Related papers: Path homologies of deep feedforward networks

Networks constitute efficient tools for assessing universal features of complex systems. In physical contexts, classical as well as quantum, networks are used to describe a wide range of phenomena, such as phase transitions, intricate…

Modifications to a neural network's input and output layers are often required to accommodate the specificities of most practical learning tasks. However, the impact of such changes on architecture's approximation capabilities is largely…

We present a unified pipeline for univariate time series classification via complex networks and persistent homology. A time series is mapped to a graph through one of five constructions across three families (visibility (natural and…

Nodes in networks that exhibit high connectivity, also called ``hubs'', play a critical role in determining the structural and functional properties of networked systems. However, there is no clear definition of what constitutes a hub node…

Recently, neural network architectures have been developed to accommodate when the data has the structure of a graph or, more generally, a hypergraph. While useful, graph structures can be potentially limiting. Hypergraph structures in…

In comparison to classical shallow representation learning techniques, deep neural networks have achieved superior performance in nearly every application benchmark. But despite their clear empirical advantages, it is still not well…

It is well known that Artificial Neural Networks are universal approximators. The classical result proves that, given a continuous function on a compact set on an n-dimensional space, then there exists a one-hidden-layer feedforward network…

Persistence diagrams (PDs), often characterized as sets of death and birth of homology class, have been known for providing a topological representation of a graph structure, which is often useful in machine learning tasks. Prior works rely…

Graph Neural Networks (GNNs) have become the de-facto standard tool for modeling relational data. However, while many real-world graphs are directed, the majority of today's GNN models discard this information altogether by simply making…

Designing efficient network structures has always been the core content of neural network research. ResNet and its variants have proved to be efficient in architecture. However, how to theoretically character the influence of network…

Over the recent years, Graph Neural Networks have become increasingly popular in network analytic and beyond. With that, their architecture noticeable diverges from the classical multi-layered hierarchical organization of the traditional…

Analyzing and characterizing the differences between networks is a fundamental and challenging problem in network science. Previously, most network comparison methods that rely on topological properties have been restricted to measuring…

A neural network with fixed topology can be regarded as a parametrization of functions, which decides on the correlations between functional variations when parameters are adapted. We propose an analysis, based on a differential geometry…

Persistent homology (PH) has been widely applied to graph data to extract topological features. However, little attention has been paid to how different distance functions on a graph affect the resulting persistence barcodes and their…

Detection of curvilinear structures in images has long been of interest. One of the most challenging aspects of this problem is inferring the graph representation of the curvilinear network. Most existing delineation approaches first…

The ability of a graph neural network (GNN) to leverage both the graph topology and graph labels is fundamental to building discriminative node and graph embeddings. Building on previous work, we theoretically show that edGNN, our model for…

We present a new distributed representation in deep neural nets wherein the information is represented in native form as a matrix. This differs from current neural architectures that rely on vector representations. We consider matrices as…

In this paper, we propose a new homological method to study weighted directed networks. Our model of such networks is a directed graph $Q$ equipped with a weight function $w$ on the set $Q_{1}$ of arrows in $Q$. We require that the range…

Hypergraph is a topological model for networks. In order to study the topology of hypergraphs, the homology of the associated simplicial complexes and the embedded homology have been invented. In this paper, we give some algorithms to…

Theoretical and empirical evidence indicates that the depth of neural networks is crucial for their success. However, training becomes more difficult as depth increases, and training of very deep networks remains an open problem. Here we…