Related papers: Persistent Topological Structures and Cohomologica…

Mobile robots require comprehensive scene understanding to operate effectively in diverse environments, enriched with contextual information such as layouts, objects, and their relationships. Although advances like neural radiation fields…

Neural models learn representations of high-dimensional data on low-dimensional manifolds. Multiple factors, including stochasticities in the training process, model architectures, and additional inductive biases, may induce different…

Neural networks can be thought of as applying a transformation to an input dataset. The way in which they change the topology of such a dataset often holds practical significance for many tasks, particularly those demanding non-homeomorphic…

One of the paramount challenges in neuroscience is to understand the dynamics of individual neurons and how they give rise to network dynamics when interconnected. Historically, researchers have resorted to graph theory, statistics, and…

Textual network embeddings aim to learn a low-dimensional representation for every node in the network so that both the structural and textual information from the networks can be well preserved in the representations. Traditionally, the…

The dynamics of neuron populations commonly evolve on low-dimensional manifolds. Thus, we need methods that learn the dynamical processes over neural manifolds to infer interpretable and consistent latent representations. We introduce a…

We develop a theoretical framework that explains how discrete symbolic structures can emerge naturally from continuous neural network training dynamics. By lifting neural parameters to a measure space and modeling training as Wasserstein…

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…

Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if…

Capturing the dynamics of active particles, i.e., small self-propelled agents that both deform and are deformed by a fluid in which they move is a formidable problem as it requires coupling fine scale hydrodynamics with large scale…

Topological structures in image data, such as connected components and loops, play a crucial role in understanding image content (e.g., biomedical objects). % Despite remarkable successes of numerous image processing methods that rely on…

The recent surge of utilizing deep neural networks for geometric processing and shape modeling has opened up exciting avenues. However, there is a conspicuous lack of research efforts on using powerful neural representations to extend the…

Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In…

Topological Machine Learning (TML) is an emerging field that leverages techniques from algebraic topology to analyze complex data structures in ways that traditional machine learning methods may not capture. This tutorial provides a…

Artificial neural networks can learn complex, salient data features to achieve a given task. On the opposite end of the spectrum, mathematically grounded methods such as topological data analysis allow users to design analysis pipelines…

Understanding how neural systems efficiently process information through distributed representations is a fundamental challenge at the interface of neuroscience and machine learning. Recent approaches analyze the statistical and geometrical…

Manifold learning aims to discover and represent low-dimensional structures underlying high-dimensional data while preserving critical topological and geometric properties. Existing methods often fail to capture local details with global…

Many dynamical systems can be described in terms of structured flows combining source/sink behavior, cyclic dynamics, and topology-constrained transport. These features arise across a wide range of domains, including physical, engineered,…

Learning the underlying patterns in data goes beyond instance-based generalization to external knowledge represented in structured graphs or networks. Deep learning that primarily constitutes neural computing stream in AI has shown…

Understanding the structure of high-dimensional data is fundamental to neuroscience and other data-intensive scientific fields. While persistent homology effectively identifies basic topological features such as "holes," it lacks the…