相关论文: A rank decomposition for the topological classific…

This paper is a cursory study on how topological features are preserved within the internal representations of neural network layers. Using techniques from topological data analysis, namely persistent homology, the topological features of a…

The rank of neural networks measures information flowing across layers. It is an instance of a key structural condition that applies across broad domains of machine learning. In particular, the assumption of low-rank feature representations…

How can we design neural networks that allow for stable universal approximation of maps between topologically interesting manifolds? The answer is with a coordinate projection. Neural networks based on topological data analysis (TDA) use…

We study how the topology of a data set $M = M_a \cup M_b \subseteq \mathbb{R}^d$, representing two classes $a$ and $b$ in a binary classification problem, changes as it passes through the layers of a well-trained neural network, i.e., with…

Understanding how neural networks learn remains one of the central challenges in machine learning research. From random at the start of training, the weights of a neural network evolve in such a way as to be able to perform a variety of…

In this paper we present an approach to determine the smallest possible number of neurons in a layer of a neural network in such a way that the topology of the input space can be learned sufficiently well. We introduce a general procedure…

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…

Despite significant advances in the field of deep learning in ap-plications to various areas, an explanation of the learning pro-cess of neural network models remains an important open ques-tion. The purpose of this paper is a comprehensive…

Deep neural networks can approximate functions on different types of data, from images to graphs, with varied underlying structure. This underlying structure can be viewed as the geometry of the data manifold. By extending recent advances…

This paper proposes a novel topological learning framework that integrates networks of different sizes and topology through persistent homology. Such challenging task is made possible through the introduction of a computationally efficient…

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…

We introduce a flexible setup allowing for a neural network to learn both its size and topology during the course of a standard gradient-based training. The resulting network has the structure of a graph tailored to the particular learning…

Despite significant advances in the field of deep learning in applications to various fields, explaining the inner processes of deep learning models remains an important and open question. The purpose of this article is to describe and…

We study how the topology of feature embedding space changes as it passes through the layers of a well-trained deep neural network (DNN) through Betti numbers. Motivated by existing studies using simplicial complexes on shallow fully…

Previous research has proven that the set of maps implemented by neural networks with a ReLU activation function is identical to the set of piecewise linear continuous maps. Furthermore, such networks induce a hyperplane arrangement…

In this paper, we study instances of complex neural networks, i.e. neural netwo rks with complex topologies. We use Self-Organizing Map neural networks whose n eighbourhood relationships are defined by a complex network, to classify handwr…

Artificial and biological agents cannon learn given completely random and unstructured data. The structure of data is encoded in the metric relationships between data points. In the context of neural networks, neuronal activity within a…

The application of network techniques to the analysis of neural data has greatly improved our ability to quantify and describe these rich interacting systems. Among many important contributions, networks have proven useful in identifying…

Convolutional neural networks (CNN's) are powerful and widely used tools. However, their interpretability is far from ideal. One such shortcoming is the difficulty of deducing a network's ability to generalize to unseen data. We use…

Neural networks have been successfully used for classification tasks in a rapidly growing number of practical applications. Despite their popularity and widespread use, there are still many aspects of training and classification that are…