We study the dihedral multi-reference alignment problem of estimating the orbit of a signal from multiple noisy observations of the signal, acted on by random elements of the dihedral group. We show that if the group elements are drawn from a generic distribution, the orbit of a generic signal is uniquely determined from the second moment of the observations. This implies that the optimal estimation rate in the high noise regime is proportional to the square of the variance of the noise. This is the first result of this type for multi-reference alignment over a non-abelian group with a non-uniform distribution of group elements. Based on tools from invariant theory and algebraic geometry, we also delineate conditions for unique orbit recovery for multi-reference alignment models over finite groups (namely, when the dihedral group is replaced by a general finite group) when the group elements are drawn from a generic distribution. Finally, we design and study numerically three computational frameworks for estimating the signal based on group synchronization, expectation-maximization, and the method of moments.

The problem of recovering a pair of signals from their blind phaseless short-time Fourier transform measurements arises in several important phase retrieval applications, including ptychography and ultra-short pulse characterization. In this paper, we prove that in order to determine a pair of generic signals uniquely, up to trivial ambiguities, the number of phaseless measurements one needs to collect is, at most, five times the number of parameters required to describe the signals. This result improves significantly upon previous papers, which required the number of measurements to be quadratic in the number of parameters rather than linear. In addition, we consider the simpler problem of recovering a pair of generic signals from their blind short-time Fourier transform, when the phases are known. In this setting, which can be understood as a special case of the blind deconvolution problem, we show that the number of measurements required to determine the two signals, up to trivial ambiguities, equals exactly the number of parameters to be recovered. As a side result, we study the classical phase retrieval problem-that is, recovering a signal from its Fourier magnitudes-when some entries of the signal are known a priori. We derive a bound on the number of required measurements as a function of the size of the set of known entries. Specifically, we show that if most of the signal's entries are known, then only a few Fourier magnitudes are necessary to determine a signal uniquely.

The problem of recovering a signal from its power spectrum is called phase retrieval. This problem appears in a variety of scientific applications, such as ultra-short laser pulse characterization and diffraction imaging. However, the problem for one-dimensional signals is ill-posed as there is no one-to-one mapping between a one-dimensional signal and its power spectrum. In the field of ultra-short laser pulse characterization, it is common to overcome this ill-posedness by using a technique called Frequency-Resolved Optical Gating (FROG). In FROG, the measured data, referred to as FROG trace, is the Fourier magnitude of the product of the underlying signal with several translated versions of itself. Therefore, in order to recover a signal from its FROG trace, one needs to invert a system of phaseless quartic equations. In this paper, we explore the symmetries and uniqueness of the FROG mapping. Our main result states that a signal bandlimited to B is determined uniquely, up to symmetries, by only 3B FROG measurements.

The purpose of this letter is to prove, for real frames, that signal reconstruction from the absolute value of the frame coefficients is equivalent to solution of a sparse signal optimization problem, namely a minimum lscrp(quasi)norm over a linear constraint. This linear constraint reflects the coefficients relationship within the range of the analysis operator