Toeplitz matrices. Write $\mathbf{FL}_{\mathrm{max}}(\mathcal{T}_n)$ for the maximal $\mathcal{T}_n$-fac…

Ceiling-bound conjecture. For $n>1$,

Quasi-Toeplitz row-equivalence conjecture. Every $n\times n$ matrix is row equivalent to a quasi-Toeplitz matrix.

Bounded invertible reduction conjecture. There exists a constant $C>0$ such that every matrix can be expressed as a product of an invertible matrix and $C$ Toeplitz matrices. In pa…

Ye–Lim's conjecture. For all $n\in\mathbb{N}$,

Let $A^{(t)}$ denote the matrix produced after $t$ steps of the uniformly random pivoting procedure, and let $A$ be the initial matrix. Write $kappa(A)$ for the condition number of…

Let $X=UV^\top$ be a full-dimensional Burer--Monteiro factorization, with gradient descent applied using sufficiently small step sizes and initialization sufficiently close to the…

Let $s(n)$ be the set of $n\times n$ stochastic matrices, and let $G$ be a generating set of a similar form to the one considered for three-dimensional stochastic matrices. Let…

Let $s(n)$ denote the set of $n\times n$ stochastic matrices, and let $N_G(S)$ be the factorization length associated with a generating set $G$ for a stochastic matrix $S$. In thre…

Nyström-initialization conjecture. The linear term in Ye et al.'s Theorem 1.1 can be removed when Nyström initialization is used.

Consider the $\ell_1$-norm rank-one symmetric matrix factorization problem and initialize the subgradient method randomly, with an initialization distribution absolutely continuous…

Let $n$ be an even natural number greater than $2$. A quasi-prime matrix is a matrix that is the product of two quasi-invertible semi-prime matrices, where a semi-prime matrix is a…

Let $\mathbf{H} \in \mathbb{R}^{n\times n}$ be a positive definite Hankel matrix with bit complexity $\ell$ and condition number bounded by $2^\ell$. An inverse symmetric positive…

Let $\mathbf{H} \in \mathbb{R}^{n\times n}$ be a positive definite Hankel matrix with bit complexity $\ell$. A symmetric positive factorization algorithm should find a representati…

Optimality conjecture. The analytical formulas for rotation-invariant estimators are optimal in the large-dimension limit, in the sense that they minimize the average mean-square e…

Cossu–Zanardo necessity conjecture. If this matrix can be written as a product of two idempotent matrices, then it must satisfy the Cossu–Zanardo conjecture.

Cossu–Zanardo conjecture. There exist $a,b,c\in\mathbb{Z}[\sqrt{D}]$ with $a(1-a)=bc$ such that

Consider gradient descent for the matrix approximation problem in the paper, where $\bSigma$ is the associated matrix and the leading eigenvalues of $\bSigma$ may or may not be ide…

Let gradient descent be applied to the matrix approximation problem considered in the paper, with $\bSigma$ denoting the associated matrix and with random initialization of moderat…

Let $G$ be a matrix-factorization parametrization, such as $G(U)=UU^\top$ or $G(U,V)=UV^\top$, and distinguish diagonal measurements, whose coordinate parametrizations commute, fro…

Let $\mathcal B\subset\mathbb R^{d_0\times d_1}\times\cdots\times\mathbb R^{d_{N-1}\times d_N}$ be a bounded initialization set, let $Q=YX^T(XX^T)^{-1/2}$, and define…