Multi-Reference Alignment in High Dimensions: Sample Complexity and Phase Transition

Romanov, Elad; Bendory, Tamir; Ordentlich, Or

doi:10.1137/20m1354994

Cited by 21 publications

(9 citation statements)

References 33 publications

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“…Our definition of the low-noise regime σ 2 K/ log K and minimax rate in this regime are most comparable to recent results of Romanov et al (2021), which studied instead the discrete MRA model in the asymptotic limit K → ∞ and (σ 2 log K)/K → 1/α ∈ (0, ∞). This work studied a Bayesian setting where each Fourier coefficient of f has a standard Gaussian prior, and showed a transition in the Bayes risk and associated sample complexity at the sharp threshold α = 2.…”

Section: Introductionsupporting

confidence: 84%

Rates of estimation for high-dimensional multi-reference alignment

Dou¹,

Zhang²,

Zhou³

2022

Preprint

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We study the continuous multi-reference alignment model of estimating a periodic function on the circle from noisy and circularly-rotated observations. Motivated by analogous high-dimensional problems that arise in cryo-electron microscopy, we establish minimax rates for estimating generic signals that are explicit in the dimension K. In a high-noise regime with noise variance σ 2 K, the rate scales as σ 6 and has no further dependence on the dimension. This rate is achieved by a bispectrum inversion procedure, and our analyses provide new stability bounds for bispectrum inversion that may be of independent interest. In a low-noise regime where σ 2 K/ log K, the rate scales instead as Kσ 2 , and we establish this rate by a sharp analysis of the maximum likelihood estimator that marginalizes over latent rotations. A complementary lower bound that interpolates between these two regimes is obtained using Assouad's hypercube lemma.

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Section: Introductionsupporting

confidence: 84%

Rates of estimation for high-dimensional multi-reference alignment

Dou¹,

Zhang²,

Zhou³

2022

Preprint

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“…MRA is a particular instance of a GMM, with exactly k = d components corresponding to different shifted versions of the same signal. While, similarly to Romanov et al (2021), the proof of our lower bound uses the mutual information method Polyanskiy and Wu (2014), here the mutual information is upper bounded using the I-MMSE relation rather than the Fano-based argument of Romanov et al (2021). More importantly, the proof of the upper bound here requires overcoming several significant hurdles not present in the MRA model.…”

Section: Prior Artmentioning

confidence: 98%

“…Our proof program closely follows that of Romanov et al (2021), which studied the sample complexity of the multi-reference alignment (MRA) problem. MRA is a particular instance of a GMM, with exactly k = d components corresponding to different shifted versions of the same signal.…”

Section: Prior Artmentioning

confidence: 99%

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On the Role of Channel Capacity in Learning Gaussian Mixture Models

Romanov¹,

Bendory²,

Ordentlich³

2022

Preprint

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This paper studies the sample complexity of learning the k unknown centers of a balanced Gaussian mixture model (GMM) in R d with spherical covariance matrix σ 2 I. In particular, we are interested in the following question: what is the maximal noise level σ 2 , for which the sample complexity is essentially the same as when estimating the centers from labeled measurements? To that end, we restrict attention to a Bayesian formulation of the problem, where the centers are uniformly distributed on the sphere √ dS d−1 . Our main results characterize the exact noise threshold σ 2 below which the GMM learning problem, in the large system limit d, k → ∞, is as easy as learning from labeled observations, and above which it is substantially harder. The threshold occurs at log k d = 1 2 log 1 + 1 σ 2 , which is the capacity of the additive white Gaussian noise (AWGN) channel. Thinking of the set of k centers as a code, this noise threshold can be interpreted as the largest noise level for which the error probability of the code over the AWGN channel is small. Previous works on the GMM learning problem have identified the minimum distance between the centers as a key parameter in determining the statistical difficulty of learning the corresponding GMM. While our results are only proved for GMMs whose centers are uniformly distributed over the sphere, they hint that perhaps it is the decoding error probability associated with the center constellation as a channel code that determines the statistical difficulty of learning the corresponding GMM, rather than just the minimum distance.

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“…The MRA problem entails estimating an image from multiple noisy and rotated copies of itself. The computational and statistical properties of the MRA problem have been analyzed thoroughly in recent years, see [7], [6], [16], [4], [2], [15], [11], [5], [3], [19], [1].…”

Section: Introductionmentioning

confidence: 99%

An accelerated expectation-maximization for multi-reference alignment

Janco,

Bendory

2021

Preprint

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The multi-reference alignment (MRA) problem entails estimating an image from multiple noisy and rotated copies of itself. If the noise level is low, one can reconstruct the image by estimating the missing rotations, aligning the images, and averaging out the noise. While accurate rotation estimation is impossible if the noise level is high, the rotations can still be approximated, and thus can provide indispensable information. In particular, learning the approximation error can be harnessed for efficient image estimation. In this paper, we propose a new computational framework, called Synch-EM, that consists of angular synchronization followed by expectationmaximization (EM). The synchronization step results in a concentrated distribution of rotations; this distribution is learned and then incorporated into the EM as a Bayesian prior. The learned distribution also dramatically reduces the search space, and thus the computational load, of the EM iterations. We show by extensive numerical experiments that the proposed framework can significantly accelerate EM for MRA in high noise levels, occasionally by a few orders of magnitude, without degrading the reconstruction quality.

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Multi-Reference Alignment in High Dimensions: Sample Complexity and Phase Transition

Cited by 21 publications

References 33 publications

Rates of estimation for high-dimensional multi-reference alignment

Rates of estimation for high-dimensional multi-reference alignment

On the Role of Channel Capacity in Learning Gaussian Mixture Models

An accelerated expectation-maximization for multi-reference alignment

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