Hypergraphs provide a superior modeling framework for representing complex multidimensional relationships in the context of real-world interactions that often occur in groups, overcoming the limitations of traditional homogeneous graphs. However, there have been few studies on hypergraphbased contrastive learning, and existing graph-based contrastive learning methods have not been able to fully exploit the highorder correlation information in hypergraphs. Here, we propose a Hypergraph Fine-grained contrastive learning (HyFi) method designed to exploit the complex high-dimensional information inherent in hypergraphs. While avoiding traditional graph augmentation methods that corrupt the hypergraph topology, the proposed method provides a simple and efficient learning augmentation function by adding noise to node features. Furthermore, we expands beyond the traditional dichotomous relationship between positive and negative samples in contrastive learning by introducing a new relationship of weak positives. It demonstrates the importance of fine-graining positive samples in contrastive learning. Therefore, HyFi is able to produce highquality embeddings, and outperforms both supervised and unsupervised baselines in average rank on node classification across 10 datasets. Our approach effectively exploits high-dimensional hypergraph information, shows significant improvement over existing graph-based contrastive learning methods, and is efficient in terms of training speed and GPU memory cost. The source code is available at https://github.com/Noverse0/HyFi.git.

Topic modeling often requires examining topics from multiple perspectives to uncover hidden patterns, especially in less explored areas. This paper presents an approach to address this need, utilizing weighted keywords from various aspects derived from a domain knowledge. The research method starts with standard topic modeling. Then, it adds a process consisting of four key steps. First, it defines keywords for each aspect. Second, it gives weights to these keywords based on their relevance. Third, it calculates relevance scores for aspect-weighted keywords and topic keywords to create aspect-topic models. Fourth, it uses these scores to tune relevant new documents. Finally, the generated topic models are interpreted and validated. The findings show that top-scoring documents are more likely to be about the same aspect of a topic. This highlights the model's effectiveness in finding the related documents to the aspects.

The research paper empirically investigates several machine learning algorithms to forecast stock prices depending on insider trading information. Insider trading offers special insights into market sentiment, pointing to upcoming changes in stock prices. This study examines the effectiveness of algorithms like decision trees, random forests, support vector machines (SVM) with different kernels, and K-Means Clustering using a dataset of Tesla stock transactions. Examining past data from April 2020 to March 2023, this study focuses on how well these algorithms identify trends and forecast stock price fluctuations. The paper uses Recursive Feature Elimination (RFE) and feature importance analysis to optimize the feature set and, hence, increase prediction accuracy. While it requires substantially greater processing time than other models, SVM with the Radial Basis Function (RBF) kernel displays the best accuracy. This paper highlights the trade-offs between accuracy and efficiency in machine learning models and proposes the possibility of pooling multiple data sources to raise prediction performance. The results of this paper aim to help financial analysts and investors in choosing strong algorithms to optimize investment strategies.

Bilevel optimization has emerged as a technique for addressing a wide range of machine learning problems that involve an outer objective implicitly determined by the minimizer of an inner problem. In this paper, we investigate the generalization properties for kernel bilevel optimization problems where the inner objective is optimized over a Reproducing Kernel Hilbert Space. This setting enables rich function approximation while providing a foundation for rigorous theoretical analysis. In this context, we establish novel generalization error bounds for the bilevel problem under finite-sample approximation. Our approach adopts a functional perspective, inspired by (Petrulionyte et al., 2024), and leverages tools from empirical process theory and maximal inequalities for degenerate $U$-processes to derive uniform error bounds. These generalization error estimates allow to characterize the statistical accuracy of gradient-based methods applied to the empirical discretization of the bilevel problem.

Model-based clustering techniques have been widely applied to various application areas, while most studies focus on canonical mixtures with unique component distribution form. However, this strict assumption is often hard to satisfy. In this paper, we consider the more flexible Copula-Based Mixture Models (CBMMs) for clustering, which allow heterogeneous component distributions composed by flexible choices of marginal and copula forms. More specifically, we propose an adaptation of the Generalized Iterative Conditional Estimation (GICE) algorithm to identify the CBMMs in an unsupervised manner, where the marginal and copula forms and their parameters are estimated iteratively. GICE is adapted from its original version developed for switching Markov model identification with the choice of realization time. Our CBMM-GICE clustering method is then tested on synthetic two-cluster data (N=2000 samples) with discussion of the factors impacting its convergence. Finally, it is compared to the Expectation Maximization identified mixture models with unique component form on the entire MNIST database (N=70000), and on real cardiac magnetic resonance data (N=276) to illustrate its value for imaging applications.

In this article we discuss an approach to doing this which one can call mathematical data science. In this paradigm, one studies mathematical objects collectively rather than individually, by creating datasets and doing machine learning experiments and interpretations. Broadly speaking, the field of data science is concerned with assembling, curating and analyzing large datasets, and developing methods which enable its users to not just answer predetermined questions about the data but to explore it, make simple descriptions and pictures, and arrive at novel insights. This certainly sounds promising as a tool for mathematical discovery! Mathematical data science is not new and has historically led to very important results. A famous example is the work of Birch and Swinnerton-Dyer leading to their conjecture [BSD65], based on computer generation of elliptic curves and linear regression analysis of the resulting data. However, the field really started to take off with the deep learning revolution and with the easy access to ML models provided by platforms such as Py-Torch and TensorFlow, and built into computer algebra systems such as Mathematica, Magma and SageMath.

This work addresses the task of generalized class discovery (GCD) in instance segmentation. The goal is to discover novel classes and obtain a model capable of segmenting instances of both known and novel categories, given labeled and unlabeled data. Since the real world contains numerous objects with long-tailed distributions, the instance distribution for each class is inherently imbalanced. To address the imbalanced distributions, we propose an instance-wise temperature assignment (ITA) method for contrastive learning and class-wise reliability criteria for pseudo-labels. The ITA method relaxes instance discrimination for samples belonging to head classes to enhance GCD. The reliability criteria are to avoid excluding most pseudo-labels for tail classes when training an instance segmentation network using pseudo-labels from GCD. Additionally, we propose dynamically adjusting the criteria to leverage diverse samples in the early stages while relying only on reliable pseudo-labels in the later stages. We also introduce an efficient soft attention module to encode object-specific representations for GCD. Finally, we evaluate our proposed method by conducting experiments on two settings: COCO$_{half}$ + LVIS and LVIS + Visual Genome. The experimental results demonstrate that the proposed method outperforms previous state-of-the-art methods.

We propose a novel online Gaussian process (GP) model that is capable of capturing long-term memory in sequential data in an online regression setting. Our model, Online HiPPO Sparse Variational Gaussian Process Regression (OHS-GPR), leverages the HiPPO (High-order Polynomial Projection Operators) framework, which is popularized in the RNN domain due to its long-range memory modeling capabilities. We interpret the HiPPO time-varying orthogonal projections as inducing variables with time-dependent orthogonal polynomial basis functions, which allows the SGPR inducing points to memorize the process history. We show that the HiPPO framework fits naturally into the interdomain GP framework and demonstrate that the kernel matrices can also be updated online in a recurrence form based on the ODE evolution of HiPPO. We evaluate our method on time series regression tasks, showing that it outperforms the existing online GP method in terms of predictive performance and computational efficiency.

Circular and non-flat data distributions are prevalent across diverse domains of data science, yet their specific geometric structures often remain underutilized in machine learning frameworks. A principled approach to accounting for the underlying geometry of such data is pivotal, particularly when extending statistical models, like the pervasive Gaussian distribution. In this work, we tackle those issue by focusing on the manifold of symmetric positive definite matrices, a key focus in information geometry. We introduced a non-isotropic wrapped Gaussian by leveraging the exponential map, we derive theoretical properties of this distribution and propose a maximum likelihood framework for parameter estimation. Furthermore, we reinterpret established classifiers on SPD through a probabilistic lens and introduce new classifiers based on the wrapped Gaussian model. Experiments on synthetic and real-world datasets demonstrate the robustness and flexibility of this geometry-aware distribution, underscoring its potential to advance manifold-based data analysis. This work lays the groundwork for extending classical machine learning and statistical methods to more complex and structured data.

Over recent years, the Transformer has become a fundamental building block for sequence modeling architectures. Yet at its core is the use of self-attention, whose memory and computational cost grow quadratically with the sequence length $N$, rendering it prohibitively expensive for long sequences. A promising approach is top-$k$ attention, which selects only the $k$ most relevant tokens and achieves performance comparable to vanilla self-attention while significantly reducing space and computational demands. However, causal masks require the current query token to only attend to past tokens, preventing the existing top-$k$ attention method from efficiently searching for the most relevant tokens in parallel, thereby limiting training efficiency. In this work, we propose ZETA, leveraging \textbf{Z}-Order Curves for \textbf{E}fficient \textbf{T}op-$k$ \textbf{A}ttention, to enable parallel querying of past tokens for entire sequences. % in both space and time complexity of $\mathcal{O}(N \log N)$. We first theoretically show that the choice of key and query dimensions involves a trade-off between the curse of dimensionality and the preservation of relative distances after projection. In light of this insight, we propose reducing the dimensionality of keys and queries in contrast to values and further leverage $Z$-order curves to map low-dimensional keys and queries into \emph{one}-dimensional space, which permits parallel sorting, thereby largely improving the efficiency for top-$k$ token selection. Experimental results demonstrate that ZETA matches the performance of standard attention on the synthetic \textsc{Multi-Query Associative Recall} task and outperforms attention and its variants on \textsc{Long Range Arena} and \textsc{WikiText-103} language modeling.