21 problems

Let $N=4M-5$ and let $A\in\mathbb C^{N\times M}$ have entries drawn independently from the standard complex Gaussian distribution, where a standard complex Gaussian has the law of…

Rathmair's conjecture. The local stability and Cheeger constants of $f$ should be finite and non-zero.

Harmonic-frame optimality conjecture. If $m\geq 4$ is even, then

Harmonic-oscillator phase-retrieval conjecture. If $|u(t,x)|=|v(t,x)|$ for every $t\in\mathbb{R}$ and every $x\in\mathbb{R}^d$, then there exists $c\in\mathbb{R}$ such that…

Let $x\in\mathcal{M}_{m_2}$ be an odd-size signal and let $\tau>0$. Write $\operatorname{supp}(\widehat{m}_2)$ for the support of the Fourier transform of the second moment, and id…

Let $g(t)=\exp(t-e^t)$, and let $\gamma(x,\xi)=\exp(-a|x|-b|\xi|)$ with $a,b>0$. Suppose that $f\in L^2(\mathbb{R})$ has exponential time-frequency concentration, meaning … Let…

Let $m\geq 4$ be an even integer, let $\boldsymbol{E}_m$ denote the vector set defined in the paper, and let $\beta_{\boldsymbol{A}}$ be the condition number associated with…

Let $L^2(\mathbb{R})$ be the Hilbert space of square-integrable functions, let $\mathcal{F}$ denote the Fourier transform, and let … A tuple of measurement operators does phase ret…

Known-support recovery conjecture. If $|S-S|>|S|$, then $I_S$ has dimension $|S|$ and degree $2|D_S|$. This is the signal-recovery step for generic signals whose support is known;…

Support-recovery conjecture. If $|S-S|=|S'-S'|\geq K$, then $I_{S,S'}$ has dimension strictly less than $K$. This would imply that a generic vector supported on $S$ cannot share it…

Almost-everywhere uniqueness conjecture. For all vectors ${\bm b}\in\mathbb{R}^m$ except for a measure zero set, this program has a unique solution. The real analogue is establishe…

Let $A$ be a finite abelian group, let $V$ be the vector space of functions $A\to\mathbb K$, and let $L_S$ be the subspace of functions supported in $S\subseteq A$. The periodic au…

Let $S\subset[0,N-1]$ and let $L_S$ be the subspace of signals supported in $S$. Let $D_S$ be the subgroup of intrinsic symmetries $D$ that preserves $L_S$. Generic signal-recovery…

Let $K$ be the sparsity level and let $a_x$ denote the periodic auto-correlation of a $K$-sparse signal $x$. Measure-zero conjecture. If $K/N<1/2$, then the set of $K$-sparse signa…

Let $N$ be even, let $K$ be the sparsity, and let $x$ be a $K$-sparse generic signal whose periodic auto-correlation is $a_x$. An intrinsic symmetry is an element of the group $D$…

Let $x_0\in\mathbb{C}^n$ be the target vector, let $A\in\mathbb{C}^{m\times n}$ be the sensing matrix, and let alternating projections reconstruct $x_0$ from the phaseless measurem…

Let $\Phi$ be the measurement matrix for signals in $\mathbb{C}^M$, with $N$ intensity measurements, and write $\mathrm{AlmInj}[\Phi,\mathbb{C}^{M\times N}]$ for almost injectivity…

Let $\Phi$ be the intensity-measurement matrix for signals in $\mathbb{C}^M$, with $N$ measurements, and write $\mathrm{Inj}[\Phi;\mathbb{C}^{M\times N}]$ for the property that the…

Let $\phi_1,\ldots,\phi_{4M-5}\in\mathbb{C}^M$, and let $L_{\Phi}$ be the Hermitian matrix subspace … Equivalently, $L_{\Phi}$ is the orthogonal complement of the span of…

Let $\mathcal{F}=\{\Phi_M\}_{M=2}^\infty$ be a family of ensembles $\Phi_M=\{\varphi_{M;n}\}_{n=1}^{2M}\subseteq\mathbb{C}^M$ which yield almost injective intensity measurements an…

Let $F$ denote the $M\times M$ discrete fractional Fourier transform defined in Candan, Kutay, and Ozaktas (2000), and let $I$ be the $M\times M$ identity matrix. The columns of a…