458 research outputs found
Tightness of the maximum likelihood semidefinite relaxation for angular synchronization
Maximum likelihood estimation problems are, in general, intractable
optimization problems. As a result, it is common to approximate the maximum
likelihood estimator (MLE) using convex relaxations. In some cases, the
relaxation is tight: it recovers the true MLE. Most tightness proofs only apply
to situations where the MLE exactly recovers a planted solution (known to the
analyst). It is then sufficient to establish that the optimality conditions
hold at the planted signal. In this paper, we study an estimation problem
(angular synchronization) for which the MLE is not a simple function of the
planted solution, yet for which the convex relaxation is tight. To establish
tightness in this context, the proof is less direct because the point at which
to verify optimality conditions is not known explicitly.
Angular synchronization consists in estimating a collection of phases,
given noisy measurements of the pairwise relative phases. The MLE for angular
synchronization is the solution of a (hard) non-bipartite Grothendieck problem
over the complex numbers. We consider a stochastic model for the data: a
planted signal (that is, a ground truth set of phases) is corrupted with
non-adversarial random noise. Even though the MLE does not coincide with the
planted signal, we show that the classical semidefinite relaxation for it is
tight, with high probability. This holds even for high levels of noise.Comment: 2 figure
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
Cram\'er-Rao bounds for synchronization of rotations
Synchronization of rotations is the problem of estimating a set of rotations
R_i in SO(n), i = 1, ..., N, based on noisy measurements of relative rotations
R_i R_j^T. This fundamental problem has found many recent applications, most
importantly in structural biology. We provide a framework to study
synchronization as estimation on Riemannian manifolds for arbitrary n under a
large family of noise models. The noise models we address encompass zero-mean
isotropic noise, and we develop tools for Gaussian-like as well as heavy-tail
types of noise in particular. As a main contribution, we derive the
Cram\'er-Rao bounds of synchronization, that is, lower-bounds on the variance
of unbiased estimators. We find that these bounds are structured by the
pseudoinverse of the measurement graph Laplacian, where edge weights are
proportional to measurement quality. We leverage this to provide interpretation
in terms of random walks and visualization tools for these bounds in both the
anchored and anchor-free scenarios. Similar bounds previously established were
limited to rotations in the plane and Gaussian-like noise
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
Toward single particle reconstruction without particle picking: Breaking the detection limit
Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray
crystallography and NMR spectroscopy as a high-resolution structural method for
biological macromolecules. In a cryo-EM experiment, the microscope produces
images called micrographs. Projections of the molecule of interest are embedded
in the micrographs at unknown locations, and under unknown viewing directions.
Standard imaging techniques first locate these projections (detection) and then
reconstruct the 3-D structure from them. Unfortunately, high noise levels
hinder detection. When reliable detection is rendered impossible, the standard
techniques fail. This is a problem especially for small molecules, which can be
particularly hard to detect. In this paper, we propose a radically different
approach: we contend that the structure could, in principle, be reconstructed
directly from the micrographs, without intermediate detection. As a result,
even small molecules should be within reach for cryo-EM. To support this claim,
we setup a simplified mathematical model and demonstrate how our
autocorrelation analysis technique allows to go directly from the micrographs
to the sought signals. This involves only one pass over the micrographs, which
is desirable for large experiments. We show numerical results and discuss
challenges that lay ahead to turn this proof-of-concept into a competitive
alternative to state-of-the-art algorithms
3D ab initio modeling in cryo-EM by autocorrelation analysis
Single-Particle Reconstruction (SPR) in Cryo-Electron Microscopy (cryo-EM) is
the task of estimating the 3D structure of a molecule from a set of noisy 2D
projections, taken from unknown viewing directions. Many algorithms for SPR
start from an initial reference molecule, and alternate between refining the
estimated viewing angles given the molecule, and refining the molecule given
the viewing angles. This scheme is called iterative refinement. Reliance on an
initial, user-chosen reference introduces model bias, and poor initialization
can lead to slow convergence. Furthermore, since no ground truth is available
for an unsolved molecule, it is difficult to validate the obtained results.
This creates the need for high quality ab initio models that can be quickly
obtained from experimental data with minimal priors, and which can also be used
for validation. We propose a procedure to obtain such an ab initio model
directly from raw data using Kam's autocorrelation method. Kam's method has
been known since 1980, but it leads to an underdetermined system, with missing
orthogonal matrices. Until now, this system has been solved only for special
cases, such as highly symmetric molecules or molecules for which a homologous
structure was already available. In this paper, we show that knowledge of just
two clean projections is sufficient to guarantee a unique solution to the
system. This system is solved by an optimization-based heuristic. For the first
time, we are then able to obtain a low-resolution ab initio model of an
asymmetric molecule directly from raw data, without 2D class averaging and
without tilting. Numerical results are presented on both synthetic and
experimental data
A Serious Game Engine for Interview Simulation: Application to the development of doctor-patient communication
International audienceIn this paper we present the architecture of a conversation engine aimed to simulate an interview process between a human and a computer player. This component is a central element of many serious games where educational goal is to develop player communication skills. We demonstrate the use of our engine in AgileDoctor, a serious game project for training medical students and general practitioners to communicate with their patients, so as to improve their long-term relationship and provide a higher quality health care. Our proposed conversation engine uses a generic method to combine the game scenario and the educational objectives. The game scenario is described by an instance of a model that formalizes the general doctor-patient interview process and the skills to develop. The conversation engine is able to use this model to engage a challenging dialogue with a human player where missing skills are focused. The proposed design methodology is not bound to the health domain and is transferable to a large range of educational usage
Design of a serious game in training non-clinical skills for professionals in health care area
International audienceMost educational games and training applications for health care professionals have been developed as simulation tools dedicated to the teaching of medical knowledge in a particular area. Non-clinical skills such as communication skills or knowledge about e-Health are insufficiently focused by such tools. A serious game with consistent educational objectives offers to the learner many possibilities to acquire multiple competences in a fun and engaging learning process. This paper presents a serious game composed with extensible educational modules that concentrate on providing high-quality health care knowledge. It is designed to respect the balance between serious and fun in both educational and game elements. The proposed architecture allows the learning objective to be clearly defined and facilitate the collaborations of actors involved in the development. A prototype of the communication skills module is presented as an example of a module design
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