Large Language Models (LLMs) excel at both informal and formal (e.g. Lean 4) mathematical reasoning but still struggle with autoformalisation, the task of transforming informal into formal mathematical statements. Autoformalisation helps pair the informal reasoning of LLMs with formal proof assistants which enable machine-verifiable generation and mitigate hallucinations. Yet, the performance of current Math LLMs is constrained by the scarcity of large-scale corpora, particularly those containing pairs of informal and formal statements. Although current models are trained to generate code from natural language instructions, structural and syntactic differences between these and formal mathematics limit effective transfer learning. We propose TopoAlign, a framework that unlocks widely available code repositories as training resources for Math LLMs. TopoAlign decomposes code into docstrings, main functions, and dependency functions, and reassembles these components into analogues that structurally mirror formal statements. This produces structurally aligned code data that can be used for training Math LLMs without requiring additional human annotation. We train two state-of-the-art models, DeepSeek-Math and Herald, and evaluate them on the minif2f, Putnam, and ProofNet benchmarks. TopoAlign provides substantial gains for DeepSeek-Math, improving performance by 17.77% on BEq@10 and 68.82% on typecheck@10. Despite introducing no new mathematical knowledge, our framework achieves gains of 0.12% and 1.09% for Herald on BEq@10 and typecheck@10, respectively, demonstrating that training on aligned code data is beneficial even for specialized models.

Fuzzy rough set theory is effective for processing datasets with complex attributes, supported by a solid mathematical foundation and closely linked to kernel methods in machine learning. Attribute reduction algorithms and classifiers based on fuzzy rough set theory exhibit promising performance in the analysis of high-dimensional multivariate complex data. However, most existing models operate at the finest granularity, rendering them inefficient and sensitive to noise, especially for high-dimensional big data. Thus, enhancing the robustness of fuzzy rough set models is crucial for effective feature selection. Muiti-garanularty granular-ball computing, a recent development, uses granular-balls of different sizes to adaptively represent and cover the sample space, performing learning based on these granular-balls. This paper proposes integrating multi-granularity granular-ball computing into fuzzy rough set theory, using granular-balls to replace sample points. The coarse-grained characteristics of granular-balls make the model more robust. Additionally, we propose a new method for generating granular-balls, scalable to the entire supervised method based on granular-ball computing. A forward search algorithm is used to select feature sequences by defining the correlation between features and categories through dependence functions. Experiments demonstrate the proposed model's effectiveness and superiority over baseline methods.

Probabilistic Graphical Models are often used to understand dynamics of a system. They can model relationships between features (nodes) and the underlying distribution. Theoretically these models can represent very complex dependency functions, but in practice often simplifying assumptions are made due to computational limitations associated with graph operations. In this work we introduce Neural Graphical Models (NGMs) which attempt to represent complex feature dependencies with reasonable computational costs. Given a graph of feature relationships and corresponding samples, we capture the dependency structure between the features along with their complex function representations by using a neural network as a multi-task learning framework. We provide efficient learning, inference and sampling algorithms. NGMs can fit generic graph structures including directed, undirected and mixed-edge graphs as well as support mixed input data types. We present empirical studies that show NGMs' capability to represent Gaussian graphical models, perform inference analysis of a lung cancer data and extract insights from a real world infant mortality data provided by Centers for Disease Control and Prevention.

Over the past few years, the use of machine learning models has emerged as a generic and powerful means for prediction purposes. At the same time, there is a growing demand for interpretability of prediction models. To determine which features of a dataset are important to predict a target variable $Y$, a Feature Importance (FI) method can be used. By quantifying how important each feature is for predicting $Y$, irrelevant features can be identified and removed, which could increase the speed and accuracy of a model, and moreover, important features can be discovered, which could lead to valuable insights. A major problem with evaluating FI methods, is that the ground truth FI is often unknown. As a consequence, existing FI methods do not give the exact correct FI values. This is one of the many reasons why it can be hard to properly interpret the results of an FI method. Motivated by this, we introduce a new global approach named the Berkelmans-Pries FI method, which is based on a combination of Shapley values and the Berkelmans-Pries dependency function. We prove that our method has many useful properties, and accurately predicts the correct FI values for several cases where the ground truth FI can be derived in an exact manner. We experimentally show for a large collection of FI methods (468) that existing methods do not have the same useful properties. This shows that the Berkelmans-Pries FI method is a highly valuable tool for analyzing datasets with complex interdependencies.

Measuring and quantifying dependencies between random variables (RV's) can give critical insights into a data-set. Typical questions are: `Do underlying relationships exist?', `Are some variables redundant?', and `Is some target variable $Y$ highly or weakly dependent on variable $X$?' Interestingly, despite the evident need for a general-purpose measure of dependency between RV's, common practice of data analysis is that most data analysts use the Pearson correlation coefficient (PCC) to quantify dependence between RV's, while it is well-recognized that the PCC is essentially a measure for linear dependency only. Although many attempts have been made to define more generic dependency measures, there is yet no consensus on a standard, general-purpose dependency function. In fact, several ideal properties of a dependency function have been proposed, but without much argumentation. Motivated by this, in this paper we will discuss and revise the list of desired properties and propose a new dependency function that meets all these requirements. This general-purpose dependency function provides data analysts a powerful means to quantify the level of dependence between variables. To this end, we also provide Python code to determine the dependency function for use in practice.