# 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…

## 23 Citations

### Multireference Alignment Is Easier With an Aperiodic Translation Distribution

- Computer ScienceIEEE Transactions on Information Theory
- 2019

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

- Computer Science2018 52nd Annual Conference on Information Sciences and Systems (CISS)
- 2018

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

- Computer ScienceSIAM J. Math. Data Sci.
- 2019

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

- MathematicsIEEE Transactions on Information Theory
- 2022

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

- Computer ScienceStat. Comput.
- 2021

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

- Computer ScienceMathematical Statistics and Learning
- 2020

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

- Computer ScienceIEEE Transactions on Signal Processing
- 2018

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

- Mathematics, Computer Science
- 2018

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

- Computer Science
- 2021

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

- Computer Science
- 2019

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.

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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.

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