Miracle-dz/UniFAR-FTUs huggingface dataset:
Well‐posedness of Bayesian inverse problems in quasi‐Banach spaces with stable priors

The Bayesian perspective on inverse problems has attracted much mathematical attention in recent years. Particular attention has been paid to Bayesian inverse problems (BIPs) in which the parameter to be inferred lies in an infinite‐dimensional space, a typical example being a scalar or tensor field coupled to some observed data via an ODE or PDE. This article gives an introduction to the framework of well‐posed BIPs in infinite‐dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451–559, 2010) and others. This framework has the advantage of ensuring uniformly well‐posed inference problems independently of the finite‐dimensional discretisation used for numerical solution. Recently, this framework has been extended to the case of a heavy‐tailed prior measure in the family of stable distributions, such as an infinite‐dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen–Loève expansion for square‐integrable random variables can be used to sample such measures on quasi‐Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger and total variation metrics upon perturbations of the misfit function and observed data. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Testing for independence of large dimensional vectors

In this paper new tests for the independence of two high-dimensional vectors are investigated. We consider the case where the dimension of the vectors increases with the sample size and propose multivariate analysis of variance-type statistics for the hypothesis of a block diagonal covariance matrix. The asymptotic properties of the new test statistics are investigated under the null hypothesis and the alternative hypothesis using random matrix theory. For this purpose we study the weak convergence of linear spectral statistics of central and (conditionally) non-central Fisher matrices. In particular, a central limit theorem for linear spectral statistics of large dimensional (conditionally) non-central Fisher matrices is derived which is then used to analyse the power of the tests under the alternative. The theoretical results are illustrated by means of a simulation study where we also compare the new tests with several alternative, in particular with the commonly used corrected likelihood ratio test. It is demonstrated that the latter test does not keep its nominal level, if the dimension of one sub-vector is relatively small compared to the dimension of the other sub-vector. On the other hand the tests proposed in this paper provide a reasonable approximation of the nominal level in such situations. Moreover, we observe that one of the proposed tests is most powerful under a variety of correlation scenarios.
Modelling spatial heterogeneity and discontinuities using Voronoi tessellations

Many methods for modelling spatial processes assume global smoothness properties; such assumptions are often violated in practice. We introduce a method for modelling spatial processes that display heterogeneity or contain discontinuities. The problem of non-stationarity is dealt with by using a combination of Voronoi tessellation to partition the input space, and a separate Gaussian process to model the data on each region of the partitioned space. Our method is highly flexible because we allow the Voronoi cells to form relationships with each other, which can enable non-convex and disconnected regions to be considered. In such problems, identifying the borders between regions is often of great importance and we propose an adaptive sampling method to gain extra information along such borders. The method is illustrated with simulation studies and application to real data.
Logistic Box-Cox Regression to Assess the Shape and Median Effect under Uncertainty about Model Specification

The shape of the relationship between a continuous exposure variable and a binary disease variable is often central to epidemiologic investigations. This paper investigates a number of issues surrounding inference and the shape of the relationship. Presuming that the relationship can be expressed in terms of regression coefficients and a shape parameter, we investigate how well the shape can be inferred in settings which might typify epidemiologic investigations and risk assessment. We also consider a suitable definition of the median effect of exposure, and investigate how precisely this can be inferred. This is done both in the case of using a model acknowledging uncertainty about the shape parameter and in the case of ignoring this uncertainty and using a two-step method, where in step one we transform the predictor and in step two we fit a simple linear model with transformed predictor. All these investigations require a family of exposure-disease relationships indexed by a shape parameter. For this purpose, we employ a family based on the Box-Cox transformation.
AEPecker: L0 Adversarial Examples are not Strong Enough

Despite the great achievements made by neural networks on tasks such as image classification, they are brittle and vulnerable to adversarial examples (AEs). By adding adversarial noise to input images, adversarial examples can be crafted to mislead neural network based image classifiers. One type of AE attack in particular, known as an L0 AE, has been used in several notable real-world incidents. Our observation is that, while L0 corruptions modify as few pixels as possible, they tend to cause large-amplitude perturbations to the modified pixels. We consider this to be an inherent limitation of L0 AEs which can be exploited. To show the weakness of L0 AEs, we thwart samples of these attacks by both detecting and rectifying them. The main novelty of the proposed detector is that we convert the AE detection problem into an image comparison problem by exploiting the inherent characteristics of L0 AEs. More concretely, given an image I, it is pre-processed to obtain another image I'. We use a Siamese network which is known to be effective in comparison, to take I and I' as the input pair. A well trained Siamese network can automatically capture the discrepancy between I and I' to detect L0 noises. In addition, the straightforward pre-processor based on heuristics can be deployed as an effective defense, having a high probability of removing the adversarial influence of L0 perturbations. The proposed technique shows not only a high accuracy but also a resilience to the adaptive adversary, which outperforms other state-of-the-art methods. We accordingly argue that L0 attacks are not strong enough.
Best subset selection, persistence in high-dimensional statistical learning and optimization under l1 constraint

Let (Y, X 1 ,., X m ) be a random vector. It is desired to predict Y based on (X 1 ,., X m ). Examples of prediction methods are regression, classification using logistic regression or separating hyperplanes, and so on. We consider the problem of best subset selection, and study it in the context m = n α , a > 1, where n is the number of observations. We investigate procedures that are based on empirical risk minimization. It is shown, that in common cases, we should aim to find the best subset among those of size which is of order o(n/log(n)). It is also shown, that in some "asymptotic sense," when assuming a certain sparsity condition, there is no loss in letting m be much larger than n, for example, m = n α , a > 1. This is in comparison to starting with the "best" subset of size smaller than n and regardless of the value of a. We then study conditions under which empirical risk minimization subject to l 1 constraint yields nearly the best subset. These results extend some recent results obtained by Greenshtein and Ritov. Finally we present a high-dimensional simulation study of a "boosting type" classification procedure.
A backward procedure for change‐point detection with applications to copy number variation detection

Change‐point detection regains much attention recently for analyzing array or sequencing data for copy number variation (CNV) detection. In such applications, the true signals are typically very short and buried in the long data sequence, which makes it challenging to identify the variations efficiently and accurately. In this article, we propose a new change‐point detection method, a backward procedure, which is not only fast and simple enough to exploit high‐dimensional data but also performs very well for detecting short signals. Although motivated by CNV detection, the backward procedure is generally applicable to assorted change‐point problems that arise in a variety of scientific applications. It is illustrated by both simulated and real CNV data that the backward detection has clear advantages over other competing methods, especially when the true signal is short. The Canadian Journal of Statistics 48: 366–385; 2020 © 2020 Statistical Society of Canada
Model Selection and estimation of Multi Screen Penalty

We propose a multi-step method, called Multi Screen Penalty (MSP), to estimate high-dimensional sparse linear models. MSP uses a series of small and adaptive penalty to iteratively estimate the regression coefficients. This structure is shown to greatly improve the model selection and estimation accuracy, i.e., it precisely selects the true model when the irrepresentable condition fails; under mild regularity conditions, MSP estimator can achieve the rate $\sqrt{q \log n /n}$ for the upper bound of l_2-norm error. At each step, we restrict the selection of MSP only on the reduced parameter space obtained from the last step; this makes its computational complexity is roughly comparable to Lasso. This algorithm is found to be stable and reaches to high accuracy over a range of small tuning parameters, hence deletes the cross-validation segment. Numerical comparisons show that the method works effectively both in model selection and estimation and nearly uniformly outperform others. We apply MSP and other methods to financial data. MSP is successful in assets selection and produces more stable and lower rates of fitted/predicted errors.
Walking into an absolute sum

We investigate a combinatorial sum that can be interpreted as the moments of a random variate, measuring the absolute distance to the origin in a symmetric Bernoulli random walk. These sums can be characterized by polynomials related to the Dumont-Foata polynomials. The sums corresponding to the odd moments have a connection to the Gandhi polynomials and Genocchi numbers.
Robust Estimation of Causal Effects via High-Dimensional Covariate Balancing Propensity Score.

In this paper, we propose a robust method to estimate the average treatment effects in observational studies when the number of potential confounders is possibly much greater than the sample size. We first use a class of penalized M-estimators for the propensity score and outcome models. We then calibrate the initial estimate of the propensity score by balancing a carefully selected subset of covariates that are predictive of the outcome. Finally, the estimated propensity score is used to construct the inverse probability weighting estimator. We prove that the proposed estimator, which has the sample boundedness property, is root-n consistent, asymptotically normal, and semiparametrically efficient when the propensity score model is correctly specified and the outcome model is linear in covariates. More importantly, we show that our estimator remains root-n consistent and asymptotically normal so long as either the propensity score model or the outcome model is correctly specified. We provide valid confidence intervals in both cases and further extend these results to the case where the outcome model is a generalized linear model. In simulation studies, we find that the proposed methodology often estimates the average treatment effect more accurately than the existing methods. We also present an empirical application, in which we estimate the average causal effect of college attendance on adulthood political participation. Open-source software is available for implementing the proposed methodology.
Bi-clustering for time-varying relational count data analysis

Relational count data are often obtained from sources such as simultaneous purchase in online shops and social networking service information. Bi-clustering such relational count data reveals the latent structure of the relationship between objects such as household items or people. When relational count data observed at multiple time points are available, it is worthwhile incorporating the time structure into the bi-clustering result to understand how objects move between the cluster over time. In this paper, we propose two bi-clustering methods for analyzing time-varying relational count data. The first model, the dynamic Poisson infinite relational model (dPIRM), handles time-varying relational count data. In the second model, which we call the dynamic zero-inflated Poisson infinite relational model, we further extend the dPIRM so that it can handle zero-inflated data. Proposing both two models is important as zero-inflated data are often encountered, especially when the time intervals are short. In addition, by explicitly deriving the relevant full conditional distributions, we describe the features of the estimated parameters and, in turn, the relationship between the two models. We show the effectiveness of both models through a simulation study and a real data example.
Multifaceted 4D Feature Segmentation and Extraction in Point and Field-based Datasets

The use of large-scale multifaceted data is common in a wide variety of scientific applications. In many cases, this multifaceted data takes the form of a field-based (Eulerian) and point/trajectory-based (Lagrangian) representation as each has a unique set of advantages in characterizing a system of study. Furthermore, studying the increasing scale and complexity of these multifaceted datasets is limited by perceptual ability and available computational resources, necessitating sophisticated data reduction and feature extraction techniques. In this work, we present a new 4D feature segmentation/extraction scheme that can operate on both the field and point/trajectory data types simultaneously. The resulting features are time-varying data subsets that have both a field and point-based component, and were extracted based on underlying patterns from both data types. This enables researchers to better explore both the spatial and temporal interplay between the two data representations and study underlying phenomena from new perspectives. We parallelize our approach using GPU acceleration and apply it to real world multifaceted datasets to illustrate the types of features that can be extracted and explored.
Structural causal models for macro-variables in time-series

We consider a bivariate time series $(X_t,Y_t)$ that is given by a simple linear autoregressive model. Assuming that the equations describing each variable as a linear combination of past values are considered structural equations, there is a clear meaning of how intervening on one particular $X_t$ influences $Y_{t'}$ at later times $t'>t$. In the present work, we describe conditions under which one can define a causal model between variables that are coarse-grained in time, thus admitting statements like `setting $X$ to $x$ changes $Y$ in a certain way' without referring to specific time instances. We show that particularly simple statements follow in the frequency domain, thus providing meaning to interventions on frequencies.
High-dimensional additive modeling

We propose a new sparsity-smoothness penalty for high-dimensional generalized additive models. The combination of sparsity and smoothness is crucial for mathematical theory as well as performance for finite-sample data. We present a computationally efficient algorithm, with provable numerical convergence properties, for optimizing the penalized likelihood. Furthermore, we provide oracle results which yield asymptotic optimality of our estimator for high dimensional but sparse additive models. Finally, an adaptive version of our sparsity-smoothness penalized approach yields large additional performance gains.
Hypocoercivity of piecewise deterministic Markov process-Monte Carlo

In this work, we establish $\mathrm{L}^2$-exponential convergence for a broad class of Piecewise Deterministic Markov Processes recently proposed in the context of Markov Process Monte Carlo methods and covering in particular the Randomized Hamiltonian Monte Carlo, the Zig-Zag process and the Bouncy Particle Sampler. The kernel of the symmetric part of the generator of such processes is non-trivial, and we follow the ideas recently introduced by (Dolbeault et al., 2009, 2015) to develop a rigorous framework for hypocoercivity in a fairly general and unifying set-up, while deriving tractable estimates of the constants involved in terms of the parameters of the dynamics. As a by-product we characterize the scaling properties of these algorithms with respect to the dimension of classes of problems, therefore providing some theoretical evidence to support their practical relevance.
Alzheimer's Disease Prediction Using Longitudinal and Heterogeneous Magnetic Resonance Imaging

Recent evidence has shown that structural magnetic resonance imaging (MRI) is an effective tool for Alzheimer's disease (AD) prediction and diagnosis. While traditional MRI-based diagnosis uses images acquired at a single time point, a longitudinal study is more sensitive and accurate in detecting early pathological changes of the AD. Two main difficulties arise in longitudinal MRI-based diagnosis: (1) the inconsistent longitudinal scans among subjects (i.e., different scanning time and different total number of scans); (2) the heterogeneous progressions of high-dimensional regions of interest (ROIs) in MRI. In this work, we propose a novel feature selection and estimation method which can be applied to extract features from the heterogeneous longitudinal MRI. A key ingredient of our method is the combination of smoothing splines and the $l_1$-penalty. We perform experiments on the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. The results corroborate the advantages of the proposed method for AD prediction in longitudinal studies.
Robust Group Comparison Using Non-Parametric Block-Based Statistics

Voxel-based analysis methods localize brain structural differences by performing voxel-wise statistical comparisons on two groups of images aligned to a common space. This procedure requires highly accurate registration as well as a sufficiently large dataset. However, in practice, the registration algorithms are not perfect due to noise, artifacts, and complex structural variations. The sample size is also limited due to low disease prevalence, recruitment difficulties, and demographic matching issues. To address these issues, in this paper, we propose a method, called block-based statistic (BBS), for robust group comparison. BBS consists of two major components: Block matching and permutation test. Specifically, based on two group of images aligned to a common space, we first perform block matching so that structural misalignments can be corrected. Then, based on results given by block matching, we conduct robust non-parametric statistical inference based on permutation test. Extensive experiments were performed on synthetic data and the real diffusion MR data of mild cognitive impairment patients. The experimental results indicate that BBS significantly improves statistical power, notwithstanding the small sample size.
Estimation of the autocovariance function with missing observations

We propose a novel estimator of the autocorrelation function in presence of missing observations. We establish the consistency, the asymptotic normality, and we derive deviation bounds for various classes of weakly dependent stationary time series, including causal or non causal models. In addition, we introduce a modified version periodogram defined from these autocorrelation estimators and derive asymptotic distribution of linear functionals of this estimator.
Relaxation Penalties and Priors for Plausible Modeling of Nonidentified Bias Sources

In designed experiments and surveys, known laws or design feat ures provide checks on the most relevant aspects of a model and identify the target parameters. In contrast, in most observational studies in the health and social sciences, the primary study data do not identify and may not even bound target parameters. Discrepancies between target and analogous identified parameters (biases) are then of paramount concern, which forces a major shift in modeling strategies. Conventional approaches are based on conditional testing of equality constraints, which correspond to implausible point-mass priors. When these constraints are not identified by available data, however, no such testing is possible. In response, implausible constraints can be relaxed into penalty functions derived from plausible prior distributions. The resulting models can be fit within familiar full or partial likelihood frameworks. The absence of identification renders all analyses part of a sensitivity analysis. In this view, results from single models are merely examples of what might be plausibly inferred. Nonetheless, just one plausible inference may suffice to demonstrate inherent limitations of the data. Points are illustrated with misclassified data from a study of sudden infant death syndrome. Extensions to confounding, selection bias and more complex data structures are outlined.
Robust graphical modeling of gene networks using classical and alternative t-distributions

Graphical Gaussian models have proven to be useful tools for exploring network structures based on multivariate data. Applications to studies of gene expression have generated substantial interest in these models, and resulting recent progress includes the development of fitting methodology involving penalization of the likelihood function. In this paper we advocate the use of multivariate $t$-distributions for more robust inference of graphs. In particular, we demonstrate that penalized likelihood inference combined with an application of the EM algorithm provides a computationally efficient approach to model selection in the $t$-distribution case. We consider two versions of multivariate $t$-distributions, one of which requires the use of approximation techniques. For this distribution, we describe a Markov chain Monte Carlo EM algorithm based on a Gibbs sampler as well as a simple variational approximation that makes the resulting method feasible in large problems.
Comments on “A note on optimal detection of a change in distribution,” by Benjamin Yakir

The purpose of this note is to show that in a widely cited paper by Yakir [Ann. Statist. 25 (1997) 2117--2126, doi:10.1214/aos/1069362390], the proof that the so-called modified Shiryayev--Roberts procedure is exactly optimal is incorrect. We also clarify the issues involved by both mathematical arguments and a simulation study. The correctness of the theorem remains in doubt.