15 problems

Consider the given BDD signal model and compare AMP and AMP-SI using Toeplitz sensing matrices with the corresponding state-evolution prediction, as well as using independent and i…

A $\tau$-matrix is a matrix in the relevant class of matrices considered in the source, and a GKK matrix is a matrix satisfying positivity of principal minors, weak sign-symmetry,…

A GKK matrix is a matrix satisfying positivity of principal minors, weak sign-symmetry, and eigenvalue monotonicity. A matrix is stable if its spectrum lies entirely in the open ri…

Generalized Fisher–Hartwig conjecture. The determinant ratio satisfies

Upstream observability conjecture. For almost every choice of entries of $A$, if there is a directed path from node $i$ to node $j$, then the time series of node $i$ can be reconst…

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…

Consider the overlapping Hankel block low-rank completion problem, with matrix-size parameter $N$, and let $\hat{X}$ denote a candidate completed matrix. Quadratic-time completion…

Let $M\times M$ 2D mesh graphs be graphs with the corresponding vertex sets $\mathbb{V}_\mathbb{G}$, and let a Hamiltonian path be a path $\mathbb{P}$ visiting every vertex exactly…

Let $(A,\mathbb{G})$ be a GIRS-$c$ pair, let $N$ denote the relevant matrix size, and let $\mathbb{V}_\mathbb{G}$ be the vertex set of $\mathbb{G}$. A graph-induced semiseparable r…

Let $(A,\mathbb{G})$ be a GIRS-$c$ pair, and let $\mathbb{V}_\mathbb{G}$ denote the vertex set of $\mathbb{G}$. A DV representation is a representation with rank parameters $r_i$ i…

A GIRS matrix is a matrix whose off-diagonal blocks associated with a graph have rank bounded by a GIRS constant; an SSS representation is a sequentially semiseparable representati…

Conjecture on the independence from $\ell$. The dependency on $\ell$ of the right-hand side of this bound is not important; equivalently, the bound should essentially be independen…

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…