Denoising linear models with permuted data

Pananjady, Ashwin; Wainwright, Martin J.; Courtade, Thomas A.

doi:10.1109/isit.2017.8006567

Cited by 55 publications

(48 citation statements)

References 20 publications

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“…Specifically, we assume that Y i relates to X = [X 1 , ..., X n ] T only through (Π i· XW) T , where X i and Y i lie on the surface of a p-dimensional unit sphere denoted by S p−1 , W ∈ R p×p is an orthogonal translation matrix satisfying WW T = I p with I p an identity matrix, and Π = [Π T 1· , ..., Π T n· ] T ∈ R n×n is a mapping matrix that corrects the potential mismatch. There is a growing literature on the shuffled linear regression problem of Y i = (Π i· XW) T + U i when Π is a permutation matrix encoding only one-to-one correspondence between X and Y and no orthogonality constraint is imposed on W (Pananjady et al 2017a,b, Slawski & Ben-David 2017, Abid et al 2017, Hsu et al 2017, Unnikrishnan et al 2018. It has been shown that the least squares estimator of W is generally inconsistent without any additional constraints imposed on Π (Pananjady et al 2017a,b, Slawski & Ben-David 2017.…”

Section: Spherical Regression With Mismatched Datamentioning

confidence: 99%

Spherical Regression Under Mismatch Corruption With Application to Automated Knowledge Translation

Shi

Cai

2020

Journal of the American Statistical Association

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Motivated by a series of applications in data integration, language translation, bioinformatics, and computer vision, we consider spherical regression with two sets of unit-length vectors when the data are corrupted by a small fraction of mismatch in the response-predictor pairs. We propose a three-step algorithm in which we initialize the parameters by solving an orthogonal Procrustes problem to estimate a translation matrix W ignoring the mismatch. We then estimate a mapping matrix aiming to correct the mismatch using hard-thresholding to induce sparsity, while incorporating potential group information. We eventually obtain a refined estimate for W by removing the estimated mismatched pairs. We derive the error bound for the initial estimate of W in both fixed and high-dimensional setting. We demonstrate that the refined estimate of W achieves an error rate that is as good as if no mismatch is present. We show that our mapping recovery method not only correctly distinguishes one-to-one and one-to-many correspondences, but also consistently identifies the matched pairs and estimates the weight vector for combined correspondence. We examine the finite sample performance of the proposed method via extensive simulation studies, and with application to the unsupervised translation of medical codes using electronic health records data.

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Section: Spherical Regression With Mismatched Datamentioning

confidence: 99%

Spherical Regression Under Mismatch Corruption With Application to Automated Knowledge Translation

Shi

Cai

2020

Journal of the American Statistical Association

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“…If we assume perfect particle picking, then the cryo-EM model (II.2) is a special case of (VI.1) when G is the group of 3-D rotations SO(3) and T is the linear operator that takes the rotated structure, integrates along the z-axis (tomographic projection), convolves with the PSF, and samples it on a Cartesian grid. The MRA model formulates many additional applications, including structure from motion in computer vision [5], localization and mapping in robotics [69], study of specimen populations [51], optical and acoustical trapping [32], and denoising of permuted data [61].…”

Section: A Multi-reference Alignmentmentioning

confidence: 99%

Single-Particle Cryo-Electron Microscopy: Mathematical Theory, Computational Challenges, and Opportunities

Bendory

Bartesaghi

Singer

2020

IEEE Signal Process. Mag.

131

129

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In recent years, an abundance of new molecular structures have been elucidated using cryo-electron microscopy (cryo-EM), largely due to advances in hardware technology and data processing techniques. Owing to these new exciting developments, cryo-EM was selected by Nature Methods as Method of the Year 2015, and the Nobel Prize in Chemistry 2017 was awarded to three pioneers in the field.The main goal of this article is to introduce the challenging and exciting computational tasks involved in reconstructing 3-D molecular structures by cryo-EM. Determining molecular structures requires a wide range of computational tools in a variety of fields, including signal processing, estimation and detection theory, high-dimensional statistics, convex and non-convex optimization, spectral algorithms, dimensionality reduction, and machine learning. The tools from these fields must be adapted to work under exceptionally challenging conditions, including extreme noise levels, the presence of missing data, and massively large datasets as large as several Terabytes.In addition, we present two statistical models: multi-reference alignment and multi-target detection, that abstract away much of the intricacies of cryo-EM, while retaining some of its essential features. Based on these abstractions, we discuss some recent intriguing results in the mathematical theory of cryo-EM, and delineate relations with group theory, invariant theory, and information theory.

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“…Nevertheless, it has only been until very recently that the problem of shuffled linear regression has been considered in its full generality. In fact, the main achievements so far have been concentrating on a theoretical understanding of the conditions that allow unique recovery of ξ * or Π * ; see [8], [28], [9], [10], [29], [11], [31], [32], [33], [18], [34], [35], [36], [19]. Letting A be drawn at random from any continuous probability distribution, [10] proved that any such ξ * can be uniquely recovered with probability 1 as long as 2 m ≥ 2n.…”

Section: Prior Artmentioning

confidence: 99%

An Algebraic-Geometric Approach for Linear Regression Without Correspondences

Tsakiris

Peng

Conca

et al. 2020

IEEE Trans. Inform. Theory

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Linear regression without correspondences is the problem of performing a linear regression fit to a dataset for which the correspondences between the independent samples and the observations are unknown. Such a problem naturally arises in diverse domains such as computer vision, data mining, communications and biology. In its simplest form, it is tantamount to solving a linear system of equations, for which the entries of the right hand side vector have been permuted. This type of data corruption renders the linear regression task considerably harder, even in the absence of other corruptions, such as noise, outliers or missing entries. Existing methods are either applicable only to noiseless data or they are very sensitive to initialization or they work only for partially shuffled data. In this paper we address these issues via an algebraic geometric approach, which uses symmetric polynomials to extract permutation-invariant constraints that the parameters ξ * ∈ R n of the linear regression model must satisfy. This naturally leads to a polynomial system of n equations in n unknowns, which contains ξ * in its root locus. Using the machinery of algebraic geometry we prove that as long as the independent samples are generic, this polynomial system is always consistent with at most n! complex roots, regardless of any type of corruption inflicted on the observations. The algorithmic implication of this fact is that one can always solve this polynomial system and use its most suitable root as initialization to the Expectation Maximization algorithm. To the best of our knowledge, the resulting method is the first working solution for small values of n able to handle thousands of fully shuffled noisy observations in milliseconds.

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Denoising linear models with permuted data

Cited by 55 publications

References 20 publications

Spherical Regression Under Mismatch Corruption With Application to Automated Knowledge Translation

Spherical Regression Under Mismatch Corruption With Application to Automated Knowledge Translation

Single-Particle Cryo-Electron Microscopy: Mathematical Theory, Computational Challenges, and Opportunities

An Algebraic-Geometric Approach for Linear Regression Without Correspondences

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