Scattering for defocusing mass sub-critical NLS

In this paper, we consider the $L_x^2$-scattering of defocusing mass sub-critical nonlinear Schr\"odinger equations with low weighted initial condition. It is known that the scattering holds with $\mathcal{F} H^1$-data, while the continuity of inverse wave operator breaks down with $L^2$-data. Moreover, for large $\mathcal{F} H^s$-data with $s<1$, there only exists the wave operator result, but scattering results are lacking. Our subject is to study the scattering in low weights space. Our results are divided into two parts. Our first result presents a systematic study on the scattering on $\mathcal{F} H^s$ for certain $s<1$, without any restrictions on smallness or radial symmetry. This extends the previous results to spaces with lower weights. Our second result is the almost sure scattering on $L^2$ by introducing a ``narrowed'' Wiener randomization in physical space. For mass subcritical NLS when $d\ge 2$, this result represents the first scattering result without imposing any conditions related to smallness, radial symmetry, or weighted properties on the initial data.Comment: 86 page

Perturbation-Assisted Sample Synthesis: A Novel Approach for Uncertainty Quantification

This paper introduces a novel generator called Perturbation-Assisted Sample Synthesis (PASS), designed for drawing reliable conclusions from complex data, especially when using advanced modeling techniques like deep neural networks. PASS utilizes perturbation to generate synthetic data that closely mirrors the distribution of raw data, encompassing numerical and unstructured data types such as gene expression, images, and text. By estimating the data-generating distribution and leveraging large pre-trained generative models, PASS enhances estimation accuracy, providing an estimated distribution of any statistic through Monte Carlo experiments. Building on PASS, we propose a generative inference framework called Perturbation-Assisted Inference (PAI), which offers a statistical guarantee of validity. In pivotal inference, PAI enables accurate conclusions without knowing a pivotal's distribution as in simulations, even with limited data. In non-pivotal situations, we train PASS using an independent holdout sample, resulting in credible conclusions. To showcase PAI's capability in tackling complex problems, we highlight its applications in three domains: image synthesis inference, sentiment word inference, and multimodal inference via stable diffusion

Hot Mums. Motherhood and Feminism in Post-socialist China

The term “hot mum” (La Ma, 辣妈) has become popular in the Chinese media in the 21st century, being regarded as a “feminist” image of the modern mother, as it breaks with the stereotype of the traditional Chinese mother. Departing from a historical framework of motherhood and feminism, as well as western theories of subjectification and individualization, the article explores the discourses of hot mums in contemporary China. Based on an analysis of more than eight hundred articles in a Chinese database, this article explores the impacts of the image of the hot mum upon practices of motherhood among contemporary Chinese women. The findings show that the notion of the hot mum has been transformed into the concept of “all-around hot mums” who take care of both their families and their careers. It is argued that this process has not changed power relations between men and women, nor the roles of father and mother. Commercial and market aspects have turned hot mums from an initial expression of women’s subjectivity with particular maternal values into subjects of consumerism. The hot mum discourse is apparently contributing to the oppression rather than empowerment of Chinese women, let alone their increased sense of individuality

STL-SGD: Speeding Up Local SGD with Stagewise Communication Period

Distributed parallel stochastic gradient descent algorithms are workhorses for large scale machine learning tasks. Among them, local stochastic gradient descent (Local SGD) has attracted significant attention due to its low communication complexity. Previous studies prove that the communication complexity of Local SGD with a fixed or an adaptive communication period is in the order of $O (N^{\frac{3}{2}} T^{\frac{1}{2}})$ and $O (N^{\frac{3}{4}} T^{\frac{3}{4}})$ when the data distributions on clients are identical (IID) or otherwise (Non-IID), where $N$ is the number of clients and $T$ is the number of iterations. In this paper, to accelerate the convergence by reducing the communication complexity, we propose \textit{ST}agewise \textit{L}ocal \textit{SGD} (STL-SGD), which increases the communication period gradually along with decreasing learning rate. We prove that STL-SGD can keep the same convergence rate and linear speedup as mini-batch SGD. In addition, as the benefit of increasing the communication period, when the objective is strongly convex or satisfies the Polyak-\L ojasiewicz condition, the communication complexity of STL-SGD is $O (N \log{T})$ and $O (N^{\frac{1}{2}} T^{\frac{1}{2}})$ for the IID case and the Non-IID case respectively, achieving significant improvements over Local SGD. Experiments on both convex and non-convex problems demonstrate the superior performance of STL-SGD.Comment: Accepted by AAAI202