Related papers: Sheaf theory: from deep geometry to deep learning

Over the past few years, we have seen fundamental breakthroughs in core problems in machine learning, largely driven by advances in deep neural networks. At the same time, the amount of data collected in a wide array of scientific domains…

Graph Neural Networks (GNNs) have become the de facto standard for learning on relational data. While traditional GNNs' message passing is well suited for vector-valued node features, there are cases in which node features are better…

This book provides an introduction to the mathematical analysis of deep learning. It covers fundamental results in approximation theory, optimization theory, and statistical learning theory, which are the three main pillars of deep neural…

Graphs are ubiquitous in encoding relational information of real-world objects in many domains. Graph generation, whose purpose is to generate new graphs from a distribution similar to the observed graphs, has received increasing attention…

Representation learning, and interpreting learned representations, are key areas of focus in machine learning and neuroscience. Both fields generally use representations as a means to understand or improve a system's computations. In this…

Deep learning constitutes a recent, modern technique for image processing and data analysis, with promising results and large potential. As deep learning has been successfully applied in various domains, it has recently entered also the…

Researchers have now achieved great success on dealing with 2D images using deep learning. In recent years, 3D computer vision and Geometry Deep Learning gain more and more attention. Many advanced techniques for 3D shapes have been…

We present a generalization of graph convolutional networks by generalizing the diffusion operation underlying this class of graph neural networks. These sheaf neural networks are based on the sheaf Laplacian, a generalization of the graph…

In this survey paper, we present \v{C}ech and sheaf cohomologies -- themes that were presented by Koszul in University of S\~ao Paulo during his visit in the late 1950s -- we present expansions for categories of generalized sheaves (i.e,…

The purpose of this paper is to give an introduction to the field of Schema Theory written by a mathematician and for mathematicians. In particular, we endeavor to to highlight areas of the field which might be of interest to a…

Recently various types of topological Laplacians have been studied from the perspective of data analysis. The spectral theory of these Laplacians has significantly extended the scope of algebraic topology and data analysis. Inspired by the…

This review gives an overview of the development of machine learning in geoscience. A thorough analysis of the co-developments of machine learning applications throughout the last 70 years relates the recent enthusiasm for machine learning…

Sheaf Neural Networks (SNNs) represent a powerful generalization of Graph Neural Networks (GNNs) that significantly improve our ability to model complex relational data. While directionality has been shown to substantially boost performance…

Learning to transfer considers learning solutions to tasks in a such way that relevant knowledge can be transferred from known task solutions to new, related tasks. This is important for general learning, as well as for improving the…

Machine learning is at the heart of managing the real-world problems associated with massive data. With the success of neural networks on such large-scale problems, more research in machine learning is being conducted now than ever before.…

Mining graph data has become a popular research topic in computer science and has been widely studied in both academia and industry given the increasing amount of network data in the recent years. However, the huge amount of network data…

Since about 100 years ago, to learn the intrinsic structure of data, many representation learning approaches have been proposed, including both linear ones and nonlinear ones, supervised ones and unsupervised ones. Particularly, deep…

The quality of learned features by representation learning determines the performance of learning algorithms and the related application tasks (such as high-dimensional data clustering). As a relatively new paradigm for representation…

A recent paper by Davies et al (2021) describes how deep learning (DL) technology was used to find plausible hypotheses that have led to two original mathematical results: one in knot theory, one in representation theory. I argue here that…

In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves…