Stability and error analysis of one-leg methods for nonlinear delay differential equations

AbstractThis paper is concerned with the numerical solution of delay differential equations (DDEs). We focus on the stability behaviour and error analysis of one-leg methods with respect to nonlinear DDEs. The new concepts of GR-stability, GAR-stability and weak GAR-stability are introduced. It is proved that a strongly A-stable one-leg method with linear interpolation is GAR-stable, and that an A-stable one-leg method with linear interpolation is GR-stable, weakly GAR-stable and D-convergent of order s, if it is consistent of order s in the classical sense

Statistical Guarantees of Generative Adversarial Networks for Distribution Estimation

Generative Adversarial Networks (GANs) have achieved great success in unsupervised learning. Despite the remarkable empirical performance, there are limited theoretical understandings on the statistical properties of GANs. This paper provides statistical guarantees of GANs for the estimation of data distributions which have densities in a H\"{o}lder space. Our main result shows that, if the generator and discriminator network architectures are properly chosen (universally for all distributions with H\"{o}lder densities), GANs are consistent estimators of the data distributions under strong discrepancy metrics, such as the Wasserstein distance. To our best knowledge, this is the first statistical theory of GANs for H\"{o}lder densities. In comparison with existing works, our theory requires minimum assumptions on data distributions. Our generator and discriminator networks utilize general weight matrices and the non-invertible ReLU activation function, while many existing works only apply to invertible weight matrices and invertible activation functions. In our analysis, we decompose the error into a statistical error and an approximation error by a new oracle inequality, which may be of independent interest

A Closer Look at Few-Shot 3D Point Cloud Classification

In recent years, research on few-shot learning (FSL) has been fast-growing in the 2D image domain due to the less requirement for labeled training data and greater generalization for novel classes. However, its application in 3D point cloud data is relatively under-explored. Not only need to distinguish unseen classes as in the 2D domain, 3D FSL is more challenging in terms of irregular structures, subtle inter-class differences, and high intra-class variances {when trained on a low number of data.} Moreover, different architectures and learning algorithms make it difficult to study the effectiveness of existing 2D FSL algorithms when migrating to the 3D domain. In this work, for the first time, we perform systematic and extensive investigations of directly applying recent 2D FSL works to 3D point cloud related backbone networks and thus suggest a strong learning baseline for few-shot 3D point cloud classification. Furthermore, we propose a new network, Point-cloud Correlation Interaction (PCIA), with three novel plug-and-play components called Salient-Part Fusion (SPF) module, Self-Channel Interaction Plus (SCI+) module, and Cross-Instance Fusion Plus (CIF+) module to obtain more representative embeddings and improve the feature distinction. These modules can be inserted into most FSL algorithms with minor changes and significantly improve the performance. Experimental results on three benchmark datasets, ModelNet40-FS, ShapeNet70-FS, and ScanObjectNN-FS, demonstrate that our method achieves state-of-the-art performance for the 3D FSL task. Code and datasets are available at https://github.com/cgye96/A_Closer_Look_At_3DFSL.Comment: Accepted by IJCV 202

Regression analysis of longitudinal data with mixed synchronous and asynchronous longitudinal covariates

In linear models, omitting a covariate that is orthogonal to covariates in the model does not result in biased coefficient estimation. This in general does not hold for longitudinal data, where additional assumptions are needed to get unbiased coefficient estimation in addition to the orthogonality between omitted longitudinal covariates and longitudinal covariates in the model. We propose methods to mitigate the omitted variable bias under weaker assumptions. A two-step estimation procedure is proposed for inference about the asynchronous longitudinal covariates, when such covariates are observed. For mixed synchronous and asynchronous longitudinal covariates, we get parametric rate of convergence for the coefficient estimation of the synchronous longitudinal covariates by the two-step method. Extensive simulation studies provide numerical support for the theoretical findings. We illustrate the performance of our method on dataset from the Alzheimers Disease Neuroimaging Initiative study

Combat molten aluminum corrosion of AISI H13 steel by lowtemperature liquid nitrocarburizing

Possibility of improving the resistance of AISI H13 steel to molten aluminum corrosion by liquid nitrocarburizing (LNC) was explored. The effects of the LNC parameters in terms of temperatures (703/723/743K) and soaking time (4/8/12h) on phase transformation, microstructure, and resistance to molten aluminum were fully studied. The surface phase compositions and the cross-sectional phase distribution of the LNC treated specimens were studied by implementable X-ray diffraction analysis. Microstructure, element distribution, microhardness, and the kinetics of the nitrocarburized case formation were fully researched. Immersion test of corrosion resistance to molten aluminum was carried out at 1023K for 30min. It is observed that an oxide layer can be produced on the top of the nitrocarburized case during LNC treatment, which cannot be regularly produced by other nitriding methods. The nitrocarburized case consists of a compound layer, a diffusion layer, and a transition layer. The growth of the nitrocarburized case is proportional to the squared treatment time and follows the Arrhenius law for the treatment temperature. The activation energy is estimated to be 195.4 kJ·mol−1. While the nitrocarburized case provided limited resistance to molten aluminum, the oxide layer formed on the top of the nitrocarburized case conferred significantly improved molten aluminum corrosion resistance, especially a duplex oxide layer produced at 743

Visual Anomaly Detection in Event Sequence Data

Anomaly detection is a common analytical task that aims to identify rare cases that differ from the typical cases that make up the majority of a dataset. When applied to the analysis of event sequence data, the task of anomaly detection can be complex because the sequential and temporal nature of such data results in diverse definitions and flexible forms of anomalies. This, in turn, increases the difficulty in interpreting detected anomalies. In this paper, we propose an unsupervised anomaly detection algorithm based on Variational AutoEncoders (VAE) to estimate underlying normal progressions for each given sequence represented as occurrence probabilities of events along the sequence progression. Events in violation of their occurrence probability are identified as abnormal. We also introduce a visualization system, EventThread3, to support interactive exploration and interpretations of anomalies within the context of normal sequence progressions in the dataset through comprehensive one-to-many sequence comparison. Finally, we quantitatively evaluate the performance of our anomaly detection algorithm and demonstrate the effectiveness of our system through a case study

LL3DA: Visual Interactive Instruction Tuning for Omni-3D Understanding, Reasoning, and Planning

Recent advances in Large Multimodal Models (LMM) have made it possible for various applications in human-machine interactions. However, developing LMMs that can comprehend, reason, and plan in complex and diverse 3D environments remains a challenging topic, especially considering the demand for understanding permutation-invariant point cloud 3D representations of the 3D scene. Existing works seek help from multi-view images, and project 2D features to 3D space as 3D scene representations. This, however, leads to huge computational overhead and performance degradation. In this paper, we present LL3DA, a Large Language 3D Assistant that takes point cloud as direct input and respond to both textual-instructions and visual-prompts. This help LMMs better comprehend human interactions and further help to remove the ambiguities in cluttered 3D scenes. Experiments show that LL3DA achieves remarkable results, and surpasses various 3D vision-language models on both 3D Dense Captioning and 3D Question Answering.Comment: Project Page: https://ll3da.github.io

Vote2Cap-DETR++: Decoupling Localization and Describing for End-to-End 3D Dense Captioning

3D dense captioning requires a model to translate its understanding of an input 3D scene into several captions associated with different object regions. Existing methods adopt a sophisticated "detect-then-describe" pipeline, which builds explicit relation modules upon a 3D detector with numerous hand-crafted components. While these methods have achieved initial success, the cascade pipeline tends to accumulate errors because of duplicated and inaccurate box estimations and messy 3D scenes. In this paper, we first propose Vote2Cap-DETR, a simple-yet-effective transformer framework that decouples the decoding process of caption generation and object localization through parallel decoding. Moreover, we argue that object localization and description generation require different levels of scene understanding, which could be challenging for a shared set of queries to capture. To this end, we propose an advanced version, Vote2Cap-DETR++, which decouples the queries into localization and caption queries to capture task-specific features. Additionally, we introduce the iterative spatial refinement strategy to vote queries for faster convergence and better localization performance. We also insert additional spatial information to the caption head for more accurate descriptions. Without bells and whistles, extensive experiments on two commonly used datasets, ScanRefer and Nr3D, demonstrate Vote2Cap-DETR and Vote2Cap-DETR++ surpass conventional "detect-then-describe" methods by a large margin. Codes will be made available at https://github.com/ch3cook-fdu/Vote2Cap-DETR