20 problems

Improved-factor conjecture. The factor $4^k$ can be replaced by $2^k$, so that

Let $\sigma_1\geq\sigma_2\geq\cdots$ be the singular values relevant to the low-rank approximation problem, let $k$ be the target rank, and let $b$ be the block size with…

Optimal low-rank covariance approximation conjecture. For almost any positive definite matrix $A$, for every iteration $m\leq n$, there exist $d(m)\ll n$ and $r(d)\leq d(m)$ such t…

Optimal local convergence-rate conjecture. The optimal local convergence rate of MS-GFEM is

Let $\mathcal{L}_2^S\subset\operatorname{Sym}_n(\mathbb{C})$ be the variety of symmetric matrices of rank at most $2$ with diagonal zero pattern $S=\{(1,1),\ldots,(s,s)\}$, where…

Let $\mathcal{L}_2^S\subset\mathbb{C}^{m\times m}$ be the variety of square matrices of rank at most $2$ with zero pattern $S=\{(1,1),\ldots,(s,s)\}$ for some $s\ge2$. Quadratic-gr…

Let $\mathcal{L}_2^S\subset\mathbb{C}^{m\times n}$ be the variety of matrices of rank at most $2$ with diagonal zero pattern $S=\{(1,1),\ldots,(s,s)\}$, where $s\in[4]$, and set…

Assume that $\mathcal{L}_r^S$ is irreducible. Let $I\subset[n]$ satisfy $|I|=m$, and let $X_{[m],I}$ and $U_{[m],I}$ denote the corresponding $m\times m$ submatrices; write…

Let $S evarnothing$ be a zero pattern, let $Uinmathbb{R}^{mtimes n}$, and let $rin[m-1]$. Denote by $Z{U,r}^{S}$ the critical-point variety and by $HU^{S}$ the corresponding affine…

Let $H^k\in\{-1,1\}^{n\times n}$ be Walsh–Hadamard matrices, where $n=2^k$. A matrix is Valiant-rigid if it cannot be changed in too few entries to obtain a matrix of substantially…

Let ${\mathbf A}\in\mathbb{C}^{M\times N}$, and consider the Cadzow algorithm for rank-1 Hankel approximation. The algorithm's fixed point is compared with an optimal rank-1 Hankel…

Let $W$ be a mask matrix, and consider the bicriteria rank required for masked low-rank approximation. Exponential bicriteria-rank conjecture. The near-linear lower bound on the bi…

Let $G$ be a $3$-colorable graph on $n$ nodes. Hardness of approximate $3$-coloring. For some fixed $\gamma > 0$, there is no polynomial time algorithm that, given $G$, returns a v…

Let $\boldsymbol{A}$ be a matrix, let $\boldsymbol{A}_k$ be its best rank-$k$ approximation, and let $\boldsymbol{U}_k$ contain the top-$k$ left singular vectors of…

Let $M \in \{0,1\}^{m \times n}$ be a binary matrix. Rank-one binary matrix factorization seeks binary vectors $u \in \{0,1\}^{m}$ and $v \in \{0,1\}^{n}$ minimizing the approximat…

Let $\mathcal{L}$ be the Helmholtz operator on a domain containing elongated scatterers, and impose a boundary condition $\mathcal{B}u=g$ on $\Gamma\cap\partial\Omega^{(k)}$ withou…

Let $m=n$, let $X^*$ be the variety of matrices of rank at most $m-1$ in $\mathbb{P}^{mn-1}$, let $Z={\rm Sing}(X\cap Q)$ be the non-transversal locus of the intersection of the Se…

Nonexistence conjecture. Generic $I\times I\times 2$ arrays of rank $I+1$ do not have a best rank-$I$ approximation.

Let $A$ be a matrix of numerical rank $\rho$, let $\mathbb T_{\rho,A}$ and $\mathbb S_{\rho,A}$ denote its left and right leading singular spaces, and let $G$ be a randomized multi…

Let $A$ be a matrix with numerical rank $\rho$ and let $\rho_+$ be a reasonably small upper bound for $\rho$. Let $G$ be a random matrix used to compute an approximate basis for a…