Single-realization recovery of a random Schr\"odinger equation with unknown source and potential

In this paper, we study an inverse scattering problem associated with the stationary Schr\"odinger equation where both the potential and the source terms are unknown. The source term is assumed to be a generalised Gaussian random distribution of the microlocally isotropic type, whereas the potential function is assumed to be deterministic. The well-posedness of the forward scattering problem is first established in a proper sense. It is then proved that the rough strength of the random source can be uniquely recovered, independent of the unknown potential, by a single realisation of the passive scattering measurement. We develop novel techniques to completely remove a restrictive geometric condition in our earlier study [25], at an unobjectionable cost of requiring the unknown potential to be deterministic. The ergodicity is used to establish the single realization recovery, and the asymptotic arguments in our analysis are based on techniques from the theory of pseudo-differential operators and the stationary phase principle.Comment: 28 page

Dynamic nexus between transportation, economic growth and environmental degradation in China: Fresh insights from the QARDL approach

The continuous growth of the transport sector and the increase in transportation carbon emissions attract policymakers’ attention. It is of great importance to understand the determinants of pollution from transportation. This study explores the dynamic nexus between transportation, growth, and environmental degradation in China. The QARDL approach is used for the empirical investigation of data series from 1995 to 2018. The findings exposed mixed results in both the long and short run. The result for freight transportation only improves the environment at upper extreme quantiles, while the results are insignificant in the short run. The results show that passenger transportation reduces CO2 emissions at the lower bottom quantiles in the long run, while the results are significant at upper extreme quantiles in the short run. In the case of GDP, the results endorsed the EKC hypothesis in the long run, while in short-run dynamics, the results for GDP2 are found insignificant, which elaborates that China’s economic growth enhances the CO2 emissions. Besides, the quantile causality test showed a bi-directional causality between all variables. The findings of this study provide concrete evidence to the policymakers of China to strengthen the sustainable transportation system by promoting eco-friendly and energy-efficient modes of transportation

Generalization Beyond Feature Alignment: Concept Activation-Guided Contrastive Learning

Learning invariant representations via contrastive learning has seen state-of-the-art performance in domain generalization (DG). Despite such success, in this paper, we find that its core learning strategy -- feature alignment -- could heavily hinder model generalization. Drawing insights in neuron interpretability, we characterize this problem from a neuron activation view. Specifically, by treating feature elements as neuron activation states, we show that conventional alignment methods tend to deteriorate the diversity of learned invariant features, as they indiscriminately minimize all neuron activation differences. This instead ignores rich relations among neurons -- many of them often identify the same visual concepts despite differing activation patterns. With this finding, we present a simple yet effective approach, Concept Contrast (CoCo), which relaxes element-wise feature alignments by contrasting high-level concepts encoded in neurons. Our CoCo performs in a plug-and-play fashion, thus it can be integrated into any contrastive method in DG. We evaluate CoCo over four canonical contrastive methods, showing that CoCo promotes the diversity of feature representations and consistently improves model generalization capability. By decoupling this success through neuron coverage analysis, we further find that CoCo potentially invokes more meaningful neurons during training, thereby improving model learning

A fixed-point formula for Dirac operators on Lie groupoids

We study equivariant families of Dirac operators on the source fibers of a Lie groupoid with a closed space of units and equipped with an action of an auxiliary compact Lie group. We use the Getzler rescaling method to derive a fixed-point formula for the pairing of a trace with the K-theory class of such a family. For the pair groupoid of a closed manifold, our formula reduces to the standard fixed-point formula for the equivariant index of a Dirac operator. Further examples involve foliations and manifolds equipped with a normal crossing divisor.Comment: 50 page

Neuron Coverage-Guided Domain Generalization

This paper focuses on the domain generalization task where domain knowledge is unavailable, and even worse, only samples from a single domain can be utilized during training. Our motivation originates from the recent progresses in deep neural network (DNN) testing, which has shown that maximizing neuron coverage of DNN can help to explore possible defects of DNN (i.e., misclassification). More specifically, by treating the DNN as a program and each neuron as a functional point of the code, during the network training we aim to improve the generalization capability by maximizing the neuron coverage of DNN with the gradient similarity regularization between the original and augmented samples. As such, the decision behavior of the DNN is optimized, avoiding the arbitrary neurons that are deleterious for the unseen samples, and leading to the trained DNN that can be better generalized to out-of-distribution samples. Extensive studies on various domain generalization tasks based on both single and multiple domain(s) setting demonstrate the effectiveness of our proposed approach compared with state-of-the-art baseline methods. We also analyze our method by conducting visualization based on network dissection. The results further provide useful evidence on the rationality and effectiveness of our approach