94 problems

Let $\mathcal F_{\bm A}$ be the phase-retrieval map associated with a matrix $\bm A\in\mathbb C^{M\times N}$, and let $N_{\min}(\mathbb C^M)$ denote the smallest $N$ for which some…

Generic sparse phase retrieval conjecture. The signal $x$ is uniquely determined, up to unavoidable ambiguities, as long as

Let $M$ be a dimension, and let $U_1$, $U_2$, and $U_3$ be unitary matrices. In phase retrieval, an intensity-measurement operator is formed as $\Phi=[U_1~U_2~U_3]$. Wright's conje…

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…

Random-initialization convergence conjecture. When $m\geq Cn$, for $C>0$ large enough, alternating projections converge to the true solution with probability at least $1-\epsilon$.

Bueckner's conjecture. If $u$ is a Hermite function, then every ambiguity partner of $u$ is trivial.

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…

Let $N,M$ be integers, let $\mathbb T$ be the unit circle in $\mathbb C$, and let $\Phi_A:\mathbb C^M(\operatorname{mod}\mathbb T)\to\mathbb R_{[?]0}^N$ be the phase retrieval map…

Let $d$ be the dimension, let $m$ be a positive integer, and let $g\in\mathbb{C}^d$ be a window satisfying the hypotheses of the strong two-window phase retrieval theorem. The curr…

Sharp-edge diffusion-field conjecture. These equations might have a broader domain of validity if $D(x,y)$ and $\widetilde{\mathbf{D}}(x,y)$ are augmented by diffusion fields induc…

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

Let $\boldsymbol{\theta}^\ast$ be the planted parameter, let $\mathcal{L}_n$ be the loss associated with the Wirtinger-flow objective … and consider the proportional regime $n=Cd$.…

Let $m$ be the number of Fourier measurements with coded diffraction patterns, let ${\bm w}\in{\mathbb R}^m$ be an additive measurement-noise vector, and let $\|{\bm w}\|_2$ denote…

Overdetermination-ratio conjecture. Regardless of the structure of the object itself, the intrinsic difficulty of phase retrieval depends on $\Omega$. The ratio measures the number…

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 $f\rho L^2(\mathbb{R}^d)$, and let $\widehat{f}$ denote its Fourier transform. A third unitary operator is a unitary map $U:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$. Wright's co…

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 $\vb{C}$ be the covariance matrix governing the biased spatial directions, let $q_c$ be the first value of the measurement ratio $q$ at which the spectral overlap $\rho(q)$ van…

Let $\tau$ denote the iteration number and consider the non-normalized hologram produced by the proposed update scheme. Non-normalized hologram convergence conjecture. The converge…

Let $A_n[\tau] \exp(j\phi_n[\tau])$ denote an element of the hologram at iteration $\tau$, and consider the update rule described in the paper. Small-iteration convergence conjectu…

Let $G$ be a group and let $\pi = m_1\pi_1\oplus \cdots \oplus m_d\pi_d$ be a unitary representation of $G$, where $\pi_i$ and $\pi_j$ are inequivalent irreducible representations…

The robust outlier bound condition (ROBC) yields a threshold $s^*$, arising from the analysis of the product of two normal random variables. Tight-threshold conjecture. The conditi…