arXiv subjects

Martin Slawski

Publications and source records attributed to Martin Slawski.

At least 19 recordsLinked to original sources

arXiv2026

Linear Regression from 1-bit Quantized Data

Motivated by the prevalence of environments in which data is abundant while resources for storage and/or transmission might be scarce, we study linear regression when predictors, their squares, and responses are subject to single-bit dithered quantization. An estimator relying on plug-in estimation of the quadratic and linear terms in the quadratic program formulation of the least squares problem is proposed. We provide a non-asymptotic bound on the $\ell_2$-estimation error of this estimator and obtain its asymptotic distribution when the number of predictors is fixed, which can be used for inference and an investigation of the mean-square error efficiency relative to the ordinary least squares estimator. It is shown that for the quantization protocol under consideration, substantial improvements over the proposed estimator cannot be expected. A compression pipeline in which the underlying data is first subject to sketching and subsequently quantization can be studied within our framework as well. We also present an extension to address high-dimensional predictors. Numerical experiments with synthetic data complement our theoretical findings.

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arXiv2025

Causal Secondary Analysis of Linked Data in the Presence of Mismatch Error

The increased prevalence of observational data and the need to integrate information from multiple sources are critical challenges in contemporary data analysis. Record linkage is a widely used tool for combining datasets in the absence of unique identifiers. The presence of linkage errors such as mismatched records, however, often hampers the analysis of data sets obtained in this way. This issue is more difficult to address in secondary analysis settings, where linkage and subsequent analysis are performed separately, and analysts have limited information about linkage quality. In this paper, we investigate the estimation of average treatment effects in the conventional potential outcome-based causal inference framework under linkage uncertainty. To mitigate the bias that would be incurred with naive analyses, we propose an approach based on estimating equations that treats the unknown match status indicators as missing data. Leveraging a variant of the Expectation-Maximization algorithm, these indicators are imputed based on a corresponding two-component mixture model. The approach is amenable to asymptotic inference. Simulation studies and a case study highlight the importance of accounting for linkage uncertainty and demonstrate the effectiveness of the proposed approach.

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arXiv2025

Relaxing the Assumption of Strongly Non-Informative Linkage Error in Secondary Regression Analysis of Linked Files

Data analysis of files that are a result of linking records from multiple sources are often affected by linkage errors. Records may be linked incorrectly, or their links may be missed. In consequence, it is essential that such errors are taken into account to ensure valid post-linkage inference. Here, we propose an extension to a general framework for regression with linked covariates and responses based on a two-component mixture model, which was developed in prior work. This framework addresses the challenging case of secondary analysis in which only the linked data is available and information about the record linkage process is limited. The extension considered herein relaxes the assumption of strongly non-informative linkage in the framework according to which linkage does not depend on the covariates used in the analysis, which may be limiting in practice. The effectiveness of the proposed extension is investigated by simulations and a case study.

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arXiv2025

Lasso Penalization for High-Dimensional Beta Regression Models: Computation, Analysis, and Inference

Beta regression is commonly employed when the outcome variable is a proportion. Since its conception, the approach has been widely used in applications spanning various scientific fields. A series of extensions have been proposed over time, several of which address variable selection and penalized estimation, e.g., with an $\ell_1$-penalty (LASSO). However, a theoretical analysis of this popular approach in the context of Beta regression with high-dimensional predictors is lacking. In this paper, we aim to close this gap. A particular challenge arises from the non-convexity of the associated negative log-likelihood, which we address by resorting to a framework for analyzing stationary points in a neighborhood of the target parameter. Leveraging this framework, we derive a non-asymptotic bound on the $\ell_1$-error of such stationary points. In addition, we propose a debiasing approach to construct confidence intervals for the regression parameters. A proximal gradient algorithm is devised for optimizing the resulting penalized negative log-likelihood function. Our theoretical analysis is corroborated via simulation studies, and a real data example concerning the prediction of county-level proportions of incarceration is presented to showcase the practical utility of our methodology.

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arXiv2025

Identifiability in Unlinked Linear Regression: Some Results and Open Problems

A tacit assumption in classical linear regression problems is the full knowledge of the existing link between the covariates and responses. In Unlinked Linear Regression (ULR) this link is either partially or completely missing. While the reasons causing such missingness can be different, a common challenge in statistical inference is the potential non-identifiability of the regression parameter. In this note, we review the existing literature on identifiability when the $d \ge 2$ components of the vector of covariates are independent and identically distributed. When these components have different distributions, we show that it is not possible to prove similar theorems in the general case. Nevertheless, we prove some identifiability results, either under additional parametric assumptions for $d \ge 2$ or conditions on the fourth moments in the case $d=2$. Finally, we draw some interesting connections between the ULR and the well established field of Independent Component Analysis (ICA).

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arXiv2024

Accounting for Mismatch Error in Small Area Estimation with Linked Data

In small area estimation different data sources are integrated in order to produce reliable estimates of target parameters (e.g., a mean or a proportion) for a collection of small subsets (areas) of a finite population. Regression models such as the linear mixed effects model or M-quantile regression are often used to improve the precision of survey sample estimates by leveraging auxiliary information for which means or totals are known at the area level. In many applications, the unit-level linkage of records from different sources is probabilistic and potentially error-prone. In this paper, we present adjustments of the small area predictors that are based on either the linear mixed effects model or M-quantile regression to account for the presence of linkage error. These adjustments are developed from a two-component mixture model that hinges on the assumption of independence of the target and auxiliary variable given incorrect linkage. Estimation and inference is based on composite likelihoods and machinery revolving around the Expectation-Maximization Algorithm. For each of the two regression methods, we propose modified small area predictors and approximations for their mean squared errors. The empirical performance of the proposed approaches is studied in both design-based and model-based simulations that include comparisons to a variety of baselines.

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arXiv2023

Tensor Completion for Causal Inference with Multivariate Longitudinal Data: A Reevaluation of COVID-19 Mandates

We propose a new method that uses tensor completion to estimate causal effects with multivariate longitudinal data, data in which multiple outcomes are observed for each unit and time period. Our motivation is to estimate the number of COVID-19 fatalities prevented by government mandates such as travel restrictions, mask-wearing directives, and vaccination requirements. In addition to COVID-19 fatalities, we observe related outcomes such as the number of fatalities from other diseases and injuries. The proposed method arranges the data as a tensor with three dimensions (unit, time, and outcome) and uses tensor completion to impute the missing counterfactual outcomes. We first prove that under general conditions, combining multiple outcomes using the proposed method improves the accuracy of counterfactual imputations. We then compare the proposed method to other approaches commonly used to evaluate COVID-19 mandates. Our main finding is that other approaches overestimate the effect of masking-wearing directives and that mask-wearing directives were not an effective alternative to travel restrictions. We conclude that while the proposed method can be applied whenever multivariate longitudinal data are available, we believe it is particularly timely as governments increasingly rely on longitudinal data to choose among policies such as mandates during public health emergencies.

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arXiv2023

A General Framework for Regression with Mismatched Data Based on Mixture Modeling

Data sets obtained from linking multiple files are frequently affected by mismatch error, as a result of non-unique or noisy identifiers used during record linkage. Accounting for such mismatch error in downstream analysis performed on the linked file is critical to ensure valid statistical inference. In this paper, we present a general framework to enable valid post-linkage inference in the challenging secondary analysis setting in which only the linked file is given. The proposed framework covers a wide selection of statistical models and can flexibly incorporate additional information about the underlying record linkage process. Specifically, we propose a mixture model for pairs of linked records whose two components reflect distributions conditional on match status, i.e., correct match or mismatch. Regarding inference, we develop a method based on composite likelihood and the EM algorithm as well as an extension towards a fully Bayesian approach. Extensive simulations and several case studies involving contemporary record linkage applications corroborate the effectiveness of our framework.

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arXiv2022

Permuted and Unlinked Monotone Regression in $\mathbb{R}^d$: an approach based on mixture modeling and optimal transport

Suppose that we have a regression problem with response variable Y in $\mathbb{R}^d$ and predictor X in $\mathbb{R}^d$, for $d \geq 1$. In permuted or unlinked regression we have access to separate unordered data on X and Y, as opposed to data on (X,Y)-pairs in usual regression. So far in the literature the case $d=1$ has received attention, see e.g., the recent papers by Rigollet and Weed [Information & Inference, 8, 619--717] and Balabdaoui et al. [J. Mach. Learn. Res., 22(172), 1--60]. In this paper, we consider the general multivariate setting with $d \geq 1$. We show that the notion of cyclical monotonicity of the regression function is sufficient for identification and estimation in the permuted/unlinked regression model. We study permutation recovery in the permuted regression setting and develop a computationally efficient and easy-to-use algorithm for denoising based on the Kiefer-Wolfowitz [Ann. Math. Statist., 27, 887--906] nonparametric maximum likelihood estimator and techniques from the theory of optimal transport. We provide explicit upper bounds on the associated mean squared denoising error for Gaussian noise. As in previous work on the case $d = 1$, the permuted/unlinked setting involves slow (logarithmic) rates of convergence rooting in the underlying deconvolution problem. Numerical studies corroborate our theoretical analysis and show that the proposed approach performs at least on par with the methods in the aforementioned prior work in the case $d = 1$ while achieving substantial reductions in terms of computational complexity.

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arXiv2021

Regularization for Shuffled Data Problems via Exponential Family Priors on the Permutation Group

In the analysis of data sets consisting of (X, Y)-pairs, a tacit assumption is that each pair corresponds to the same observation unit. If, however, such pairs are obtained via record linkage of two files, this assumption can be violated as a result of mismatch error rooting, for example, in the lack of reliable identifiers in the two files. Recently, there has been a surge of interest in this setting under the term "Shuffled data" in which the underlying correct pairing of (X, Y)-pairs is represented via an unknown index permutation. Explicit modeling of the permutation tends to be associated with substantial overfitting, prompting the need for suitable methods of regularization. In this paper, we propose a flexible exponential family prior on the permutation group for this purpose that can be used to integrate various structures such as sparse and locally constrained shuffling. This prior turns out to be conjugate for canonical shuffled data problems in which the likelihood conditional on a fixed permutation can be expressed as product over the corresponding (X,Y)-pairs. Inference is based on the EM algorithm in which the intractable E-step is approximated by the Fisher-Yates algorithm. The M-step is shown to admit a significant reduction from $n^2$ to $n$ terms if the likelihood of (X,Y)-pairs has exponential family form as in the case of generalized linear models. Comparisons on synthetic and real data show that the proposed approach compares favorably to competing methods.

stat.ML↗

arXiv2020

Estimation in exponential family Regression based on linked data contaminated by mismatch error

Identification of matching records in multiple files can be a challenging and error-prone task. Linkage error can considerably affect subsequent statistical analysis based on the resulting linked file. Several recent papers have studied post-linkage linear regression analysis with the response variable in one file and the covariates in a second file from the perspective of the "Broken Sample Problem" and "Permuted Data". In this paper, we present an extension of this line of research to exponential family response given the assumption of a small to moderate number of mismatches. A method based on observation-specific offsets to account for potential mismatches and $\ell_1$-penalization is proposed, and its statistical properties are discussed. We also present sufficient conditions for the recovery of the correct correspondence between covariates and responses if the regression parameter is known. The proposed approach is compared to established baselines, namely the methods by Lahiri-Larsen and Chambers, both theoretically and empirically based on synthetic and real data. The results indicate that substantial improvements over those methods can be achieved even if only limited information about the linkage process is available.

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arXiv2020

Asynchronous Online Federated Learning for Edge Devices with Non-IID Data

Federated learning (FL) is a machine learning paradigm where a shared central model is learned across distributed edge devices while the training data remains on these devices. Federated Averaging (FedAvg) is the leading optimization method for training non-convex models in this setting with a synchronized protocol. However, the assumptions made by FedAvg are not realistic given the heterogeneity of devices. In particular, the volume and distribution of collected data vary in the training process due to different sampling rates of edge devices. The edge devices themselves also vary in their available communication bandwidth and system configurations, such as memory, processor speed, and power requirements. This leads to vastly different training times as well as model/data transfer times. Furthermore, availability issues at edge devices can lead to a lack of contribution from specific edge devices to the federated model. In this paper, we present an Asynchronous Online Federated Learning (ASO-Fed) framework, where the edge devices perform online learning with continuous streaming local data and a central server aggregates model parameters from clients. Our framework updates the central model in an asynchronous manner to tackle the challenges associated with both varying computational loads at heterogeneous edge devices and edge devices that lag behind or dropout. We perform extensive experiments on a simulated benchmark image dataset and three real-world non-IID streaming datasets. The results demonstrate the effectiveness of \model~on converging fast and maintaining good prediction performance.

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arXiv2020

The Benefits of Diversity: Permutation Recovery in Unlabeled Sensing from Multiple Measurement Vectors

In "Unlabeled Sensing", one observes a set of linear measurements of an underlying signal with incomplete or missing information about their ordering, which can be modeled in terms of an unknown permutation. Previous work on the case of a single noisy measurement vector has exposed two main challenges: 1) a high requirement concerning the \emph{signal-to-noise ratio} ($\snr$), i.e., approximately of the order of $n^{5}$, and 2) a massive computational burden in light of NP-hardness in general. In this paper, we study the case of \emph{multiple} noisy measurement vectors (MMVs) resulting from a \emph{common} permutation and investigate to what extent the number of MMVs $m$ facilitates permutation recovery by "borrowing strength". The above two challenges have at least partially been resolved within our work. First, we show that a large stable rank of the signal significantly reduces the required snr which can drop from a polynomial in $n$ for $m = 1$ to a constant for $m = Ω(\log n)$, where $m$ denotes the number of MMVs and $n$ denotes the number of measurements per MV. This bound is shown to be sharp and is associated with a phase transition phenomenon. Second, we propose a computational scheme for recovering the unknown permutation in practice. For the "oracle case" with the known signal, the maximum likelihood (ML) estimator reduces to a linear assignment problem whose global optimum can be obtained efficiently. For the case in which both the signal and permutation are unknown, the problem is reformulated as a bi-convex optimization problem with an auxiliary variable, which can be solved by the Alternating Direction Method of Multipliers (ADMM). Numerical experiments based on the proposed computational scheme confirm the tightness of our theoretical analysis.

cs.IT↗

arXiv2020

A Two-Stage Approach to Multivariate Linear Regression with Sparsely Mismatched Data

A tacit assumption in linear regression is that (response, predictor)-pairs correspond to identical observational units. A series of recent works have studied scenarios in which this assumption is violated under terms such as ``Unlabeled Sensing and ``Regression with Unknown Permutation''. In this paper, we study the setup of multiple response variables and a notion of mismatches that generalizes permutations in order to allow for missing matches as well as for one-to-many matches. A two-stage method is proposed under the assumption that most pairs are correctly matched. In the first stage, the regression parameter is estimated by handling mismatches as contaminations, and subsequently the generalized permutation is estimated by a basic variant of matching. The approach is both computationally convenient and equipped with favorable statistical guarantees. Specifically, it is shown that the conditions for permutation recovery become considerably less stringent as the number of responses $m$ per observation increase. Particularly, for $m = Ω(\log n)$, the required signal-to-noise ratio no longer depends on the sample size $n$. Numerical results on synthetic and real data are presented to support the main findings of our analysis.

stat.ML↗

arXiv2019

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

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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arXiv2018

A Note on Coding and Standardization of Categorical Variables in (Sparse) Group Lasso Regression

Categorical regressor variables are usually handled by introducing a set of indicator variables, and imposing a linear constraint to ensure identifiability in the presence of an intercept, or equivalently, using one of various coding schemes. As proposed in Yuan and Lin [J. R. Statist. Soc. B, 68 (2006), 49-67], the group lasso is a natural and computationally convenient approach to perform variable selection in settings with categorical covariates. As pointed out by Simon and Tibshirani [Stat. Sin., 22 (2011), 983-1001], "standardization" by means of block-wise orthonormalization of column submatrices each corresponding to one group of variables can substantially boost performance. In this note, we study the aspect of standardization for the special case of categorical predictors in detail. The main result is that orthonormalization is not required; column-wise scaling of the design matrix followed by re-scaling and centering of the coefficients is shown to have exactly the same effect. Similar reductions can be achieved in the case of interactions. The extension to the so-called sparse group lasso, which additionally promotes within-group sparsity, is considered as well. The importance of proper standardization is illustrated via extensive simulations.

stat.CO↗

arXiv2017

Linear Regression with Sparsely Permuted Data

In regression analysis of multivariate data, it is tacitly assumed that response and predictor variables in each observed response-predictor pair correspond to the same entity or unit. In this paper, we consider the situation of "permuted data" in which this basic correspondence has been lost. Several recent papers have considered this situation without further assumptions on the underlying permutation. In applications, the latter is often to known to have additional structure that can be leveraged. Specifically, we herein consider the common scenario of "sparsely permuted data" in which only a small fraction of the data is affected by a mismatch between response and predictors. However, an adverse effect already observed for sparsely permuted data is that the least squares estimator as well as other estimators not accounting for such partial mismatch are inconsistent. One approach studied in detail herein is to treat permuted data as outliers which motivates the use of robust regression formulations to estimate the regression parameter. The resulting estimate can subsequently be used to recover the permutation. A notable benefit of the proposed approach is its computational simplicity given the general lack of procedures for the above problem that are both statistically sound and computationally appealing.

math.ST↗

arXiv2017

On Principal Components Regression, Random Projections, and Column Subsampling

Principal Components Regression (PCR) is a traditional tool for dimension reduction in linear regression that has been both criticized and defended. One concern about PCR is that obtaining the leading principal components tends to be computationally demanding for large data sets. While random projections do not possess the optimality properties of the leading principal subspace, they are computationally appealing and hence have become increasingly popular in recent years. In this paper, we present an analysis showing that for random projections satisfying a Johnson-Lindenstrauss embedding property, the prediction error in subsequent regression is close to that of PCR, at the expense of requiring a slightly large number of random projections than principal components. Column sub-sampling constitutes an even cheaper way of randomized dimension reduction outside the class of Johnson-Lindenstrauss transforms. We provide numerical results based on synthetic and real data as well as basic theory revealing differences and commonalities in terms of statistical performance.

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