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We propose a robust and reliable evaluation metric for generative models by introducing topological and statistical treatments for rigorous support estimation. Existing metrics, such as Inception Score (IS), Frechet Inception Distance (FID), and the variants of Precision and Recall (P&R), heavily rely on supports that are estimated from sample features. However, the reliability of their estimation has not been seriously discussed (and overlooked) even though the quality of the evaluation entirely depends on it. In this paper, we propose Topological Precision and Recall (TopP&R, pronounced 'topper'), which provides a systematic approach to estimating supports, retaining only topologically and statistically important features with a certain level of confidence. This not only makes TopP&R strong for noisy features, but also provides statistical consistency. Our theoretical and experimental results show that TopP&R is robust to outliers and non-independent and identically distributed (Non-IID) perturbations, while accurately capturing the true trend of change in samples. To the best of our knowledge, this is the first evaluation metric focused on the robust estimation of the support and provides its statistical consistency under noise.

We present a new approach to semiparametric inference using corrected posterior distributions. The method allows us to leverage the adaptivity, regularization and predictive power of nonparametric Bayesian procedures to estimate low-dimensional functionals of interest without being restricted by the holistic Bayesian formalism. Starting from a conventional nonparametric posterior, we target the functional of interest by transforming the entire distribution with a Bayesian bootstrap correction. We provide conditions for the resulting $\textit{one-step posterior}$ to possess calibrated frequentist properties and specialize the results for several canonical examples: the integrated squared density, the mean of a missing-at-random outcome, and the average causal treatment effect on the treated. The procedure is computationally attractive, requiring only a simple, efficient post-processing step that can be attached onto any arbitrary posterior sampling algorithm. Using the ACIC 2016 causal data analysis competition, we illustrate that our approach can outperform the existing state-of-the-art through the propagation of Bayesian uncertainty.

In compact settings, the convergence rate of the empirical optimal transport cost to its population value is well understood for a wide class of spaces and cost functions. In unbounded settings, however, hitherto available results require strong assumptions on the ground costs and the concentration of the involved measures. In this work, we pursue a decomposition-based approach to generalize the convergence rates found in compact spaces to unbounded settings under generic moment assumptions that are sharp up to an arbitrarily small $\epsilon > 0$. Hallmark properties of empirical optimal transport on compact spaces, like the recently established adaptation to lower complexity, are shown to carry over to the unbounded case.

Large-scale administrative or observational datasets are increasingly used to inform decision making. While this effort aims to ground policy in real-world evidence, challenges have arise as that selection bias and other forms of distribution shift often plague observational data. Previous attempts to provide robust inferences have given guarantees depending on a user-specified amount of possible distribution shift (e.g., the maximum KL divergence between the observed and target distributions). However, decision makers will often have additional knowledge about the target distribution which constrains the kind of shifts which are possible. To leverage such information, we proposed a framework that enables statistical inference in the presence of distribution shifts which obey user-specified constraints in the form of functions whose expectation is known under the target distribution. The output is high-probability bounds on the value an estimand takes on the target distribution. Hence, our method leverages domain knowledge in order to partially identify a wide class of estimands. We analyze the computational and statistical properties of methods to estimate these bounds, and show that our method can produce informative bounds on a variety of simulated and semisynthetic tasks.

We examine the problem of variance components testing in general mixed effects models using the likelihood ratio test. We account for the presence of nuisance parameters, i.e. the fact that some untested variances might also be equal to zero. Two main issues arise in this context leading to a non regular setting. First, under the null hypothesis the true parameter value lies on the boundary of the parameter space. Moreover, due to the presence of nuisance parameters the exact location of these boundary points is not known, which prevents from using classical asymptotic theory of maximum likelihood estimation. Then, in the specific context of nonlinear mixed-effects models, the Fisher information matrix is singular at the true parameter value. We address these two points by proposing a shrinked parametric bootstrap procedure, which is straightforward to apply even for nonlinear models. We show that the procedure is consistent, solving both the boundary and the singularity issues, and we provide a verifiable criterion for the applicability of our theoretical results. We show through a simulation study that, compared to the asymptotic approach, our procedure has a better small sample performance and is more robust to the presence of nuisance parameters. A real data application is also provided.

Elliptical distribution is a basic assumption underlying many multivariate statistical methods. For example, in sufficient dimension reduction and statistical graphical models, this assumption is routinely imposed to simplify the data dependence structure. Before applying such methods, we need to decide whether the data are elliptically distributed. Currently existing tests either focus exclusively on spherical distributions, or rely on bootstrap to determine the null distribution, or require specific forms of the alternative distribution. In this paper, we introduce a general nonparametric test for elliptical distribution based on kernel embedding of the probability measure that embodies the two properties that characterize an elliptical distribution: namely, after centering and rescaling, (1) the direction and length of the random vector are independent, and (2) the directional vector is uniformly distributed on the unit sphere. We derive the null asymptotic distribution of the test statistic via von-Mises expansion, develop the sample-level procedure to determine the rejection region, and establish the consistency and validity of the proposed test. We apply our test to a SENIC dataset with and without a transformation aimed to achieve ellipticity.

Nonparametric density estimation is an unsupervised learning problem. In this work we propose a two-step procedure that casts the density estimation problem in the first step into a supervised regression problem. The advantage is that we can afterwards apply supervised learning methods. Compared to the standard nonparametric regression setting, the proposed procedure creates, however, dependence among the training samples. To derive statistical risk bounds, one can therefore not rely on the well-developed theory for i.i.d. data. To overcome this, we prove an oracle inequality for this specific form of data dependence. As an application, it is shown that under a compositional structure assumption on the underlying density the proposed two-step method achieves faster convergence rates. A simulation study illustrates the finite sample performance.

An algorithm is said to be adaptive to a certain parameter (of the problem) if it does not need a priori knowledge of such a parameter but performs competitively to those that know it. This dissertation presents our work on adaptive algorithms in following scenarios: 1. In the stochastic optimization setting, we only receive stochastic gradients and the level of noise in evaluating them greatly affects the convergence rate. Tuning is typically required when without prior knowledge of the noise scale in order to achieve the optimal rate. Considering this, we designed and analyzed noise-adaptive algorithms that can automatically ensure (near)-optimal rates under different noise scales without knowing it. 2. In training deep neural networks, the scales of gradient magnitudes in each coordinate can scatter across a very wide range unless normalization techniques, like BatchNorm, are employed. In such situations, algorithms not addressing this problem of gradient scales can behave very poorly. To mitigate this, we formally established the advantage of scale-free algorithms that adapt to the gradient scales and presented its real benefits in empirical experiments. 3. Traditional analyses in non-convex optimization typically rely on the smoothness assumption. Yet, this condition does not capture the properties of some deep learning objective functions, including the ones involving Long Short-Term Memory networks and Transformers. Instead, they satisfy a much more relaxed condition, with potentially unbounded smoothness. Under this condition, we show that a generalized SignSGD algorithm can theoretically match the best-known convergence rates obtained by SGD with gradient clipping but does not need explicit clipping at all, and it can empirically match the performance of Adam and beat others. Moreover, it can also be made to automatically adapt to the unknown relaxed smoothness.

We propose a novel measure of statistical depth, the metric spatial depth, for data residing in an arbitrary metric space. The measure assigns high (low) values for points located near (far away from) the bulk of the data distribution, allowing quantifying their centrality/outlyingness. This depth measure is shown to have highly interpretable geometric properties, making it appealing in object data analysis where standard descriptive statistics are difficult to compute. The proposed measure reduces to the classical spatial depth in a Euclidean space. In addition to studying its theoretical properties, to provide intuition on the concept, we explicitly compute metric spatial depths in several different metric spaces. Finally, we showcase the practical usefulness of the metric spatial depth in outlier detection, non-convex depth region estimation and classification.

We consider the problem of detecting multiple changes in multiple independent time series. The search for the best segmentation can be expressed as a minimization problem over a given cost function. We focus on dynamic programming algorithms that solve this problem exactly. When the number of changes is proportional to data length, an inequality-based pruning rule encoded in the PELT algorithm leads to a linear time complexity. Another type of pruning, called functional pruning, gives a close-to-linear time complexity whatever the number of changes, but only for the analysis of univariate time series. We propose a few extensions of functional pruning for multiple independent time series based on the use of simple geometric shapes (balls and hyperrectangles). We focus on the Gaussian case, but some of our rules can be easily extended to the exponential family. In a simulation study we compare the computational efficiency of different geometric-based pruning rules. We show that for small dimensions (2, 3, 4) some of them ran significantly faster than inequality-based approaches in particular when the underlying number of changes is small compared to the data length.