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Graph foundation models (GFMs) seek transferable representations across graph domains but are limited by structural heterogeneity and incompatible node feature spaces. We propose Structure-Centric Graph Foundation Models (SCGFM), which…

Geometric graphs are a special kind of graph with geometric features, which are vital to model many scientific problems. Unlike generic graphs, geometric graphs often exhibit physical symmetries of translations, rotations, and reflections,…

Recent advances in diffusion models have demonstrated their remarkable ability to capture complex image distributions, but the geometric properties of the learned data manifold remain poorly understood. We address this gap by introducing a…

With the emergence of powerful representations of continuous data in the form of neural fields, there is a need for discretization invariant learning: an approach for learning maps between functions on continuous domains without being…

Real-world data generation often involves certain geometries (e.g., graphs) that induce instance-level interdependence. This characteristic makes the generalization of learning models more difficult due to the intricate interdependent…

Kernel methods approximate nonlinear maps in a data-driven manner by projecting the target map onto a finite-dimensional Hilbert space called the solution space. Traditionally, this space is a subspace of a fixed ambient reproducing kernel…

The predominant success of diffusion models in generative modeling has spurred significant interest in understanding their theoretical foundations. In this work, we propose a feature learning framework aimed at analyzing and comparing the…

We consider the problem of learning a set from random samples. We show how relevant geometric and topological properties of a set can be studied analytically using concepts from the theory of reproducing kernel Hilbert spaces. A new kind of…

Spectral clustering and diffusion maps are celebrated dimensionality reduction algorithms built on eigen-elements related to the diffusive structure of the data. The core of these procedures is the approximation of a Laplacian through a…

Spectral kernel methods are techniques for transforming data into a coordinate system that efficiently reveals the geometric structure - in particular, the "connectivity" - of the data. These methods depend on certain tuning parameters. We…

Geometric data analysis and learning has emerged as a distinct and rapidly developing research area, increasingly recognized for its effectiveness across diverse applications. At the heart of this field lies curvature, a powerful and…

Kernelization studies polynomial-time preprocessing algorithms. Over the last 20 years, the most celebrated positive results of the field have been linear kernels for classical NP-hard graph problems on sparse graph classes. In this paper,…

Diffusion models have emerged as powerful generative models for graph generation, yet their use for conditional graph generation remains a fundamental challenge. In particular, guiding diffusion models on graphs under arbitrary reward…

Effective analysis in neuroscience benefits significantly from robust conceptual frameworks. Traditional metrics of interbrain synchrony in social neuroscience typically depend on fixed, correlation-based approaches, restricting their…

It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential…

Functional connectivity (FC) refers to the investigation of interactions between brain regions to understand integration of neural activity in several regions. FC is often estimated using functional magnetic resonance images (fMRI). There…

Feature learning in neural networks is crucial for their expressive power and inductive biases, motivating various theoretical approaches. Some approaches describe network behavior after training through a change in kernel scale from…

Graph convolutional neural networks (GCNs) operate by aggregating messages over local neighborhoods given the prediction task under interest. Many GCNs can be understood as a form of generalized diffusion of input features on the graph, and…

Neural operators have been applied in various scientific fields, such as solving parametric partial differential equations, dynamical systems with control, and inverse problems. However, challenges arise when dealing with input functions…

We study a general framework of distributional computational graphs: computational graphs whose inputs are probability distributions rather than point values. We analyze the discretization error that arises when these graphs are evaluated…