However,in manyapplications there are several different parameters one might wish to vary: for example, scale and density. In contrast to the one-parameter setting, techniques for applying statistics and machine learning in the setting of multiparameter persistence are not well understood due to the lack of a concise representationoftheresults.

A series of methods have been developed to map a persistence diagram to a vector representation so as to facilitate the downstream use of machine learning tools.

A series of methods have been developed to map a persistence diagram to a vector representation so as to facilitate the downstream use of machine learning tools.

Thank you for your comments.

The paper uses a'learned' weight function for persistence images, i.e. a descriptor of topological features that can be vectorised. The weight function is seen to be crucial for obtaining good performance values. While I have some more comments/suggestions, I suggest acceptance.

Representation learning on graphs is a fundamental problem that can be crucial in various tasks. Graph neural networks, the dominant approach for graph representation learning, are limited in their representation power. Therefore, it can be beneficial to explicitly extract and incorporate high-order topological and geometric information into these models. In this paper, we propose a principled approach to extract the rich connectivity information of graphs based on the theory of persistent homology. Our method utilizes the topological features to enhance the representation learning of graph neural networks and achieve state-of-the-art performance on various node classification and link prediction benchmarks. We also explore the option of end-to-end learning of the topological features, i.e., treating topological computation as a differentiable operator during learning. Our theoretical analysis and empirical study provide insights and potential guidelines for employing topological features in graph learning tasks.

Building upon [2308.02636], this article investigates the potential constraining power of persistent homology for cosmological parameters and primordial non-Gaussianity amplitudes in a likelihood-free inference pipeline. We evaluate the ability of persistence images (PIs) to infer parameters, compared to the combined Power Spectrum and Bispectrum (PS/BS), and we compare two types of models: neural-based, and tree-based. PIs consistently lead to better predictions compared to the combined PS/BS when the parameters can be constrained (i.e., for $\{\Omega_{\rm m}, \sigma_8, n_{\rm s}, f_{\rm NL}^{\rm loc}\}$). PIs perform particularly well for $f_{\rm NL}^{\rm loc}$, showing the promise of persistent homology in constraining primordial non-Gaussianity. Our results show that combining PIs with PS/BS provides only marginal gains, indicating that the PS/BS contains little extra or complementary information to the PIs. Finally, we provide a visualization of the most important topological features for $f_{\rm NL}^{\rm loc}$ and for $\Omega_{\rm m}$. This reveals that clusters and voids (0-cycles and 2-cycles) are most informative for $\Omega_{\rm m}$, while $f_{\rm NL}^{\rm loc}$ uses the filaments (1-cycles) in addition to the other two types of topological features.

Understanding the decision-making processes of large language models (LLMs) is critical given their widespread applications. Towards this goal, describing the topological and geometrical properties of internal representations has recently provided valuable insights. For a more comprehensive characterization of these inherently complex spaces, we present a novel framework based on zigzag persistence, a method in topological data analysis (TDA) well-suited for describing data undergoing dynamic transformations across layers. Within this framework, we introduce persistence similarity, a new metric that quantifies the persistence and transformation of topological features such as $p$-cycles throughout the model layers. Unlike traditional similarity measures, our approach captures the entire evolutionary trajectory of these features, providing deeper insights into the internal workings of LLMs. As a practical application, we leverage persistence similarity to identify and prune redundant layers, demonstrating comparable performance to state-of-the-art methods across several benchmark datasets. Additionally, our analysis reveals consistent topological behaviors across various models and hyperparameter settings, suggesting a universal structure in LLM internal representations.

Machine learning has emerged as a potent computational tool for expediting research and development in solid oxide fuel cell electrodes. The effective application of machine learning for performance prediction requires transforming electrode microstructure into a format compatible with artificial neural networks. Input data may range from a comprehensive digital material representation of the electrode to a selected set of microstructural parameters. The chosen representation significantly influences the performance and results of the network. Here, we show a novel approach utilizing persistence representation derived from computational topology. Using 500 microstructures and current-voltage characteristics obtained with 3D first-principles simulations, we have prepared an artificial neural network model that can replicate current-voltage characteristics of unseen microstructures based on their persistent image representation. The artificial neural network can accurately predict the polarization curve of solid oxide fuel cell electrodes. The presented method incorporates complex microstructural information from the digital material representation while requiring substantially less computational resources (preprocessing and prediction time approximately 1 min) compared to our high-fidelity simulations (simulation time approximately 1 hour) to obtain a single current-potential characteristic for one microstructure.