Sample complexity of the boolean multireference alignment problem

@article{Abbe2017SampleCO,
  title={Sample complexity of the boolean multireference alignment problem},
  author={Emmanuel Abbe and Jo{\~a}o M. Pereira and Amit Singer},
  journal={2017 IEEE International Symposium on Information Theory (ISIT)},
  year={2017},
  pages={1316-1320}
}
The Boolean multireference alignment problem consists in recovering a Boolean signal from multiple shifted and noisy observations. In this paper we obtain an expression for the error exponent of the maximum A posteriori decoder. This expression is used to characterize the number of measurements needed for signal recovery in the low SNR regime, in terms of higher order autocorrelations of the signal. The characterization is explicit for various signal dimensions, such as prime and even… 

Multireference Alignment Is Easier With an Aperiodic Translation Distribution

It is shown that in the low SNR regime, in the same regime the sample complexity for any aperiodic translation distribution scales as <inline-formula> <tex-math notation="LaTeX">$\omega (1/ \mathrm {SNR}^{2})$ </tex-Math></inline- formula>.

Heterogeneous multireference alignment: A single pass approach

This paper proposes an algorithm which estimates the K signals without estimating either the shifts or the classes of the observations, and designs a smooth, non-convex optimization problem to compute a set of signals which are consistent with the estimated averaged features.

The sample complexity of multi-reference alignment

This work considers multi-reference alignment (MRA), a simple model that captures fundamental aspects of the statistical and algorithmic challenges arising in cryo-EM and related problems, and proves that it rises to a surprising $1/SNR^3 in the low SNR regime.

Dihedral Multi-Reference Alignment

It is shown 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, which implies that the optimal estimation rate in the high noise regime is proportional to the square of the variance of the noise.

Rank-one multi-reference factor analysis

It is shown that an accurate estimation of the signal from its noisy observations is possible, and a procedure is derived which is proved to consistently estimate the signal.

Optimal rates of estimation for multi-reference alignment

In this paper, we establish optimal rates of adaptive estimation of a vector in the multi-reference alignment model, a problem with important applications in fields such as signal processing, image

Bispectrum Inversion With Application to Multireference Alignment

This work considers the problem of estimating a signal from noisy circularly translated versions of itself, called multireference alignment, and proposes and analyzes a method based on estimating the signal directly, using features of the signal that are invariant under translations.

Statistical estimation in the presence of group actions

This thesis studies two statistical models for estimation in the presence of group actions, the synchronization model and the orbit recovery model, in which noisy copies of a hidden signal are observed and each of which is acted upon by a random group element.

An adaptive variational model for multireference alignment with mixed noise

An adaptive variational model is derived by combining maximum a posteriori (MAP) estimation and soft-max method which has a more impressive performance than the existing methods when one Gaussian noise is large and the other is small.

Sample complexity is a concept at the cornerstone of statistics and machine learning with far reaching implications for experimental design and data collection strate

This work considers multireference alignment (MRA), a simple model that captures fundamental aspects of the statistical and algorithmic challenges arising in cryo-EM and related problems, and proves that it rises to a surprising 1/SNR in the low SNR regime.

References

SHOWING 1-10 OF 17 REFERENCES

The sample complexity of multi-reference alignment

This work considers multi-reference alignment (MRA), a simple model that captures fundamental aspects of the statistical and algorithmic challenges arising in cryo-EM and related problems, and proves that it rises to a surprising $1/SNR^3 in the low SNR regime.

Multireference alignment using semidefinite programming

This work provides a semidefinite program (SDP) based relaxation which approximates the maximum likelihood estimator (MLE) for the multireference alignment problem and shows that if certain positivity constraints in the relaxation are dropped, its solution becomes equivalent to performing phase correlation.

Angular Synchronization by Eigenvectors and Semidefinite Programming.

  • A. Singer
  • Computer Science
    Applied and computational harmonic analysis
  • 2011

Fundamental Limits in Multi-Image Alignment

This work derives and analyzes the Cramér-Rao and Ziv-Zakai lower bounds under different statistical models for the underlying image, and shows the existence of different behavior zones depending on the difficulty level of the problem, given by the SNR conditions of the input images.

Shift- and rotation-invariant object reconstruction using the bispectrum

Recursive and least-squares fast-Fourier-transform-based algorithms for reconstructing a two-dimensional discrete Fourier transform from the bispectrum are reviewed in detail and results presented include application of the method to a sequence of infrared images.

Shift And Rotation Invariant Object Reconstruction Using The Bispectrum

Triple correlations and their Fourier transforms, called bispectra, have properties desirable for image sequence analysis. Specifically, the triple correlation of a 2-d sequence is shift -invariant,

Elements of Information Theory

The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.

Lectures in Abstract Algebra : vol. III, Theory of Fields and Galois Theory. By N. Jacobson. Pp. xi, 323. 76s. (Van Nostrand)

This is an attractively written and truly elementary introduction to the theory of groups. In the first few chapters the reader is gently guided towards the group concept and, although the pace

The Bispectrum as a Source of Phase-Sensitive Invariants for Fourier Descriptors: A Group-Theoretic Approach

  • R. Kakarala
  • Mathematics
    Journal of Mathematical Imaging and Vision
  • 2012
The main theoretical result shows that the bispectrum serves as a complete source of invariants for homogeneous spaces of compact groups, including such important domains as the sphere S2.

Orthogonal Polynomials

In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.