In this work, we propose Generative and Explainable Adversarial Data Augmentation (GEADA), a novel framework designed to tackle the single-domain generalization challenge in image classification. The framework consists of two competing components: an augmentor to synthesize diverse yet semantically consistent augmentations, and a projector to learn domain-invariant representations from the augmented samples. The augmentor leverages a generative network for style transformations and an attribution-based cropping module for explainable geometric augmentations. We further incorporate theoretically-grounded contrastive loss functions, inspired by the geometric properties of unit hyperspheres, to promote the diversity of generated augmentations and the robustness of learned representations. Extensive experiments on multiple standard domain generalization benchmarks demonstrate the effectiveness of our approach against domain shifts.

Under non-additive probabilities, cluster points of the empirical average have been proved to quasi-surely fall into the interval constructed by either the lower and upper expectations or the lower and upper Choquet expectations. In this paper, based on the initiated notion of independence, we obtain a different Marcinkiewicz-Zygmund type strong law of large numbers. Then the Kolmogorov type strong law of large numbers can be derived from it directly, stating that the closed interval between the lower and upper expectations is the smallest one that covers cluster points of the empirical average quasi-surely.

In this article, we propose a novel variable screening method for linear models named as conditional screening via ordinary least squares projection (COLP). COLP can take advantage of prior knowledge concerning certain active predictors by eliminating the adverse impact of their coefficients in the estimation of remaining ones and thus significantly enhance the screening accuracy. We prove its sure-screening property under reasonable assumptions and demonstrate its utility in an application to a leukemia dataset. Moreover, based on the conditional approach, we introduce an iterative algorithm named as forward screening via ordinary least squares projection (FOLP), which not only could exploit the prior information more effectively, but also has promising performance when no prior knowledge is available using a data-driven conditioning set. Extensive simulation studies are carried out to demonstrate the competence of both proposed methods.

In this paper, the notion of negatively associated random variables is introduced under nonlinear probabilities, which is weaker than some existing conditions of independence.Within this framework, a law of the logarithm for arrays of rowwise negatively associated random variables is established by investigating their convergence properties.In addition, a weak law of the logarithm with the convergence of capacity is obtained.

受Peng-中心极限定理的启发,本文主要应用G-正态分布的概念,放宽Peng-中心极限定理的条件,在次线性期望下得到形式更为一般的中心极限定理.首先,将均值条件E[Xn]=E[Xn]=0放宽为|E[Xn]|+|E[Xn]|=O(1/n);其次,应用随机变量截断的方法,放宽随机变量的2阶矩与2+δ阶矩条件;最后,将该定理的Peng-独立性条件进行放宽,得到卷积独立随机变量的中心极限定理.

In this paper, the authors generalize the concept of asymptotically almost negatively associated random variables from the classic probability space to the upper ex- pectation space. Within the framework, the authors prove some different types of Rosen- thal’s inequalities for sub-additive expectations. Finally, the authors prove a strong law of large numbers as the application of Rosenthal’s inequalities.

本文在Peng建立的次线性期望空间下证明了Bernstein不等式, Kolmogorov不等式以及Rademacher不等式.进一步,本文分别应用Bernstein不等式、Kolmogorov不等式以及Rademacher不等式对次线性期望空间下随机变量列的拟必然收敛性质进行了深入研究,并得到了相应的强收敛定理.

Fan and Lv (2008) proposed the path-breaking theory of sure independencescreening (SIS) and an iterative algorithm (ISIS) to effectively reduce thepredictor dimension for further variable selection approaches. Fan et al.(2009) extended ISIS to generalized linear models and introduced the VanillaISIS (Van-ISIS) algorithm, allowing selected predictors to be screened out inupcoming iterations. The success of SIS depends on its sure screening property,which was obtained by Fan and Lv (2008) under the marginal correlationassumption. However, despite wide applications of ISIS and Van-ISIS in variousscientific fields, their sure screening properties have not been proved duringthe past decade. To fill this gap, we prove the sure screening properties ofthree different types of iterative algorithms for linear models without relyingon the marginal correlation assumption, where ISIS and Van-ISIS can be regardedas two special cases of them.

In this paper, on the sublinear expectation space, we establish a comparison theorem between independent and convolutionary random vectors, which states that the partial sums of those two sequences of random vectors are identically distributed. Under the sublinear framework, through the comparison theorem, several fundamental limit theorems for convolutionary random vectors are obtained, including the law of large numbers, the central limit theorem and the law of iterated logarithm.

In this paper, based on the initiation of the notion of negatively associated random variables under nonlinear probability, a strong limit theorem for weighted sums of random variables within the same frame is achieved without assumptions of independence and identical distribution, from which the Marcinkiewich-Zygmund type and Kolmogorov type strong laws of large numbers are derived. In addition, as applications of our results, Stranssen type invariance principles of negatively associated random variables and vertically independent random variables are proposed respectively.

The purpose of this paper is to establish a Fubini-like theorem of real-valued Choquet integrals for set-valued mappings in the frame of capacity theory. To this, we introduce the comonotonic random sets and slice-comonotonic set-valued mappings, which to make good use of the comonotonic additivity of Choquet integrals.

In this paper, we introduce the concepts of upper-lower set-valued probabilities and related upper-lower expectations for random variables. With a new concept of independence for random variables, we show a strong law of large numbers for upper-lower set-valued probabilities. Furthermore, we extend those concepts and theorem to the case of fuzzy-set.