Determinantal conditions for homomorphic sensing

Tsakiris, Manolis C.

doi:10.48550/arxiv.1812.07966

Cited by 12 publications

(28 citation statements)

References 14 publications

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“…The works [2,44] discuss alternating minimization as well as the use of the Expectation-Maximization (EM) Algorithm (combined with MCMC sampling) in which Π * constitutes missing data. The recent paper [38] discusses a branch-and-bound scheme with promising empirical performance on small data sets; the theoretical properties of the approaches [2,38,44] remain to be investigated.…”

Section: Introductionmentioning

confidence: 99%

A Pseudo-Likelihood Approach to Linear Regression with Partially Shuffled Data

Slawski¹,

Diao²,

Ben-David³

2019

Preprint

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Recently, there has been significant interest in linear regression in the situation where predictors and responses are not observed in matching pairs corresponding to the same statistical unit as a consequence of separate data collection and uncertainty in data integration. Mismatched pairs can considerably impact the model fit and disrupt the estimation of regression parameters. In this paper, we present a method to adjust for such mismatches under "partial shuffling" in which a sufficiently large fraction of (predictors, response)-pairs are observed in their correct correspondence. The proposed approach is based on a pseudo-likelihood in which each term takes the form of a two-component mixture density. Expectation-Maximization schemes are proposed for optimization, which (i) scale favorably in the number of samples, and (ii) achieve excellent statistical performance relative to an oracle that has access to the correct pairings as certified by simulations and case studies. In particular, the proposed approach can tolerate considerably larger fraction of mismatches than existing approaches, and enables estimation of the noise level as well as the fraction of mismatches. Inference for the resulting estimator (standard errors, confidence intervals) can be based on established theory for composite likelihood estimation. Along the way, we also propose a statistical test for the presence of mismatches and establish its consistency under suitable conditions.

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Section: Introductionmentioning

confidence: 99%

A Pseudo-Likelihood Approach to Linear Regression with Partially Shuffled Data

Slawski¹,

Diao²,

Ben-David³

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…The oversampling factor of 2 was shown to be sufficient as well in the more challenging case of unlabeled sensing where only a subvector of y is known [3]. These results were then generalized to arbitrary linear transformations beyond permutations and down-samplings by [4] and [5], see also [6]. Bringing back the noise ǫ into the picture [7] obtained SNR conditions under which recovery of Π * is possible from the maximal likelihood estimator…”

Section: Introductionmentioning

confidence: 93%

Linear Regression without Correspondences via Concave Minimization

Peng,

Tsakiris

2020

Preprint

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Linear regression without correspondences concerns the recovery of a signal in the linear regression setting, where the correspondences between the observations and the linear functionals are unknown. The associated maximum likelihood function is NP-hard to compute when the signal has dimension larger than one. To optimize this objective function we reformulate it as a concave minimization problem, which we solve via branch-and-bound. This is supported by a computable search space to branch, an effective lower bounding scheme via convex envelope minimization and a refined upper bound, all naturally arising from the concave minimization reformulation. The resulting algorithm outperforms state-of-the-art methods for fully shuffled data and remains tractable for up to 8-dimensional signals, an untouched regime in prior work.

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“…In [12], the authors characterized a necessary condition on the dimension of the observation vector for uniquely recovering the original data in the noiseless case. A generalized framework of unlabeled sensing was presented in [15]- [17]. The estimation of a sorted vector based on noisy observations was proposed in [18], where the MMSE estimator on sorted data was characterized as a linear combination of estimators on the unsorted data.…”

Section: A Related Workmentioning

confidence: 99%

Recovering Data Permutations From Noisy Observations: The Linear Regime

Jeong

Dytso

Cardone

et al. 2020

IEEE J. Sel. Areas Inf. Theory

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This paper considers the problem of recovering the permutation of an n-dimensional random vector X observed in Gaussian noise. First, a general expression for the probability of error is derived when a linear decoder (i.e., linear estimator followed by a sorting operation) is used. The derived expression holds with minimal assumptions on the distribution of X and when the noise has memory. Second, for the case of isotropic noise (i.e., noise with a diagonal scalar covariance matrix), the rates of convergence of the probability of error are characterized in the high and low noise regimes. In the low noise regime, for every dimension n, the probability of error is shown to behave proportionally to σ, where σ is the noise standard deviation. Moreover, the slope is computed exactly for several distributions and it is shown to behave quadratically in n. In the high noise regime, for every dimension n, the probability of correctness is shown to behave as 1/σ, and the exact expression for the rate of convergence is also provided.

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Determinantal conditions for homomorphic sensing

Cited by 12 publications

References 14 publications

A Pseudo-Likelihood Approach to Linear Regression with Partially Shuffled Data

A Pseudo-Likelihood Approach to Linear Regression with Partially Shuffled Data

Linear Regression without Correspondences via Concave Minimization

Recovering Data Permutations From Noisy Observations: The Linear Regime

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