Principal Component Analysis (PCA) is a widely utilized technique for dimensionality reduction; however, its inherent lack of interpretability-stemming from dense linear combinations of all feature-limits its applicability in many domains. In this paper, we propose a novel sparse PCA algorithm that imposes sparsity through a smooth L1 penalty and leverages a Hamiltonian formulation solved via geometric integration techniques. Specifically, we implement two distinct numerical methods-one based on the Proximal Gradient (ISTA) approach and another employing a leapfrog (fourth-order Runge-Kutta) scheme-to minimize the energy function that balances variance maximization with sparsity enforcement. To extract a subset of sparse principal components, we further incorporate a deflation technique and subsequently transform the original high-dimensional face data into a lower-dimensional feature space. Experimental evaluations on a face recognition dataset-using both k-nearest neighbor and kernel ridge regression classifiers-demonstrate that the proposed sparse PCA methods consistently achieve higher classification accuracy than conventional PCA. Future research will extend this framework to integrate sparse PCA with modern deep learning architectures for multimodal recognition tasks.

We investigate the problem of learning a Single Index Model (SIM)- a popular model for studying the ability of neural networks to learn features - from anisotropic Gaussian inputs by training a neuron using vanilla Stochastic Gradient Descent (SGD). While the isotropic case has been extensively studied, the anisotropic case has received less attention and the impact of the covariance matrix on the learning dynamics remains unclear. For instance, Mousavi-Hosseini et al. (2023b) proposed a spherical SGD that requires a separate estimation of the data covariance matrix, thereby oversimplifying the influence of covariance. In this study, we analyze the learning dynamics of vanilla SGD under the SIM with anisotropic input data, demonstrating that vanilla SGD automatically adapts to the data's covariance structure. Leveraging these results, we derive upper and lower bounds on the sample complexity using a notion of effective dimension that is determined by the structure of the covariance matrix instead of the input data dimension.

Understanding how the brain's complex nonlinear dynamics give rise to adaptive cognition and behavior is a central challenge in neuroscience. These dynamics exhibit scale-free and multifractal properties, influencing the reconfiguration of neural networks. However, conventional neuroimaging models are constrained by linear and stationary assumptions, limiting their ability to capture these processes. Transformer-based architectures, known for capturing long-range dependencies, align well with the brain's hierarchical and temporal organization. We introduce Multi-Band Brain Net (MBBN), a transformer-based framework that models frequency-specific spatiotemporal brain dynamics from fMRI by integrating scale-free network principles with frequency-resolved multi-band self-attention. Trained on three large-scale neuroimaging cohorts (UK Biobank, ABCD, ABIDE) totaling 45,951 individuals, MBBN reveals previously undetectable frequency-dependent network interactions, shedding light on connectivity disruptions in psychiatric conditions (ADHD, ASD, depression). This validation shows robust generalizability and highlights core neural principles conserved across populations. MBBN achieves up to 30.59% higher predictive accuracy than state-of-the-art methods, demonstrating the advantage of frequency-informed spatiotemporal modeling in capturing latent neural computations. MBBN's interpretability uncovers novel frequency-specific biomarkers for neurodevelopmental disorders, providing insights into the hierarchical organization of brain function. By offering an interpretable framework for spatiotemporal learning, MBBN provides insights into how neural computations underpin cognitive function and psychiatric vulnerability, with implications for brain decoding, cognitive neuroscience, and precision psychiatry.

Federated learning (FL) has emerged as a promising paradigm in machine learning, enabling collaborative model training across decentralized devices without the need for raw data sharing. In FL, a global model is trained iteratively on local datasets residing on individual devices, each contributing to the model's improvement. However, the heterogeneous nature of these local datasets, stemming from diverse user behaviours, device capabilities, and data distributions, poses a significant challenge. The inherent heterogeneity in federated learning gives rise to various issues, including model performance discrepancies, convergence challenges, and potential privacy concerns. As the global model progresses through rounds of training, the disparities in local data quality and quantity can impede the overall effectiveness of federated learning systems. Moreover, maintaining fairness and privacy across diverse user groups becomes a paramount concern. To address this issue, this paper introduces a novel FL framework, ClusterGuardFL, that employs dissimilarity scores, k-means clustering, and reconciliation confidence scores to dynamically assign weights to client updates. The dissimilarity scores between global and local models guide the formation of clusters, with cluster size influencing the weight allocation. Within each cluster, a reconciliation confidence score is calculated for individual data points, and a softmax layer generates customized weights for clients. These weights are utilized in the aggregation process, enhancing the model's robustness and privacy. Experimental results demonstrate the efficacy of the proposed approach in achieving improved model performance in diverse datasets.

The density ratio is an important metric for evaluating the relative likelihood of two probability distributions, with extensive applications in statistics and machine learning. However, existing estimation theories for density ratios often depend on stringent regularity conditions, mainly focusing on density ratio functions with bounded domains and ranges. In this paper, we study density ratio estimators using loss functions based on least squares and logistic regression. We establish upper bounds on estimation errors with standard minimax optimal rates, up to logarithmic factors. Our results accommodate density ratio functions with unbounded domains and ranges. We apply our results to nonparametric regression and conditional flow models under covariate shift and identify the tail properties of the density ratio as crucial for error control across domains affected by covariate shift. We provide sufficient conditions under which loss correction is unnecessary and demonstrate effective generalization capabilities of a source estimator to any suitable target domain. Our simulation experiments support these theoretical findings, indicating that the source estimator can outperform those derived from loss correction methods, even when the true density ratio is known.

The main objective of fair statistical modeling and machine learning is to minimize or eliminate biases that may arise from the data or the model itself, ensuring that predictions and decisions are not unjustly influenced by sensitive attributes such as race, gender, age, or other protected characteristics. In this paper, we introduce a Fair Sufficient Representation Learning (FSRL) method that balances sufficiency and fairness. Sufficiency ensures that the representation should capture all necessary information about the target variables, while fairness requires that the learned representation remains independent of sensitive attributes. FSRL is based on a convex combination of an objective function for learning a sufficient representation and an objective function that ensures fairness. Our approach manages fairness and sufficiency at the representation level, offering a novel perspective on fair representation learning. We implement this method using distance covariance, which is effective for characterizing independence between random variables. We establish the convergence properties of the learned representations. Experiments conducted on healthcase and text datasets with diverse structures demonstrate that FSRL achieves a superior trade-off between fairness and accuracy compared to existing approaches.

Real-world event sequences are often generated by different temporal point processes (TPPs) and thus have clustering structures. Nonetheless, in the modeling and prediction of event sequences, most existing TPPs ignore the inherent clustering structures of the event sequences, leading to the models with unsatisfactory interpretability. In this study, we learn structure-enhanced TPPs with the help of Gromov-Wasserstein (GW) regularization, which imposes clustering structures on the sequence-level embeddings of the TPPs in the maximum likelihood estimation framework.In the training phase, the proposed method leverages a nonparametric TPP kernel to regularize the similarity matrix derived based on the sequence embeddings. In large-scale applications, we sample the kernel matrix and implement the regularization as a Gromov-Wasserstein (GW) discrepancy term, which achieves a trade-off between regularity and computational efficiency.The TPPs learned through this method result in clustered sequence embeddings and demonstrate competitive predictive and clustering performance, significantly improving the model interpretability without compromising prediction accuracy.

Adapting pre-trained deep learning models to new and unknown environments is a difficult challenge in underwater acoustic localization. We show that although pre-trained models have performance that suffers from mismatch between the training and test data, they generally exhibit a higher ``implied uncertainty'' in environments where there is more mismatch. Leveraging this notion of implied uncertainty, we partition the test samples into more certain and less certain sets, and implement an estimation method using the certain samples to improve the labeling for uncertain samples, which helps to adapt the model. We use an efficient method to quantify model prediction uncertainty, and an innovative approach to adapt a pre-trained model to unseen underwater environments at test time. This eliminates the need for labeled data from the target environment or the original training data. This adaptation is enhanced by integrating an independent estimate based on the received signal energy. We validate the approach extensively using real experimental data, as well as synthetic data consisting of model-generated signals with real ocean noise. The results demonstrate significant improvements in model prediction accuracy, underscoring the potential of the method to enhance underwater acoustic localization in diverse, noisy, and unknown environments.

Motivation: In recent years, protein function prediction has broken through the bottleneck of sequence features, significantly improving prediction accuracy using high-precision protein structures predicted by AlphaFold2. While single-species protein function prediction methods have achieved remarkable success, multi-species protein function prediction methods are still in the stage of using PPI networks and sequence features. Providing effective cross-species label propagation for species with sparse protein annotations remains a challenging issue. To address this problem, we propose the MSNGO model, which integrates structural features and network propagation methods. Our validation shows that using structural features can significantly improve the accuracy of multi-species protein function prediction. Results: We employ graph representation learning techniques to extract amino acid representations from protein structure contact maps and train a structural model using a graph convolution pooling module to derive protein-level structural features. After incorporating the sequence features from ESM-2, we apply a network propagation algorithm to aggregate information and update node representations within a heterogeneous network. The results demonstrate that MSNGO outperforms previous multi-species protein function prediction methods that rely on sequence features and PPI networks. Availability: https://github.com/blingbell/MSNGO.

Likelihood-free approaches are appealing for performing inference on complex dependence models, either because it is not possible to formulate a likelihood function, or its evaluation is very computationally costly. This is the case for several models available in the multivariate extremes literature, particularly for the most flexible tail models, including those that interpolate between the two key dependence classes of `asymptotic dependence' and `asymptotic independence'. We focus on approaches that leverage neural networks to approximate Bayes estimators. In particular, we explore the properties of neural Bayes estimators for parameter inference for several flexible but computationally expensive models to fit, with a view to aiding their routine implementation. Owing to the absence of likelihood evaluation in the inference procedure, classical information criteria such as the Bayesian information criterion cannot be used to select the most appropriate model. Instead, we propose using neural networks as neural Bayes classifiers for model selection. Our goal is to provide a toolbox for simple, fast fitting and comparison of complex extreme-value dependence models, where the best model is selected for a given data set and its parameters subsequently estimated using neural Bayes estimation. We apply our classifiers and estimators to analyse the pairwise extremal behaviour of changes in horizontal geomagnetic field fluctuations at three different locations.