We consider the problem of linear estimation, and establish an extension of the Gauss-Markov theorem, in which the bias operator is allowed to be non-zero but bounded with respect to a matrix norm of Schatten type. We derive simple and explicit formulas for the optimal estimator in the cases of Nuclear and Spectral norms (with the Frobenius case recovering ridge regression). Additionally, we analytically derive the generalization error in multiple random matrix ensembles, and compare with Ridge regression. Finally, we conduct an extensive simulation study, in which we show that the cross-validated Nuclear and Spectral regressors can outperform Ridge in several circumstances. Linear models are among the most used of all machine-learning models in applications to science and engineering. In addition to this practical interest, they are also of great theoretical interest. Indeed, the empirical successes of neural networks have proven somewhat at odds with traditional received wisdom from classical statistical learning theory(Belkin et al., 2019). This has begun to be reconciled by careful analysis of well-chosen linear "model systems" such as high-dimensional regression (Hastie et al., 2019), kernel ridge regression (Canatar et al., 2021; Jacot et al., 2020), or linear neural networks (Saxe et al., 2013), which can reproduce many qualitative properties of learning dynamics of neural networks. Thus, the careful study of linear models, especially in the limit of a large number of features and observations can prove highly valuable for qualitative understanding of non-linear models.

Typically, a graph (or network) is represented by an adjacency matrix A of size, where A(,) denotes the edge weight between the th and th vertices. Alternatively, the graph can be stored in an edgelist E of size 3, with the first two columns indicating the vertex indices of each edge and the last column representing the edge weight. Community detection, also known as vertex clustering or graph partitioning, is a fundamental problem in graph analysis [6, 8, 10, 13]. The primary objective is to identify natural groups of vertices where intra-group connections are stronger than inter-group connections. Over the years, various approaches have been proposed, including modularitybased methods [2, 22], spectral-based methods [15, 21], and likelihood-based techniques [1, 7], among others.

Many engineering applications rely on the evaluation of expensive, non-linear high-dimensional functions. In this paper, we propose the RONAALP algorithm (Reduced Order Nonlinear Approximation with Active Learning Procedure) to incrementally learn a fast and accurate reduced-order surrogate model of a target function on-the-fly as the application progresses. First, the combination of nonlinear auto-encoder, community clustering and radial basis function networks allows to learn an efficient and compact surrogate model with limited training data. Secondly, the active learning procedure overcome any extrapolation issue when evaluating the surrogate model outside of its initial training range during the online stage. This results in generalizable, fast and accurate reduced-order models of high-dimensional functions. The method is demonstrated on three direct numerical simulations of hypersonic flows in chemical nonequilibrium. Accurate simulations of these flows rely on detailed thermochemical gas models that dramatically increase the cost of such calculations. Using RONAALP to learn a reduced-order thermodynamic model surrogate on-the-fly, the cost of such simulation was reduced by up to 75% while maintaining an error of less than 10% on relevant quantities of interest.

This paper addresses a gap in the current state of the art by providing a solution for modeling causal relationships that evolve over time and occur at different time scales. Specifically, we introduce the multiscale non-stationary directed acyclic graph (MN-DAG), a framework for modeling multivariate time series data. Our contribution is twofold. Firstly, we expose a probabilistic generative model by leveraging results from spectral and causality theories. Our model allows sampling an MN-DAG according to user-specified priors on the time-dependence and multiscale properties of the causal graph. Secondly, we devise a Bayesian method named Multiscale Non-stationary Causal Structure Learner (MN-CASTLE) that uses stochastic variational inference to estimate MN-DAGs. The method also exploits information from the local partial correlation between time series over different time resolutions. The data generated from an MN-DAG reproduces well-known features of time series in different domains, such as volatility clustering and serial correlation. Additionally, we show the superior performance of MN-CASTLE on synthetic data with different multiscale and non-stationary properties compared to baseline models. Finally, we apply MN-CASTLE to identify the drivers of the natural gas prices in the US market. Causal relationships have strengthened during the COVID-19 outbreak and the Russian invasion of Ukraine, a fact that baseline methods fail to capture. MN-CASTLE identifies the causal impact of critical economic drivers on natural gas prices, such as seasonal factors, economic uncertainty, oil prices, and gas storage deviations.

Human migration and mobility drives major societal phenomena including epidemics, economies, innovation, and the diffusion of ideas. Although human mobility and migration have been heavily constrained by geographic distance throughout the history, advances and globalization are making other factors such as language and culture increasingly more important. Advances in neural embedding models, originally designed for natural language, provide an opportunity to tame this complexity and open new avenues for the study of migration. Here, we demonstrate the ability of the model word2vec to encode nuanced relationships between discrete locations from migration trajectories, producing an accurate, dense, continuous, and meaningful vector-space representation. The resulting representation provides a functional distance between locations, as well as a digital double that can be distributed, re-used, and itself interrogated to understand the many dimensions of migration. We show that the unique power of word2vec to encode migration patterns stems from its mathematical equivalence with the gravity model of mobility. Focusing on the case of scientific migration, we apply word2vec to a database of three million migration trajectories of scientists derived from the affiliations listed on their publication records. Using techniques that leverage its semantic structure, we demonstrate that embeddings can learn the rich structure that underpins scientific migration, such as cultural, linguistic, and prestige relationships at multiple levels of granularity. Our results provide a theoretical foundation and methodological framework for using neural embeddings to represent and understand migration both within and beyond science.

A realistic human kinematic model that satisfies anatomical constraints is essential for human-robot interaction, biomechanics and robot-assisted rehabilitation. Modeling realistic joint constraints, however, is challenging as human arm motion is constrained by joint limits, inter- and intra-joint dependencies, self-collisions, individual capabilities and muscular or neurological constraints which are difficult to represent. Hence, physicians and researchers have relied on simple box-constraints, ignoring important anatomical factors. In this paper, we propose a data-driven method to learn realistic anatomically constrained upper-limb range of motion (RoM) boundaries from motion capture data. This is achieved by fitting a one-class support vector machine to a dataset of upper-limb joint space exploration motions with an efficient hyper-parameter tuning scheme. Our approach outperforms similar works focused on valid RoM learning. Further, we propose an impairment index (II) metric that offers a quantitative assessment of capability/impairment when comparing healthy and impaired arms. We validate the metric on healthy subjects physically constrained to emulate hemiplegia and different disability levels as stroke patients.

Many countries such as Saudi Arabia, Morocco and Tunisia are among the top exporters and consumers of palm date fruits. Date fruit production plays a major role in the economies of the date fruit exporting countries. Date fruits are susceptible to disease just like any fruit and early detection and intervention can end up saving the produce. However, with the vast farming lands, it is nearly impossible for farmers to observe date trees on a frequent basis for early disease detection. In addition, even with human observation the process is prone to human error and increases the date fruit cost. With the recent advances in computer vision, machine learning, drone technology, and other technologies; an integrated solution can be proposed for the automatic detection of date fruit disease. In this paper, a hybrid features based method with the standard classifiers is proposed based on the extraction of L*a*b color features, statistical features, and Discrete Wavelet Transform (DWT) texture features for the early detection and classification of date fruit disease. A dataset was developed for this work consisting of 871 images divided into the following classes; Healthy date, Initial stage of disease, Malnourished date, and Parasite infected. The extracted features were input to common classifiers such as the Random Forest (RF), Multilayer Perceptron (MLP), Na\"ive Bayes (NB), and Fuzzy Decision Trees (FDT). The highest average accuracy was achieved when combining the L*a*b, Statistical, and DWT Features.

We propose Tree Variational Autoencoder (TreeVAE), a new generative hierarchical clustering model that learns a flexible tree-based posterior distribution over latent variables. TreeVAE hierarchically divides samples according to their intrinsic characteristics, shedding light on hidden structures in the data. It adapts its architecture to discover the optimal tree for encoding dependencies between latent variables. The proposed tree-based generative architecture enables lightweight conditional inference and improves generative performance by utilizing specialized leaf decoders. We show that TreeVAE uncovers underlying clusters in the data and finds meaningful hierarchical relations between the different groups on a variety of datasets, including real-world imaging data. We present empirically that Tree-VAE provides a more competitive log-likelihood lower bound than the sequential counterparts. Finally, due to its generative nature, TreeVAE is able to generate new samples from the discovered clusters via conditional sampling.

Contrastive learning, a dominant self-supervised technique, emphasizes similarity in representations between augmentations of the same input and dissimilarity for different ones. Although low contrastive loss often correlates with high classification accuracy, recent studies challenge this direct relationship, spotlighting the crucial role of inductive biases. We delve into these biases from a clustering viewpoint, noting that contrastive learning creates locally dense clusters, contrasting the globally dense clusters from supervised learning. To capture this discrepancy, we introduce the "RLD (Relative Local Density)" metric. While this cluster property can hinder linear classification accuracy, leveraging a Graph Convolutional Network (GCN) based classifier mitigates this, boosting accuracy and reducing parameter requirements. The code is available \href{https://github.com/xsgxlz/How-does-Contrastive-Learning-Organize-Images/tree/main}{here}.

Pre-trained Transformers are challenging human performances in many NLP tasks. The massive datasets used for pre-training seem to be the key to their success on existing tasks. In this paper, we explore how a range of pre-trained Natural Language Understanding models perform on definitely unseen sentences provided by classification tasks over a DarkNet corpus. Surprisingly, results show that syntactic and lexical neural networks perform on par with pre-trained Transformers even after fine-tuning. Only after what we call extreme domain adaptation, that is, retraining with the masked language model task on all the novel corpus, pre-trained Transformers reach their standard high results. This suggests that huge pre-training corpora may give Transformers unexpected help since they are exposed to many of the possible sentences.