相关论文: Spikes, Roots, and Modulations: Phase Retrieval fo…
In this note we show that stable recovery of complex-valued signals $x\in\mathbb{C}^n$ up to global sign can be achieved from the magnitudes of $4n-1$ Fourier measurements when a certain "symmetrization and zero-padding" is performed before…
In coherent diffraction imaging (CDI) experiments, the intensity of the scattered wave impinging on an object is measured on an array of detectors. This signal can be interpreted as the square of the modulus of the Fourier transform of the…
A complex frame is a collection of vectors that span $\mathbb{C}^M$ and define measurements, called intensity measurements, on vectors in $\mathbb{C}^M$. In purely mathematical terms, the problem of phase retrieval is to recover a complex…
We investigate a class of moment problems, namely recovering a measure supported on the graph of a function from partial knowledge of its moments, as for instance in some problems of optimal transport or density estimation. We show that the…
This paper presents some results on a well-known problem in Algebraic Signal Sampling and in other areas of applied mathematics: reconstruction of piecewise-smooth functions from their integral measurements (like moments, Fourier…
In this paper we tackle the problem of recovering the phase of complex linear measurements when only magnitude information is available and we control the input. We are motivated by the recent development of dedicated optics-based hardware…
We study the multifractal analysis of self-similar measures arising from random homogeneous iterated function systems. Under the assumption of the uniform strong separation condition, we see that this analysis parallels that of the…
A conjecture is proposed concerning the recovery of a discrete magnitude spectrum through a nonlinear transformation involving the Newman's phase sequence. Given a discrete magnitude spectrum sampled from a continuous function, consider the…
Consider the rank-1 spiked model: $\bf{X}=\sqrt{\nu}\xi \bf{u}+ \bf{Z}$, where $\nu$ is the spike intensity, $\bf{u}\in\mathbb{S}^{k-1}$ is an unknown direction and $\xi\sim \mathcal{N}(0,1),\bf{Z}\sim \mathcal{N}(\bf{0},\bf{I})$. Motivated…
In this paper, we show that sparse signals f representable as a linear combination of a finite number N of spikes at arbitrary real locations or as a finite linear combination of B-splines of order m with arbitrary real knots can be almost…
The conjugate phase retrieval problem concerns the determination of a complex-valued function, up to a unimodular constant and conjugation, from its magnitude observations. It can also be considered as a conjugate phaseless sampling and…
Recovering probability measures from moments is a central theme in statistics and optimization. In particular, we focus on the recovery of measures from moments and pseudo-moments, which may come from solving the moment-SOS hierarchy in one…
A recent line of work shows that a deep neural network with ReLU nonlinearities arises from a finite sequence of cascaded sparse coding models, the outputs of which, except for the last element in the cascade, are sparse and unobservable.…
In this paper, we address the problem of reconstruction of support of a measure from its moments. More precisely, given a finite subset of the moments of a measure, we develop a semidefinite program for approximating the support of measure…
Suppose we wish to recover a signal x in C^n from m intensity measurements of the form |<x,z_i>|^2, i = 1, 2,..., m; that is, from data in which phase information is missing. We prove that if the vectors z_i are sampled independently and…
Recovery of support of a sparse vector from simple measurements is a widely-studied problem, considered under the frameworks of compressed sensing, 1-bit compressed sensing, and more general single index models. We consider generalizations…
Phase retrieval in dynamical sampling is a novel research direction, where an unknown signal has to be recovered from the phaseless measurements with respect to a dynamical frame, i.e. a sequence of sampling vectors constructed by the…
This paper considers the recovery of continuous signals in infinite dimensional spaces from the magnitude of their frequency samples. It proposes a sampling scheme which involves a combination of oversampling and modulations with complex…
Prony's method is a prototypical eigenvalue analysis based method for the reconstruction of a finitely supported complex measure on the unit circle from its moments up to a certain degree. In this note, we give a generalization of this…
This paper develops a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging and many other applications. Our approach combines multiple structured illuminations…