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Representation learning plays a crucial role in automated feature selection, particularly in the context of high-dimensional data, where non-parametric methods often struggle. In this study, we focus on supervised learning scenarios where the pertinent information resides within a lower-dimensional linear subspace of the data, namely the multi-index model. If this subspace were known, it would greatly enhance prediction, computation, and interpretation. To address this challenge, we propose a novel method for linear feature learning with non-parametric prediction, which simultaneously estimates the prediction function and the linear subspace. Our approach employs empirical risk minimisation, augmented with a penalty on function derivatives, ensuring versatility. Leveraging the orthogonality and rotation invariance properties of Hermite polynomials, we introduce our estimator, named RegFeaL. By utilising alternative minimisation, we iteratively rotate the data to improve alignment with leading directions and accurately estimate the relevant dimension in practical settings. We establish that our method yields a consistent estimator of the prediction function with explicit rates. Additionally, we provide empirical results demonstrating the performance of RegFeaL in various experiments.

We present a meshless Schwarz-type non-overlapping domain decomposition method based on artificial neural networks for solving forward and inverse problems involving partial differential equations (PDEs). To ensure the consistency of solutions across neighboring subdomains, we adopt a generalized Robin-type interface condition, assigning unique Robin parameters to each subdomain. These subdomain-specific Robin parameters are learned to minimize the mismatch on the Robin interface condition, facilitating efficient information exchange during training. Our method is applicable to both the Laplace's and Helmholtz equations. It represents local solutions by an independent neural network model which is trained to minimize the loss on the governing PDE while strictly enforcing boundary and interface conditions through an augmented Lagrangian formalism. A key strength of our method lies in its ability to learn a Robin parameter for each subdomain, thereby enhancing information exchange with its neighboring subdomains. We observe that the learned Robin parameters adapt to the local behavior of the solution, domain partitioning and subdomain location relative to the overall domain. Extensive experiments on forward and inverse problems, including one-way and two-way decompositions with crosspoints, demonstrate the versatility and performance of our proposed approach.

Various goodness-of-fit tests are designed based on the so-called information matrix equivalence: if the assumed model is correctly specified, two information matrices that are derived from the likelihood function are equivalent. In the literature, this principle has been established for the likelihood function with fully observed data, but it has not been verified under the likelihood for censored data. In this manuscript, we prove the information matrix equivalence in the framework of semiparametric copula models for multivariate censored survival data. Based on this equivalence, we propose an information ratio (IR) test for the specification of the copula function. The IR statistic is constructed via comparing consistent estimates of the two information matrices. We derive the asymptotic distribution of the IR statistic and propose a parametric bootstrap procedure for the finite-sample $P$-value calculation. The performance of the IR test is investigated via a simulation study and a real data example.

The goal of 3D mesh watermarking is to embed the message in 3D meshes that can withstand various attacks imperceptibly and reconstruct the message accurately from watermarked meshes. Traditional methods are less robust against attacks. Recent DNN-based methods either introduce excessive distortions or fail to embed the watermark without the help of texture information. However, embedding the watermark in textures is insecure because replacing the texture image can completely remove the watermark. In this paper, we propose a robust deep 3D mesh watermarking WM-NET, which leverages attention-based convolutions in watermarking tasks to embed binary messages in vertex distributions without texture assistance. Furthermore, our WM-NET exploits the property that simplified meshes inherit similar relations from the original ones, where the relation is the offset vector directed from one vertex to its neighbor. By doing so, our method can be trained on simplified meshes(limited data) but remains effective on large-sized meshes (size adaptable) and unseen categories of meshes (geometry adaptable). Extensive experiments demonstrate our method brings 50% fewer distortions and 10% higher bit accuracy compared to previous work. Our watermark WM-NET is robust against various mesh attacks, e.g. Gauss, rotation, translation, scaling, and cropping.

Medical image segmentation is a crucial task that relies on the ability to accurately identify and isolate regions of interest in medical images. Thereby, generative approaches allow to capture the statistical properties of segmentation masks that are dependent on the respective structures. In this work we propose a conditional score-based generative modeling framework to represent the signed distance function (SDF) leading to an implicit distribution of segmentation masks. The advantage of leveraging the SDF is a more natural distortion when compared to that of binary masks. By learning the score function of the conditional distribution of SDFs we can accurately sample from the distribution of segmentation masks, allowing for the evaluation of statistical quantities. Thus, this probabilistic representation allows for the generation of uncertainty maps represented by the variance, which can aid in further analysis and enhance the predictive robustness. We qualitatively and quantitatively illustrate competitive performance of the proposed method on a public nuclei and gland segmentation data set, highlighting its potential utility in medical image segmentation applications.

Having efficient testing strategies is a core challenge that needs to be overcome for the release of automated driving. This necessitates clear requirements as well as suitable methods for testing. In this work, the requirements for perception modules are considered with respect to relevance. The concept of relevance currently remains insufficiently defined and specified. In this paper, we propose a novel methodology to overcome this challenge by exemplary application to collision safety in the highway domain. Using this general system and use case specification, a corresponding concept for relevance is derived. Irrelevant objects are thus defined as objects which do not limit the set of safe actions available to the ego vehicle under consideration of all uncertainties. As an initial step, the use case is decomposed into functional scenarios with respect to collision relevance. For each functional scenario, possible actions of both the ego vehicle and any other dynamic object are formalized as equations. This set of possible actions is constrained by traffic rules, yielding relevance criteria. As a result, we present a conservative estimation which dynamic objects are relevant for perception and need to be considered for a complete evaluation. The estimation provides requirements which are applicable for offline testing and validation of perception components. A visualization is presented for examples from the highD dataset, showing the plausibility of the results. Finally, a possibility for a future validation of the presented relevance concept is outlined.

Labeling of multivariate biomedical time series data is a laborious and expensive process. Self-supervised contrastive learning alleviates the need for large, labeled datasets through pretraining on unlabeled data. However, for multivariate time series data, the set of input channels often varies between applications, and most existing work does not allow for transfer between datasets with different sets of input channels. We propose learning one encoder to operate on all input channels individually. We then use a message passing neural network to extract a single representation across channels. We demonstrate the potential of this method by pretraining our model on a dataset with six EEG channels and then fine-tuning it on a dataset with two different EEG channels. We compare models with and without the message passing neural network across different contrastive loss functions. We show that our method, combined with the TS2Vec loss, outperforms all other methods in most settings.

This paper is devoted to the statistical and numerical properties of the geometric median, and its applications to the problem of robust mean estimation via the median of means principle. Our main theoretical results include (a) an upper bound for the distance between the mean and the median for general absolutely continuous distributions in R^d, and examples of specific classes of distributions for which these bounds do not depend on the ambient dimension d; (b) exponential deviation inequalities for the distance between the sample and the population versions of the geometric median, which again depend only on the trace-type quantities and not on the ambient dimension. As a corollary, we deduce improved bounds for the (geometric) median of means estimator that hold for large classes of heavy-tailed distributions. Finally, we address the error of numerical approximation, which is an important practical aspect of any statistical estimation procedure. We demonstrate that the objective function minimized by the geometric median satisfies a "local quadratic growth" condition that allows one to translate suboptimality bounds for the objective function to the corresponding bounds for the numerical approximation to the median itself, and propose a simple stopping rule applicable to any optimization method which yields explicit error guarantees. We conclude with the numerical experiments including the application to estimation of mean values of log-returns for S&P 500 data.

Over the past few years, the rapid development of deep learning technologies for computer vision has greatly promoted the performance of medical image segmentation (MedISeg). However, the recent MedISeg publications usually focus on presentations of the major contributions (e.g., network architectures, training strategies, and loss functions) while unwittingly ignoring some marginal implementation details (also known as "tricks"), leading to a potential problem of the unfair experimental result comparisons. In this paper, we collect a series of MedISeg tricks for different model implementation phases (i.e., pre-training model, data pre-processing, data augmentation, model implementation, model inference, and result post-processing), and experimentally explore the effectiveness of these tricks on the consistent baseline models. Compared to paper-driven surveys that only blandly focus on the advantages and limitation analyses of segmentation models, our work provides a large number of solid experiments and is more technically operable. With the extensive experimental results on both the representative 2D and 3D medical image datasets, we explicitly clarify the effect of these tricks. Moreover, based on the surveyed tricks, we also open-sourced a strong MedISeg repository, where each of its components has the advantage of plug-and-play. We believe that this milestone work not only completes a comprehensive and complementary survey of the state-of-the-art MedISeg approaches, but also offers a practical guide for addressing the future medical image processing challenges including but not limited to small dataset learning, class imbalance learning, multi-modality learning, and domain adaptation. The code has been released at: //github.com/hust-linyi/MedISeg

Modeling multivariate time series has long been a subject that has attracted researchers from a diverse range of fields including economics, finance, and traffic. A basic assumption behind multivariate time series forecasting is that its variables depend on one another but, upon looking closely, it is fair to say that existing methods fail to fully exploit latent spatial dependencies between pairs of variables. In recent years, meanwhile, graph neural networks (GNNs) have shown high capability in handling relational dependencies. GNNs require well-defined graph structures for information propagation which means they cannot be applied directly for multivariate time series where the dependencies are not known in advance. In this paper, we propose a general graph neural network framework designed specifically for multivariate time series data. Our approach automatically extracts the uni-directed relations among variables through a graph learning module, into which external knowledge like variable attributes can be easily integrated. A novel mix-hop propagation layer and a dilated inception layer are further proposed to capture the spatial and temporal dependencies within the time series. The graph learning, graph convolution, and temporal convolution modules are jointly learned in an end-to-end framework. Experimental results show that our proposed model outperforms the state-of-the-art baseline methods on 3 of 4 benchmark datasets and achieves on-par performance with other approaches on two traffic datasets which provide extra structural information.