76 problems

Let $Y=AX+E$ with $Y\in\mathbb{R}^{n\times p}$, $A\in\mathbb{R}^{m\times n}$, and $X\in\mathbb{R}^{m\times p}$, where $p\geq\max\\{n,m\\}$ and typically $m\geq n$. In the square no…

Information-theoretic impossibility conjecture. Recovery is information-theoretically impossible no matter the sample size in the sub-linear sparsification regime where $d=o(p)$.

Sharp-bound conjecture. For every $t$ satisfying $0<t<4/3$,

Optimal sampling-rate conjecture. One may conjecture that

Curvelet counting conjecture. As $\varepsilon\rightarrow 0$, this term estimates the length of the discontinuity.

Logarithmic-factor conjecture. The factor $\sqrt{\log m}$ in this lower bound is simply a proof artifact.

Tight measurement complexity conjecture. The lower bound can be extended to every

Let $A\in\mathbb{R}^{m\times n}$ have rank $m$, let $({\bm u}^*,{\bm v}^*)$ be a global minimizer of $g({\bm u},{\bm v})$ with ${\bm z}^*={\bm u}^*\circ{\bm v}^*$, and let…

Splittability conjecture. All full-spark frames are $s$-splittable with sufficiently small $\beta>0$.

Let $\kappa$ denote the target sparsity level, and consider a high dimension-based sparse recovery algorithm using the described greedy approach, with each iteration selecting mult…

Let $x\in\mathbb{R}^N$ be a signal supported on a set $S\subset [1,N]$ with respect to the standard basis, and let $|S-S|$ denote the cardinality of its difference set. Bendory et…

Let $\spr$ denote sparse phase retrieval with a $k$-sparse signal, and let $n$ be the sample size. Sparse phase retrieval computational infeasibility conjecture. Sparse phase retri…

For $\alpha=0$, let $\mathcal{B}_0$ denote the Legendre polynomial basis and, for $r>\frac12$, let $\mathcal{A}_{0,1}^r$ be the corresponding nonperiodic Wiener-type class. Writing…

Bendory et al.'s conjecture. A generic $K$-sparse vector in $\mathbb{R}^N$ can be recovered, up to unavoidable ambiguities, from its power spectrum whenever

Let $A\in\{0,1\}^{m\times n}$ be a matrix whose entries are independent and identically distributed Bernoulli $p$ random variables. The robust nullspace property conjecture. There…

Robustness conjecture. The approximation results should remain valid under perturbations of the PDE by possibly nonsymmetric or indefinite lower-order terms with bounded coefficien…

Extension conjecture. The theoretical results should extend to LU factorizations of elliptic operators whose leading order term is self-adjoint and positive, without requiring the…

Plonka et al.'s four-line conjecture. Recovery of the parameters is always possible from $O(N)$ Fourier coefficients that do not depend on the data and are supported on four predet…

Conjecture that $C(u^m)=1$. For every such $u^m$,

Let MSRA denote Multi-sequence Spreading Random Access, and consider the associated multiple-measurement-vector (MMV) problem for activity detection. A well-conditioned MMV problem…

For $q obreaktext{ satisfying } obreak 1<q eq?$—no, the relevant range is $qrightin(1, obreakinfty]$—consider minimization of the ratio $lVert zrVert1/lVert zrVertq$ subject to th…

Higher-order norm recovery conjecture. The dictionary can be recovered via maximizing any $ℓ_p$-norm with $p>2$.

Let $s$ denote the sparsity level, let $p$ be in the range specified by Theorem, and let $w^{\text{BP}}$ be the model-error vector produced by basis pursuit. Descent-speed insensit…

The DPCP problem seeks a vector orthogonal to as many data points as possible, typically by minimizing a nonsmooth or smooth objective over the unit sphere. Benign global geometry…

Let $K_a$ be the number of active users, $K_\text{tot}$ the number of potentially active users, and $L$ the signal dimension in the covariance-based activity-detection problem. A l…