Bispectrum Inversion with Application to Multireference Alignment

We consider the problem of estimating a signal from noisy circularly-translated versions of itself, called multireference alignment (MRA). One natural approach to MRA could be to estimate the shifts of the observations first, and infer the signal by aligning and averaging the data. In contrast, we consider a method based on estimating the signal directly, using features of the signal that are invariant under translations. Specifically, we estimate the power spectrum and the bispectrum of the signal from the observations. Under mild assumptions, these invariant features contain enough information to infer the signal. In particular, the bispectrum can be used to estimate the Fourier phases. To this end, we propose and analyze a few algorithms. Our main methods consist of non-convex optimization over the smooth manifold of phases. Empirically, in the absence of noise, these non-convex algorithms appear to converge to the target signal with random initialization. The algorithms are also robust to noise. We then suggest three additional methods. These methods are based on frequency marching, semidefinite relaxation and integer programming. The first two methods provably recover the phases exactly in the absence of noise. In the high noise level regime, the invariant features approach for MRA results in stable estimation if the number of measurements scales like the cube of the noise variance, which is the information-theoretic rate. Additionally, it requires only one pass over the data which is important at low signal-to-noise ratio when the number of observations must be large

Идентификация цифровых последовательностей с интегральным биспектральным преобразованием

The paper presents algorithms for identifying signals and determining the threshold of false identification based on the integral bispectrum conversion and the Euclidean distance calculation. The analytical calculation of statistical characteristics in the form of the average probability of identification error, identification error of a known signal and a new signal is carried out. The advantages of bispectral signal conversion over spectral power density in identifying signals with their strong mutual correlation (from 0.5 to 0.9) are shown. Mathematical and computer modeling of the signal identification procedure and the optimal threshold for determining a new signal is performed. The simulation results confirmed the coincidence with the theoretical values of the probability of signal identification error. Manokhin A. E. Identification of Digital Signals with Integrated Bispectral Conversion. Ural Radio Engineering Journal. 2023;7(1):56–71. (In Russ.) DOI 10.15826/urej.2023.7.1.004.В работе представлены алгоритмы идентификации сигналов и определения порога ложной идентификации на основе формирования интегрального биспектра и вычисления евклидового расстояния. Проведен аналитический расчет статистических характеристик в виде средней вероятности ошибки идентификации, ошибки идентификации известного сигнала и нового сигнала. Показаны преимущества биспектрального преобразования сигнала перед спектральной плотностью мощности в идентификации сигналов при их сильной взаимной корреляции (от 0,5 до 0,9). Выполнено математическое и компьютерное моделирование процедуры идентификации сигналов и формирования оптимального порога, позволяющего определить новый сигнал. Результаты моделирования подтвердили совпадение с теоретическими значениями вероятности ошибки идентификации сигнала. Манохин А. Е. Идентификация цифровых последовательностей с интегральным биспектральным преобразованием. Ural Radio Engineering Journal. 2023;7(1):56–71. DOI 10.15826/urej.2023.7.1.004

Estimation in the group action channel

We analyze the problem of estimating a signal from multiple measurements on a \mbox{group action channel} that linearly transforms a signal by a random group action followed by a fixed projection and additive Gaussian noise. This channel is motivated by applications such as multi-reference alignment and cryo-electron microscopy. We focus on the large noise regime prevalent in these applications. We give a lower bound on the mean square error (MSE) of any asymptotically unbiased estimator of the signal's orbit in terms of the signal's moment tensors, which implies that the MSE is bounded away from 0 when $N/\sigma^{2d}$ is bounded from above, where $N$ is the number of observations, $\sigma$ is the noise standard deviation, and $d$ is the so-called \mbox{moment order cutoff}. In contrast, the maximum likelihood estimator is shown to be consistent if $N /\sigma^{2d}$ diverges.Comment: 5 pages, conferenc

Identification of Digital Signals with Integrated Bispectral Conversion

Поступила: 31.10.2022. Принята в печать: 07.03.2023.Received: 31.10.2022. Accepted: 07.03.2023.В работе представлены алгоритмы идентификации сигналов и определения порога ложной идентификации на основе формирования интегрального биспектра и вычисления евклидового расстояния. Проведен аналитический расчет статистических характеристик в виде средней вероятности ошибки идентификации, ошибки идентификации известного сигнала и нового сигнала. Показаны преимущества биспектрального преобразования сигнала перед спектральной плотностью мощности в идентификации сигналов при их сильной взаимной корреляции (от 0,5 до 0,9). Выполнено математическое и компьютерное моделирование процедуры идентификации сигналов и формирования оптимального порога, позволяющего определить новый сигнал. Результаты моделирования подтвердили совпадение с теоретическими значениями вероятности ошибки идентификации сигнала.The paper presents algorithms for identifying signals and determining the threshold of false identification based on the integral bispectrum conversion and the Euclidean distance calculation. The analytical calculation of statistical characteristics in the form of the average probability of identification error, identification error of a known signal and a new signal is carried out. The advantages of bispectral signal conversion over spectral power density in identifying signals with their strong mutual correlation (from 0.5 to 0.9) are shown. Mathematical and computer modeling of the signal identification procedure and the optimal threshold for determining a new signal is performed. The simulation results confirmed the coincidence with the theoretical values of the probability of signal identification error

Image recovery from rotational and translational invariants

We introduce a framework for recovering an image from its rotationally and translationally invariant features based on autocorrelation analysis. This work is an instance of the multi-target detection statistical model, which is mainly used to study the mathematical and computational properties of single-particle reconstruction using cryo-electron microscopy (cryo-EM) at low signal-to-noise ratios. We demonstrate with synthetic numerical experiments that an image can be reconstructed from rotationally and translationally invariant features and show that the reconstruction is robust to noise. These results constitute an important step towards the goal of structure determination of small biomolecules using cryo-EM.Comment: 5 pages, 3 figure