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We study approaches for compressing the empirical measure in the context of finite dimensional reproducing kernel Hilbert spaces (RKHSs).In this context, the empirical measure is contained within a natural convex set and can be approximated using convex optimization methods. Such an approximation gives under certain conditions rise to a coreset of data points. A key quantity that controls how large such a coreset has to be is the size of the largest ball around the empirical measure that is contained within the empirical convex set. The bulk of our work is concerned with deriving high probability lower bounds on the size of such a ball under various conditions. We complement this derivation of the lower bound by developing techniques that allow us to apply the compression approach to concrete inference problems such as kernel ridge regression. We conclude with a construction of an infinite dimensional RKHS for which the compression is poor, highlighting some of the difficulties one faces when trying to move to infinite dimensional RKHSs.

This paper establishes the asymptotic independence between the quadratic form and maximum of a sequence of independent random variables. Based on this theoretical result, we find the asymptotic joint distribution for the quadratic form and maximum, which can be applied into the high-dimensional testing problems. By combining the sum-type test and the max-type test, we propose the Fisher's combination tests for the one-sample mean test and two-sample mean test. Under this novel general framework, several strong assumptions in existing literature have been relaxed. Monte Carlo simulation has been done which shows that our proposed tests are strongly robust to both sparse and dense data.

We investigate the feature compression of high-dimensional ridge regression using the optimal subsampling technique. Specifically, based on the basic framework of random sampling algorithm on feature for ridge regression and the A-optimal design criterion, we first obtain a set of optimal subsampling probabilities. Considering that the obtained probabilities are uneconomical, we then propose the nearly optimal ones. With these probabilities, a two step iterative algorithm is established which has lower computational cost and higher accuracy. We provide theoretical analysis and numerical experiments to support the proposed methods. Numerical results demonstrate the decent performance of our methods.

This paper considers the problem of inference in cluster randomized experiments when cluster sizes are non-ignorable. Here, by a cluster randomized experiment, we mean one in which treatment is assigned at the level of the cluster; by non-ignorable cluster sizes we mean that "large" clusters and "small" clusters may be heterogeneous, and, in particular, the effects of the treatment may vary across clusters of differing sizes. In order to permit this sort of flexibility, we consider a sampling framework in which cluster sizes themselves are random. In this way, our analysis departs from earlier analyses of cluster randomized experiments in which cluster sizes are treated as non-random. We distinguish between two different parameters of interest: the equally-weighted cluster-level average treatment effect, and the size-weighted cluster-level average treatment effect. For each parameter, we provide methods for inference in an asymptotic framework where the number of clusters tends to infinity and treatment is assigned using simple random sampling. We additionally permit the experimenter to sample only a subset of the units within each cluster rather than the entire cluster and demonstrate the implications of such sampling for some commonly used estimators. A small simulation study shows the practical relevance of our theoretical results.

Many existing algorithms for streaming geometric data analysis have been plagued by exponential dependencies in the space complexity, which are undesirable for processing high-dimensional data sets. In particular, once $d\geq\log n$, there are no known non-trivial streaming algorithms for problems such as maintaining convex hulls and L\"owner-John ellipsoids of $n$ points, despite a long line of work in streaming computational geometry since [AHV04]. We simultaneously improve these results to $\mathrm{poly}(d,\log n)$ bits of space by trading off with a $\mathrm{poly}(d,\log n)$ factor distortion. We achieve these results in a unified manner, by designing the first streaming algorithm for maintaining a coreset for $\ell_\infty$ subspace embeddings with $\mathrm{poly}(d,\log n)$ space and $\mathrm{poly}(d,\log n)$ distortion. Our algorithm also gives similar guarantees in the \emph{online coreset} model. Along the way, we sharpen results for online numerical linear algebra by replacing a log condition number dependence with a $\log n$ dependence, answering a question of [BDM+20]. Our techniques provide a novel connection between leverage scores, a fundamental object in numerical linear algebra, and computational geometry. For $\ell_p$ subspace embeddings, we give nearly optimal trade-offs between space and distortion for one-pass streaming algorithms. For instance, we give a deterministic coreset using $O(d^2\log n)$ space and $O((d\log n)^{1/2-1/p})$ distortion for $p>2$, whereas previous deterministic algorithms incurred a $\mathrm{poly}(n)$ factor in the space or the distortion [CDW18]. Our techniques have implications in the offline setting, where we give optimal trade-offs between the space complexity and distortion of subspace sketch data structures. To do this, we give an elementary proof of a "change of density" theorem of [LT80] and make it algorithmic.

Randomized Maximum Likelihood (RML) is an approximate posterior sampling methodology, widely used in Bayesian inverse problems with complex forward models, particularly in petroleum engineering applications. The procedure involves solving a multi-objective optimization problem, which can be challenging in high-dimensions and when there are constraints on computational costs. We propose a new methodology for tackling the RML optimization problem based on the high-dimensional Bayesian optimization literature. By sharing data between the different objective functions, we are able to implement RML at a greatly reduced computational cost. We demonstrate the benefits of our methodology in comparison with the solutions obtained by alternative optimization methods on a variety of synthetic and real-world problems, including medical and fluid dynamics applications. Furthermore, we show that the samples produced by our method cover well the high-posterior density regions in all of the experiments.

An important challenge in statistical analysis lies in controlling the estimation bias when handling the ever-increasing data size and model complexity. For example, approximate methods are increasingly used to address the analytical and/or computational challenges when implementing standard estimators, but they often lead to inconsistent estimators. So consistent estimators can be difficult to obtain, especially for complex models and/or in settings where the number of parameters diverges with the sample size. We propose a general simulation-based estimation framework that allows to construct consistent and bias corrected estimators for parameters of increasing dimensions. The key advantage of the proposed framework is that it only requires to compute a simple inconsistent estimator multiple times. The resulting Just Identified iNdirect Inference estimator (JINI) enjoys nice properties, including consistency, asymptotic normality, and finite sample bias correction better than alternative methods. We further provide a simple algorithm to construct the JINI in a computationally efficient manner. Therefore, the JINI is especially useful in settings where standard methods may be challenging to apply, for example, in the presence of misclassification and rounding. We consider comprehensive simulation studies and analyze an alcohol consumption data example to illustrate the excellent performance and usefulness of the method.

In this work, we study the transfer learning problem under high-dimensional generalized linear models (GLMs), which aim to improve the fit on target data by borrowing information from useful source data. Given which sources to transfer, we propose a transfer learning algorithm on GLM, and derive its $\ell_1/\ell_2$-estimation error bounds as well as a bound for a prediction error measure. The theoretical analysis shows that when the target and source are sufficiently close to each other, these bounds could be improved over those of the classical penalized estimator using only target data under mild conditions. When we don't know which sources to transfer, an algorithm-free transferable source detection approach is introduced to detect informative sources. The detection consistency is proved under the high-dimensional GLM transfer learning setting. We also propose an algorithm to construct confidence intervals of each coefficient component, and the corresponding theories are provided. Extensive simulations and a real-data experiment verify the effectiveness of our algorithms. We implement the proposed GLM transfer learning algorithms in a new R package glmtrans, which is available on CRAN.

A High-dimensional and sparse (HiDS) matrix is frequently encountered in a big data-related application like an e-commerce system or a social network services system. To perform highly accurate representation learning on it is of great significance owing to the great desire of extracting latent knowledge and patterns from it. Latent factor analysis (LFA), which represents an HiDS matrix by learning the low-rank embeddings based on its observed entries only, is one of the most effective and efficient approaches to this issue. However, most existing LFA-based models perform such embeddings on a HiDS matrix directly without exploiting its hidden graph structures, thereby resulting in accuracy loss. To address this issue, this paper proposes a graph-incorporated latent factor analysis (GLFA) model. It adopts two-fold ideas: 1) a graph is constructed for identifying the hidden high-order interaction (HOI) among nodes described by an HiDS matrix, and 2) a recurrent LFA structure is carefully designed with the incorporation of HOI, thereby improving the representa-tion learning ability of a resultant model. Experimental results on three real-world datasets demonstrate that GLFA outperforms six state-of-the-art models in predicting the missing data of an HiDS matrix, which evidently supports its strong representation learning ability to HiDS data.

Kernel smooth is the most fundamental technique for data density and regression estimation. However, time-consuming is the biggest obstacle for the application that the direct evaluation of kernel smooth for $N$ samples needs ${O}\left( {{N}^{2}} \right)$ operations. People have developed fast smooth algorithms using the idea of binning with FFT. Unfortunately, the accuracy is not controllable, and the implementation for multivariable and its bandwidth selection for the fast method is not available. Hence, we introduce a new MATLAB toolbox for fast multivariate kernel regression with the idea of non-uniform FFT (NUFFT), which implemented the algorithm for $M$ gridding points with ${O}\left( N+M\log M \right)$ complexity and accuracy controllability. The bandwidth selection problem utilizes the Fast Monte-Carlo algorithm to estimate the degree of freedom (DF), saving enormous cross-validation time even better when data share the same grid space for multiple regression. Up to now, this is the first toolbox for fast-binning high-dimensional kernel regression. Moreover, the estimation for local polynomial regression, the conditional variance for the heteroscedastic model, and the complex-valued datasets are also implemented in this toolbox. The performance is demonstrated with simulations and an application on the quantitive EEG.