arXiv subjects

Matthew Reimherr

Publications and source records attributed to Matthew Reimherr.

At least 37 records · Page 2Linked to original sources

arXiv2022

Formal Privacy for Partially Private Data

Differential privacy (DP) quantifies privacy loss by analyzing noise injected into output statistics. For non-trivial statistics, this noise is necessary to ensure finite privacy loss. However, data curators frequently release collections of statistics where some use DP mechanisms and others are released as-is, i.e., without additional randomized noise. Consequently, DP alone cannot characterize the privacy loss attributable to the entire collection of releases. In this paper, we present a privacy formalism, $(ε, \{ Θ_z\}_{z \in \mathcal{Z}})$-Pufferfish ($ε$-TP for short when $\{ Θ_z\}_{z \in \mathcal{Z}}$ is implied), a collection of Pufferfish mechanisms indexed by realizations of a random variable $Z$ representing public information not protected with DP noise. First, we prove that this definition has similar properties to DP. Next, we introduce mechanisms for releasing partially private data (PPD) satisfying $ε$-TP and prove their desirable properties. We provide algorithms for sampling from the posterior of a parameter given PPD. We then compare this inference approach to the alternative where noisy statistics are deterministically combined with Z. We derive mild conditions under which using our algorithms offers both theoretical and computational improvements over this more common approach. Finally, we demonstrate all the effects above on a case study on COVID-19 data.

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arXiv2022

Fast and Fair Simultaneous Confidence Bands for Functional Parameters

Quantifying uncertainty using confidence regions is a central goal of statistical inference. Despite this, methodologies for confidence bands in Functional Data Analysis are still underdeveloped compared to estimation and hypothesis testing. In this work, we present a new methodology for constructing simultaneous confidence bands for functional parameter estimates. Our bands possess a number of positive qualities: (1) they are not based on resampling and thus are fast to compute, (2) they are constructed under the fairness constraint of balanced false positive rates across partitions of the bands' domain which facilitates the typical global, but also novel local interpretations, and (3) they do not require an estimate of the full covariance function and thus can be used in the case of fragmentary functional data. Simulations show the excellent finite-sample behavior of our bands in comparison to existing alternatives. The practical use of our bands is demonstrated in two case studies on sports biomechanics and fragmentary growth curves.

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arXiv2022

Shape And Structure Preserving Differential Privacy

It is common for data structures such as images and shapes of 2D objects to be represented as points on a manifold. The utility of a mechanism to produce sanitized differentially private estimates from such data is intimately linked to how compatible it is with the underlying structure and geometry of the space. In particular, as recently shown, utility of the Laplace mechanism on a positively curved manifold, such as Kendall's 2D shape space, is significantly influences by the curvature. Focusing on the problem of sanitizing the Fréchet mean of a sample of points on a manifold, we exploit the characterisation of the mean as the minimizer of an objective function comprised of the sum of squared distances and develop a K-norm gradient mechanism on Riemannian manifolds that favors values that produce gradients close to the the zero of the objective function. For the case of positively curved manifolds, we describe how using the gradient of the squared distance function offers better control over sensitivity than the Laplace mechanism, and demonstrate this numerically on a dataset of shapes of corpus callosa. Further illustrations of the mechanism's utility on a sphere and the manifold of symmetric positive definite matrices are also presented.

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arXiv2022

Exact Privacy Guarantees for Markov Chain Implementations of the Exponential Mechanism with Artificial Atoms

Implementations of the exponential mechanism in differential privacy often require sampling from intractable distributions. When approximate procedures like Markov chain Monte Carlo (MCMC) are used, the end result incurs costs to both privacy and accuracy. Existing work has examined these effects asymptotically, but implementable finite sample results are needed in practice so that users can specify privacy budgets in advance and implement samplers with exact privacy guarantees. In this paper, we use tools from ergodic theory and perfect simulation to design exact finite runtime sampling algorithms for the exponential mechanism by introducing an intermediate modified target distribution using artificial atoms. We propose an additional modification of this sampling algorithm that maintains its $ε$-DP guarantee and has improved runtime at the cost of some utility. We then compare these methods in scenarios where we can explicitly calculate a $δ$ cost (as in $(ε, δ)$-DP) incurred when using standard MCMC techniques. Much as there is a well known trade-off between privacy and utility, we demonstrate that there is also a trade-off between privacy guarantees and runtime.

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arXiv2021

Differential Privacy Over Riemannian Manifolds

In this work we consider the problem of releasing a differentially private statistical summary that resides on a Riemannian manifold. We present an extension of the Laplace or K-norm mechanism that utilizes intrinsic distances and volumes on the manifold. We also consider in detail the specific case where the summary is the Fréchet mean of data residing on a manifold. We demonstrate that our mechanism is rate optimal and depends only on the dimension of the manifold, not on the dimension of any ambient space, while also showing how ignoring the manifold structure can decrease the utility of the sanitized summary. We illustrate our framework in two examples of particular interest in statistics: the space of symmetric positive definite matrices, which is used for covariance matrices, and the sphere, which can be used as a space for modeling discrete distributions.

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arXiv2021

KNG: The K-Norm Gradient Mechanism

This paper presents a new mechanism for producing sanitized statistical summaries that achieve \emph{differential privacy}, called the \emph{K-Norm Gradient} Mechanism, or KNG. This new approach maintains the strong flexibility of the exponential mechanism, while achieving the powerful utility performance of objective perturbation. KNG starts with an inherent objective function (often an empirical risk), and promotes summaries that are close to minimizing the objective by weighting according to how far the gradient of the objective function is from zero. Working with the gradient instead of the original objective function allows for additional flexibility as one can penalize using different norms. We show that, unlike the exponential mechanism, the noise added by KNG is asymptotically negligible compared to the statistical error for many problems. In addition to theoretical guarantees on privacy and utility, we confirm the utility of KNG empirically in the settings of linear and quantile regression through simulations.

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arXiv2021

Elliptical Perturbations for Differential Privacy

We study elliptical distributions in locally convex vector spaces, and determine conditions when they can or cannot be used to satisfy differential privacy (DP). A requisite condition for a sanitized statistical summary to satisfy DP is that the corresponding privacy mechanism must induce equivalent measures for all possible input databases. We show that elliptical distributions with the same dispersion operator, $C$, are equivalent if the difference of their means lies in the Cameron-Martin space of $C$. In the case of releasing finite-dimensional projections using elliptical perturbations, we show that the privacy parameter $\ep$ can be computed in terms of a one-dimensional maximization problem. We apply this result to consider multivariate Laplace, $t$, Gaussian, and $K$-norm noise. Surprisingly, we show that the multivariate Laplace noise does not achieve $\ep$-DP in any dimension greater than one. Finally, we show that when the dimension of the space is infinite, no elliptical distribution can be used to give $\ep$-DP; only $(ε,δ)$-DP is possible.

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arXiv2021

Prior sample size extensions for assessing prior impact and prior--likelihood discordance

This paper outlines a framework for quantifying the prior's contribution to posterior inference in the presence of prior-likelihood discordance, a broader concept than the usual notion of prior-likelihood conflict. We achieve this dual purpose by extending the classic notion of \textit{prior sample size}, $M$, in three directions: (I) estimating $M$ beyond conjugate families; (II) formulating $M$ as a relative notion, i.e., as a function of the likelihood sample size $k, M(k),$ which also leads naturally to a graphical diagnosis; and (III) permitting negative $M$, as a measure of prior-likelihood conflict, i.e., harmful discordance. Our asymptotic regime permits the prior sample size to grow with the likelihood data size, hence making asymptotic arguments meaningful for investigating the impact of the prior relative to that of likelihood. It leads to a simple asymptotic formula for quantifying the impact of a proper prior that only involves computing a centrality and a spread measure of the prior and the posterior. We use simulated and real data to illustrate the potential of the proposed framework, including quantifying how weak is a "weakly informative" prior adopted in a study of lupus nephritis. Whereas we take a pragmatic perspective in assessing the impact of a prior on a given inference problem under a specific evaluative metric, we also touch upon conceptual and theoretical issues such as using improper priors and permitting priors with asymptotically non-vanishing influence.

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arXiv2020

Modern Multiple Imputation with Functional Data

This work considers the problem of fitting functional models with sparsely and irregularly sampled functional data. It overcomes the limitations of the state-of-the-art methods, which face major challenges in the fitting of more complex non-linear models. Currently, many of these models cannot be consistently estimated unless the number of observed points per curve grows sufficiently quickly with the sample size, whereas, we show numerically that a modified approach with more modern multiple imputation methods can produce better estimates in general. We also propose a new imputation approach that combines the ideas of {\it MissForest} with {\it Local Linear Forest} and compare their performance with {\it PACE} and several other multivariate multiple imputation methods. This work is motivated by a longitudinal study on smoking cessation, in which the Electronic Health Records (EHR) from Penn State PaTH to Health allow for the collection of a great deal of data, with highly variable sampling. To illustrate our approach, we explore the relation between relapse and diastolic blood pressure. We also consider a variety of simulation schemes with varying levels of sparsity to validate our methods.

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arXiv2020

An Efficient Semi-smooth Newton Augmented Lagrangian Method for Elastic Net

Feature selection is an important and active research area in statistics and machine learning. The Elastic Net is often used to perform selection when the features present non-negligible collinearity or practitioners wish to incorporate additional known structure. In this article, we propose a new Semi-smooth Newton Augmented Lagrangian Method to efficiently solve the Elastic Net in ultra-high dimensional settings. Our new algorithm exploits both the sparsity induced by the Elastic Net penalty and the sparsity due to the second order information of the augmented Lagrangian. This greatly reduces the computational cost of the problem. Using simulations on both synthetic and real datasets, we demonstrate that our approach outperforms its best competitors by at least an order of magnitude in terms of CPU time. We also apply our approach to a Genome Wide Association Study on childhood obesity.

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arXiv2019

Highly Irregular Functional Generalized Linear Regression with Electronic Health Records

This work presents a new approach, called MISFIT, for fitting generalized functional linear regression models with sparsely and irregularly sampled data. Current methods do not allow for consistent estimation unless one assumes that the number of observed points per curve grows sufficiently quickly with the sample size. In contrast, MISFIT is based on a multiple imputation framework, which has the potential to produce consistent estimates without such an assumption. Just as importantly, it propagates the uncertainty of not having completely observed curves, allowing for a more accurate assessment of the uncertainty of parameter estimates, something that most methods currently cannot accomplish. This work is motivated by a longitudinal study on macrocephaly, or atypically large head size, in which electronic medical records allow for the collection of a great deal of data. However, the sampling is highly variable from child to child. Using MISFIT we are able to clearly demonstrate that the development of pathologic conditions related to macrocephaly is associated with both the overall head circumference of the children as well as the velocity of their head growth.

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arXiv2019

Adaptive Function-on-Scalar Regression with a Smoothing Elastic Net

This paper presents a new methodology, called AFSSEN, to simultaneously select significant predictors and produce smooth estimates in a high-dimensional function-on-scalar linear model with a sub-Gaussian errors. Outcomes are assumed to lie in a general real separable Hilbert space, H, while parameters lie in a subspace known as a Cameron Martin space, K, which are closely related to Reproducing Kernel Hilbert Spaces, so that parameter estimates inherit particular properties, such as smoothness or periodicity, without enforcing such properties on the data. We propose a regularization method in the style of an adaptive Elastic Net penalty that involves mixing two types of functional norms, providing a fine tune control of both the smoothing and variable selection in the estimated model. Asymptotic theory is provided in the form of a functional oracle property, and the paper concludes with a simulation study demonstrating the advantage of using AFSSEN over existing methods in terms of prediction error and variable selection.

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arXiv2019

Optimal Function-on-Scalar Regression over Complex Domains

In this work we consider the problem of estimating function-on-scalar regression models when the functions are observed over multi-dimensional or manifold domains and with potentially multivariate output. We establish the minimax rates of convergence and present an estimator based on reproducing kernel Hilbert spaces that achieves the minimax rate. To better interpret the derived rates, we extend well-known links between RKHS and Sobolev spaces to the case where the domain is a compact Riemannian manifold. This is accomplished using an interesting connection to Weyl's Law from partial differential equations. We conclude with a numerical study and an application to 3D facial imaging.

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arXiv2019

Benefits and Pitfalls of the Exponential Mechanism with Applications to Hilbert Spaces and Functional PCA

The exponential mechanism is a fundamental tool of Differential Privacy (DP) due to its strong privacy guarantees and flexibility. We study its extension to settings with summaries based on infinite dimensional outputs such as with functional data analysis, shape analysis, and nonparametric statistics. We show that one can design the mechanism with respect to a specific base measure over the output space, such as a Guassian process. We provide a positive result that establishes a Central Limit Theorem for the exponential mechanism quite broadly. We also provide an apparent negative result, showing that the magnitude of the noise introduced for privacy is asymptotically non-negligible relative to the statistical estimation error. We develop an \ep-DP mechanism for functional principal component analysis, applicable in separable Hilbert spaces. We demonstrate its performance via simulations and applications to two datasets.

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arXiv2019

Formal Privacy for Functional Data with Gaussian Perturbations

Motivated by the rapid rise in statistical tools in Functional Data Analysis, we consider the Gaussian mechanism for achieving differential privacy with parameter estimates taking values in a, potentially infinite-dimensional, separable Banach space. Using classic results from probability theory, we show how densities over function spaces can be utilized to achieve the desired differential privacy bounds. This extends prior results of Hall et al (2013) to a much broader class of statistical estimates and summaries, including "path level" summaries, nonlinear functionals, and full function releases. By focusing on Banach spaces, we provide a deeper picture of the challenges for privacy with complex data, especially the role regularization plays in balancing utility and privacy. Using an application to penalized smoothing, we explicitly highlight this balance in the context of mean function estimation. Simulations and an application to diffusion tensor imaging are briefly presented, with extensive additions included in a supplement.

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arXiv2018

Optimal Prediction for Additive Function-on-Function Regression

As with classic statistics, functional regression models are invaluable in the analysis of functional data. While there are now extensive tools with accompanying theory available for linear models, there is still a great deal of work to be done concerning nonlinear models for functional data. In this work we consider the Additive Function-on-Function Regression model, a type of nonlinear model that uses an additive relationship between the functional outcome and functional covariate. We present an estimation methodology built upon Reproducing Kernel Hilbert Spaces, and establish optimal rates of convergence for our estimates in terms of prediction error. We also discuss computational challenges that arise with such complex models, developing a representer theorem for our estimate as well as a more practical and computationally efficient approximation. Simulations and an application to Cumulative Intraday Returns around the 2008 financial crisis are also provided.

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arXiv2018

Testing Separability of Functional Time Series

We derive and study a significance test for determining if a panel of functional time series is separable. In the context of this paper, separability means that the covariance structure factors into the product of two functions, one depending only on time and the other depending only on the coordinates of the panel. Separability is a property which can dramatically improve computational efficiency by substantially reducing model complexity. It is especially useful for functional data as it implies that the functional principal components are the same for each member of the panel. However such an assumption must be verified before proceeding with further inference. Our approach is based on functional norm differences and provides a test with well controlled size and high power. We establish our procedure quite generally, allowing one to test separability of autocovariances as well. In addition to an asymptotic justification, our methodology is validated by a simulation study. It is applied to functional panels of particulate pollution and stock market data.

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arXiv2017

Manifold Data Analysis with Applications to High-Frequency 3D Imaging

Many scientific areas are faced with the challenge of extracting information from large, complex, and highly structured data sets. A great deal of modern statistical work focuses on developing tools for handling such data. This paper presents a new subfield of functional data analysis, FDA, which we call Manifold Data Analysis, or MDA. MDA is concerned with the statistical analysis of samples where one or more variables measured on each unit is a manifold, thus resulting in as many manifolds as we have units. We propose a framework that converts manifolds into functional objects, an efficient 2-step functional principal component method, and a manifold-on-scalar regression model. This work is motivated by an anthropological application involving 3D facial imaging data, which is discussed extensively throughout the paper. The proposed framework is used to understand how individual characteristics, such as age and genetic ancestry, influence the shape of the human face.

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