7 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
Rotationally Invariant Image Representation for Viewing Direction Classification in Cryo-EM
We introduce a new rotationally invariant viewing angle classification method
for identifying, among a large number of Cryo-EM projection images, similar
views without prior knowledge of the molecule. Our rotationally invariant
features are based on the bispectrum. Each image is denoised and compressed
using steerable principal component analysis (PCA) such that rotating an image
is equivalent to phase shifting the expansion coefficients. Thus we are able to
extend the theory of bispectrum of 1D periodic signals to 2D images. The
randomized PCA algorithm is then used to efficiently reduce the dimensionality
of the bispectrum coefficients, enabling fast computation of the similarity
between any pair of images. The nearest neighbors provide an initial
classification of similar viewing angles. In this way, rotational alignment is
only performed for images with their nearest neighbors. The initial nearest
neighbor classification and alignment are further improved by a new
classification method called vector diffusion maps. Our pipeline for viewing
angle classification and alignment is experimentally shown to be faster and
more accurate than reference-free alignment with rotationally invariant K-means
clustering, MSA/MRA 2D classification, and their modern approximations
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
Multi-Reference Alignment for sparse signals, Uniform Uncertainty Principles and the Beltway Problem
Motivated by cutting-edge applications like cryo-electron microscopy
(cryo-EM), the Multi-Reference Alignment (MRA) model entails the learning of an
unknown signal from repeated measurements of its images under the latent action
of a group of isometries and additive noise of magnitude . Despite
significant interest, a clear picture for understanding rates of estimation in
this model has emerged only recently, particularly in the high-noise regime
that is highly relevant in applications. Recent investigations
have revealed a remarkable asymptotic sample complexity of order for
certain signals whose Fourier transforms have full support, in stark contrast
to the traditional that arise in regular models. Often prohibitively
large in practice, these results have prompted the investigation of variations
around the MRA model where better sample complexity may be achieved. In this
paper, we show that \emph{sparse} signals exhibit an intermediate
sample complexity even in the classical MRA model. Our results explore and
exploit connections of the MRA estimation problem with two classical topics in
applied mathematics: the \textit{beltway problem} from combinatorial
optimization, and \textit{uniform uncertainty principles} from harmonic
analysis