9 research outputs found
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
is bounded from above, where is the number of observations,
is the noise standard deviation, and is the so-called
\mbox{moment order cutoff}. In contrast, the maximum likelihood estimator is
shown to be consistent if diverges.Comment: 5 pages, conferenc
Heterogeneous multireference alignment: a single pass approach
Multireference alignment (MRA) is the problem of estimating a signal from
many noisy and cyclically shifted copies of itself. In this paper, we consider
an extension called heterogeneous MRA, where signals must be estimated, and
each observation comes from one of those signals, unknown to us. This is a
simplified model for the heterogeneity problem notably arising in cryo-electron
microscopy. We propose an algorithm which estimates the signals without
estimating either the shifts or the classes of the observations. It requires
only one pass over the data and is based on low-order moments that are
invariant under cyclic shifts. Given sufficiently many measurements, one can
estimate these invariant features averaged over the signals. We then design
a smooth, non-convex optimization problem to compute a set of signals which are
consistent with the estimated averaged features. We find that, in many cases,
the proposed approach estimates the set of signals accurately despite
non-convexity, and conjecture the number of signals that can be resolved as
a function of the signal length is on the order of .Comment: 6 pages, 3 figure
A Modified Cross Correlation Algorithm for Reference-free Image Alignment of Non-Circular Projections in Single-Particle Electron Microscopy
In this paper we propose a modified cross correlation method to align images
from the same class in single-particle electron microscopy of highly
non-spherical structures. In this new method, First we coarsely align
projection images, and then re-align the resulting images using the cross
correlation (CC) method. The coarse alignment is obtained by matching the
centers of mass and the principal axes of the images. The distribution of
misalignment in this coarse alignment can be quantified based on the
statistical properties of the additive background noise. As a consequence, the
search space for re-alignment in the cross correlation method can be reduced to
achieve better alignment. In order to overcome problems associated with false
peaks in the cross correlations function, we use artificially blurred images
for the early stage of the iterative cross correlation method and segment the
intermediate class average from every iteration step. These two additional
manipulations combined with the reduced search space size in the cross
correlation method yield better alignments for low signal-to-noise ratio images
than both classical cross correlation and maximum likelihood(ML) methods.Comment: 29page
Multireference Alignment is Easier with an Aperiodic Translation Distribution
In the multireference alignment model, a signal is observed by the action of
a random circular translation and the addition of Gaussian noise. The goal is
to recover the signal's orbit by accessing multiple independent observations.
Of particular interest is the sample complexity, i.e., the number of
observations/samples needed in terms of the signal-to-noise ratio (the signal
energy divided by the noise variance) in order to drive the mean-square error
(MSE) to zero. Previous work showed that if the translations are drawn from the
uniform distribution, then, in the low SNR regime, the sample complexity of the
problem scales as . In this work, using a
generalization of the Chapman--Robbins bound for orbits and expansions of the
divergence at low SNR, we show that in the same regime the sample
complexity for any aperiodic translation distribution scales as
. This rate is achieved by a simple spectral algorithm.
We propose two additional algorithms based on non-convex optimization and
expectation-maximization. We also draw a connection between the multireference
alignment problem and the spiked covariance model
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