  This survey provides a comprehensive exploration of applications of
Topological Data Analysis (TDA) within neural network analysis. Using TDA tools
such as persistent homology and Mapper, we delve into the intricate structures
and behaviors of neural networks and their datasets. We discuss different
strategies to obtain topological information from data and neural networks by
means of TDA. Additionally, we review how topological information can be
leveraged to analyze properties of neural networks, such as their
generalization capacity or expressivity. We explore practical implications of
deep learning, specifically focusing on areas like adversarial detection and
model selection. Our survey organizes the examined works into four broad
domains: 1. Characterization of neural network architectures; 2. Analysis of
decision regions and boundaries; 3. Study of internal representations,
activations, and parameters; 4. Exploration of training dynamics and loss
functions. Within each category, we discuss several articles, offering
background information to aid in understanding the various methodologies. We
conclude with a synthesis of key insights gained from our study, accompanied by
a discussion of challenges and potential advancements in the field.
