Related papers: Geometric framework to predict structure from func…

Real-world networks often benefit from capturing both local and global interactions. Inspired by multi-modal analysis in brain imaging, where structural and functional connectivity offer complementary views of network organization, we…

Many physical systems--from mechanical lattices and electrical circuits to biological tissues and architected metamaterials--can be understood as networks transmitting physical quantities. We present a unified mathematical framework for…

We present a model, based on symmetry and geometry, for proteins. Using elementary ideas from mathematics and physics, we derive the geometries of discrete helices and sheets. We postulate a compatible solvent-mediated emergent pairwise…

Neural networks have succeeded in many reasoning tasks. Empirically, these tasks require specialized network structures, e.g., Graph Neural Networks (GNNs) perform well on many such tasks, but less structured networks fail. Theoretically,…

Recent years have seen the emergence of many new neural network structures (architectures and layers). To solve a given task, a network requires a certain set of abilities reflected in its structure. The required abilities depend on each…

Graph neural networks (GNNs) have emerged as powerful tools for learning protein structures by capturing spatial relationships at the residue level. However, existing GNN-based methods often face challenges in learning multiscale…

Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this…

Neural generative models can be used to learn complex probability distributions from data, to sample from them, and to produce probability density estimates. We propose a computational framework for developing neural generative models…

The predictive power of neural networks often costs model interpretability. Several techniques have been developed for explaining model outputs in terms of input features; however, it is difficult to translate such interpretations into…

We review the class of continuous latent space (statistical) models for network data, paying particular attention to the role of the geometry of the latent space. In these models, the presence/absence of network dyadic ties are assumed to…

Most complex systems can be captured by graphs or networks. Networks connect nodes (e.g.\ neurons) through edges (synapses), thus summarizing the system's structure. A popular way of interrogating graphs is community detection, which…

This paper investigates the foundations of deep learning through insight of geometry, algebra and differential calculus. At is core, artificial intelligence relies on assumption that data and its intrinsic structure can be embedded into…

Alignment, the tendency of adjacent weight matrices in deep networks to develop compatible subspace orientations, underlies gradient flow, Neural Collapse, and representation similarity across architectures. Despite extensive empirical…

In neuroscience, functional brain connectivity describes the connectivity between brain regions that share functional properties. Neuroscientists often characterize it by a time series of covariance matrices between functional measurements…

Real networks are finite metric spaces. Yet the geometry induced by shortest path distances in a network is definitely not its only geometry. Other forms of network geometry are the geometry of latent spaces underlying many networks, and…

Adversarial examples are often cited by neuroscientists and machine learning researchers as an example of how computational models diverge from biological sensory systems. Recent work has proposed adding biologically-inspired components to…

Heterogeneous molecular entities and their interactions, commonly depicted as a network, are crucial for advancing our systems-level understanding of biology. With recent advancements in high-throughput data generation and a significant…

Training neural networks means solving a high-dimensional optimization problem. Normally the goal is to minimize a loss function that depends on what is called the network function, or in other words the function that gives the network…

One important question in neuroscience is how global behavior in a brain network emerges from the interplay between network connectivity and the neural dynamics of individual nodes. To better understand this theoretical relationship, we…

Geometric trees are characterized by their tree-structured layout and spatially constrained nodes and edges, which significantly impacts their topological attributes. This inherent hierarchical structure plays a crucial role in domains such…