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Information-theoretic quantities, such as conditional entropy and mutual information, are critical data summaries for quantifying uncertainty. Current widely used approaches for computing such quantities rely on nearest neighbor methods and exhibit both strong performance and theoretical guarantees in certain simple scenarios. However, existing approaches fail in high-dimensional settings and when different features are measured on different scales.We propose decision forest-based adaptive nearest neighbor estimators and show that they are able to effectively estimate posterior probabilities, conditional entropies, and mutual information even in the aforementioned settings.We provide an extensive study of efficacy for classification and posterior probability estimation, and prove certain forest-based approaches to be consistent estimators of the true posteriors and derived information-theoretic quantities under certain assumptions. In a real-world connectome application, we quantify the uncertainty about neuron type given various cellular features in the Drosophila larva mushroom body, a key challenge for modern neuroscience.

This paper considers inference in a linear regression model with random right censoring and outliers. The number of outliers can grow with the sample size while their proportion goes to zero. The model is semiparametric and we make only very mild assumptions on the distribution of the error term, contrary to most other existing approaches in the literature. We propose to penalize the estimator proposed by Stute for censored linear regression by the l1-norm. We derive rates of convergence and establish asymptotic normality of the estimator of the regression coefficients. Our estimator has the same asymptotic variance as Stute's estimator in the censored linear model without outliers. Hence, there is no loss of efficiency as a result of robustness. Tests and confidence sets can therefore rely on the theory developed by Stute. The outlined procedure is also computationally advantageous, since it amounts to solving a convex optimization program. We also propose a second estimator which uses the proposed penalized Stute estimator as a first step to detect outliers. It has similar theoretical properties but better performance in finite samples as assessed by simulations.

In this paper, we derive improved a priori error estimates for families of hybridizable interior penalty discontinuous Galerkin (H-IP) methods using a variable penalty for second-order elliptic problems. The strategy is to use a penalization function of the form $\mathcal{O}(1/h^{1+\delta})$, where $h$ denotes the mesh size and $\delta$ is a user-dependent parameter. We then quantify its direct impact on the convergence analysis, namely, the (strong) consistency, discrete coercivity, and boundedness (with $h^{\delta}$-dependency), and we derive updated error estimates for both discrete energy- and $L^{2}$-norms. The originality of the error analysis relies specifically on the use of conforming interpolants of the exact solution. All theoretical results are supported by numerical evidence.

Real-world applications of machine learning tools in high-stakes domains are often regulated to be fair, in the sense that the predicted target should satisfy some quantitative notion of parity with respect to a protected attribute. However, the exact tradeoff between fairness and accuracy with a real-valued target is not entirely clear. In this paper, we characterize the inherent tradeoff between statistical parity and accuracy in the regression setting by providing a lower bound on the error of any fair regressor. Our lower bound is sharp, algorithm-independent, and admits a simple interpretation: when the moments of the target differ between groups, any fair algorithm has to make an error on at least one of the groups. We further extend this result to give a lower bound on the joint error of any (approximately) fair algorithm, using the Wasserstein distance to measure the quality of the approximation. With our novel lower bound, we also show that the price paid by a fair regressor that does not take the protected attribute as input is less than that of a fair regressor with explicit access to the protected attribute. On the upside, we establish the first connection between individual fairness, accuracy parity, and the Wasserstein distance by showing that if a regressor is individually fair, it also approximately verifies the accuracy parity, where the gap is given by the Wasserstein distance between the two groups. Inspired by our theoretical results, we develop a practical algorithm for fair regression through the lens of representation learning, and conduct experiments on a real-world dataset to corroborate our findings.

The noncentral Wishart distribution has become more mainstream in statistics as the prevalence of applications involving sample covariances with underlying multivariate Gaussian populations as dramatically increased since the advent of computers. Multiple sources in the literature deal with local approximations of the noncentral Wishart distribution with respect to its central counterpart. However, no source has yet developed explicit local approximations for the (central) Wishart distribution in terms of a normal analogue, which is important since Gaussian distributions are at the heart of the asymptotic theory for many statistical methods. In this paper, we prove a precise asymptotic expansion for the ratio of the Wishart density to the symmetric matrix-variate normal density with the same mean and covariances. The result is then used to derive an upper bound on the total variation between the corresponding probability measures and to find the pointwise variance of a new density estimator on the space of positive definite matrices with a Wishart asymmetric kernel. For the sake of completeness, we also find expressions for the pointwise bias of our new estimator, the pointwise variance as we move towards the boundary of its support, the mean squared error, the mean integrated squared error away from the boundary, and we prove its asymptotic normality.

We propose a novel method for estimating heterogeneous treatment effects based on the fused lasso. By first ordering samples based on the propensity or prognostic score, we match units from the treatment and control groups. We then run the fused lasso to obtain piecewise constant treatment effects with respect to the ordering defined by the score. Similar to the existing methods based on discretizing the score, our methods yields interpretable subgroup effects. However, the existing methods fixed the subgroup a priori, but our causal fused lasso forms data-adaptive subgroups. We show that the estimator consistently estimates the treatment effects conditional on the score under very general conditions on the covariates and treatment. We demonstrate the performance of our procedure using extensive experiments that show that it can outperform state-of-the-art methods.

We provide a posteriori error estimates in the energy norm for temporal semi-discretisations of wave maps into spheres that are based on the angular momentum formulation. Our analysis is based on novel weak-strong stability estimates which we combine with suitable reconstructions of the numerical solution. We present time-adaptive numerical simulations based on the a posteriori error estimators for solutions involving blow-up.

Distributional data analysis, concerned with statistical analysis and modeling for data objects consisting of random probability density functions (PDFs) in the framework of functional data analysis (FDA), has received considerable interest in recent years. However, many important aspects remain unexplored, such as outlier detection and robustness. Existing functional outlier detection methods are mainly used for ordinary functional data and usually perform poorly when applied to PDFs. To fill this gap, this study focuses on PDF-valued outlier detection, as well as its application in robust distributional regression. Similar to ordinary functional data, detecting the shape outlier masked by the "curve net" formed by the bulk of the PDFs is the major challenge in PDF-outlier detection. To this end, we propose a tree-structured transformation system for feature extraction as well as converting the shape outliers to easily detectable magnitude outliers, relevant outlier detectors are designed for the specific transformed data. A multiple detection strategy is also proposed to account for detection uncertainties and to combine different detectors to form a more reliable detection tool. Moreover, we propose a distributional-regression-based approach for detecting the abnormal associations of PDF-valued two-tuples. As a specific application, the proposed outlier detection methods are applied to robustify a distribution-to-distribution regression method, and we develop a robust estimator for the regression operator by downweighting the detected outliers. The proposed methods are validated and evaluated by extensive simulation studies or real data applications. Relevant comparative studies demonstrate the superiority of the developed outlier detection method with other competitors in distributional outlier detection.

Heatmap-based methods dominate in the field of human pose estimation by modelling the output distribution through likelihood heatmaps. In contrast, regression-based methods are more efficient but suffer from inferior performance. In this work, we explore maximum likelihood estimation (MLE) to develop an efficient and effective regression-based methods. From the perspective of MLE, adopting different regression losses is making different assumptions about the output density function. A density function closer to the true distribution leads to a better regression performance. In light of this, we propose a novel regression paradigm with Residual Log-likelihood Estimation (RLE) to capture the underlying output distribution. Concretely, RLE learns the change of the distribution instead of the unreferenced underlying distribution to facilitate the training process. With the proposed reparameterization design, our method is compatible with off-the-shelf flow models. The proposed method is effective, efficient and flexible. We show its potential in various human pose estimation tasks with comprehensive experiments. Compared to the conventional regression paradigm, regression with RLE bring 12.4 mAP improvement on MSCOCO without any test-time overhead. Moreover, for the first time, especially on multi-person pose estimation, our regression method is superior to the heatmap-based methods. Our code is available at //github.com/Jeff-sjtu/res-loglikelihood-regression

We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.