Prior studies [18, 12] show that GAN often experiences generation failures with severely degraded generation performance when only limited training data is available. Specifically, with limited training data, the discriminator tends to discriminate via meaningless shortcuts by merely focusing on easy-to-discriminate image locations and spectra instead of holistic understanding of images. This can be viewed clearly in Figure 1, where the Gini Coefficient [4] of discriminator's spatial attentions increases quickly along the training iteration (when only limited training data is available). Note that the Gini coefficient [4] is negatively correlated with equality, i.e., the discriminator will pay more unevenly distributed attention to each spatial location while the Gini coefficient increases from '0' to '1'. For image generation with GAN, the large Gini coefficient (of discriminator's spatial attentions) thus means that the discriminator starts to focus on certain spatial locations (easy to discriminate) while ignoring other spatial locations (hard to discriminate), ultimately leading to an over-confident discriminator and training collapse. In another word, the Gini coefficient [4] of '0' expresses perfect equality where all values are the same (i.e., where the discriminator pays the same attention to every spatial location) while '1' expresses maximal inequality among values (i.e., the discriminator focuses on only one location while all others are ignored).

In uncertainty quantification, evaluating sensitivity measures under specific conditions (i.e., conditional Sobol' indices) is essential for systems with parameterized responses, such as spatial fields or varying operating conditions. Traditional approaches often rely on point-wise modeling, which is computationally expensive and may lack consistency across the parameter space. This paper demonstrates that for a pre-trained global Polynomial Chaos Expansion (PCE) model, the analytical conditional Sobol' indices are inherently embedded within its basis functions. By leveraging the tensor-product property of PCE bases, we reformulate the global expansion into a set of analytical coefficient fields that depend on the conditioning variables. Based on the preservation of orthogonality under conditional probability measures, we derive closed-form expressions for conditional variances and Sobol' indices. This framework bypasses the need for repetitive modeling or additional sampling, transforming conditional sensitivity analysis into a purely algebraic post-processing step. Numerical benchmarks indicate that the proposed method ensures physical coherence and offers superior numerical robustness and computational efficiency compared to conventional point-wise approaches.

Recently, attention-based networks, such as the Vision Transformer, have also become popular.

Wefirst implement global augmentation approaches including random rotation and random scaling on two LiDAR scans separately and thenconcatenate themfortraining. The more copies the better segmentation performance as shown in ' 1, 2, 3' in the table, which indicates the effectiveness of the approach in enriching data distribution. In this section, we conducted experiments to analyze how PolarMix benefits LiDAR point cloud learning. As a comparison, PolarMix is more robust to the instance spatial location without much performance drop. PolarMix improves the robustness of the baseline clearly with respect to the angular variations of instances (i.e.

This is a challenging inference task given the need to reason beyond the local appearance of hands. The lack of training annotations indicating which object or parts of an object the hand is in contact with further complicates the task.