Pull-back Geometry of Persistent Homology Encodings
Authors: Shuang Liang, Renata Turkes, Jiayi Li, Nina Otter, Guido Montufar
TMLR 2024 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | Empirical studies that investigate the properties of PH, such as its sensitivity to perturbations or ability to detect a feature of interest, commonly rely on training and testing an additional model on the basis of the PH representation. Experimentally, the insights gained through our methodology align well with the existing knowledge about PH. Moreover, we show that the pull-back norm correlates with the performance on downstream tasks, and can therefore guide the choice of a suitable PH encoding. ... In Section 4, we use synthetic data for experimental demonstration. ... In Section 5 we utilize the brain artery tree data, to show that our proposed methodology can be used to select appropriate PH encodings in practice. |
| Researcher Affiliation | Academia | Shuang Liang EMAIL Department of Statistics & Data Science UCLA Renata Turkeš EMAIL Department of Mathematics & Computer Science University of Antwerp Jiayi Li EMAIL Department of Statistics & Data Science UCLA Nina Otter EMAIL Data Shape, Inria-Saclay; Laboratoire de Mathématiques d Orsay, Université Paris-Saclay Guido Montúfar EMAIL Departments of Mathematics and Statistics & Data Science, UCLA; Max Planck Institute for Mathematics in the Sciences |
| Pseudocode | No | The paper describes methods and processes verbally and with diagrams (e.g., Figure 1 outlines the pipeline), but it does not include any clearly labeled 'Pseudocode' or 'Algorithm' blocks with structured, code-like steps. |
| Open Source Code | Yes | Data and code developed in this research are available at https://github.com/shuangliang15/ pullback-geometry-persistent-homology. |
| Open Datasets | Yes | Synthetic data Throughout this section we consider a synthetic data set of point clouds in R2 sampled from curves in the Radial Frequency Pattern (RFP) family. ... Real-world data In this section we utilize the brain artery tree data (Bendich et al., 2016). ... Human body data We utilize the benchmark mesh segmentation data (Chen et al., 2009). |
| Dataset Splits | Yes | We evaluate the performance in terms of validation accuracy and robust validation accuracy using cross-validation, which are presented in the lower left and lower right plots in Figure 11. ... We use a 7-fold cross validation. And for robust evaluation in Section 5.1.3, we add identically and independent distributed Gaussian noise with variance 10 2 to each coordinate of each point in input point clouds. |
| Hardware Specification | No | The paper does not provide specific hardware details (e.g., CPU, GPU models, or memory) used for running the experiments. It only mentions software dependencies like TensorFlow, which implies computational resources but without specific specifications. |
| Software Dependencies | Yes | To compute the Jacobian in our experiments, we use the Gudhi library (The GUDHI Project, 2020) version 3.8.0 and Tensorflow version 2.12.0. ... The implementation utilized python library Open3D (Zhou et al., 2018) version 0.17.0 and POT (Flamary et al., 2021) version 0.9.0. |
| Experiment Setup | Yes | For the Vietoris-Rips filtration, we set maximal_edge_length as 1. For the DTM filtration, we set maximal_edge_length as 0.5 and parameter m as 0.02. For the Height filtration, we set maximal_edge_length as 0.1. ... For the construction of PI, we set the resolution P as 20, variance γ2 of the Gaussian kernel as 10 4, and the range of the image as [0,1] [0,1]. The weighting function is set as α(b,l) = l. ... We trained the model with a batch size of 32 for 100 epochs, using a learning rate of 0.001. The optimization algorithm used was Adam, and the model was trained using the cross-entropy loss function. ... We set the baseline PI parameters as P = 20 and γ2 = 3 10 5. For the weighting function, we consider the beta weighting function... We set s2 as 0.065 and κ as 1. |