Generating Topologically and Geometrically Diverse Manifold Data in Dimensions Four and Below

This work aims to define the concept of manifold, which has a very important place in the topology, on digital images. So, a general perspective is provided for two and three-dimensional imaging studies on digital curves and digital surfaces. Throughout the study, the features present in topological manifolds but that are not satisfied in the discrete version are specifically underlined. In addition, other concepts closely related to manifolds such as submanifold, orientation...

Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as the three-circle model for $3 \times 3$ pixel patches from natural images, or the configuration space of the cyclo-octane molecule, which is a sphere with a Klein bottle attached via two circles of singularity. In these studies of global shap...

Machine-learning (ML) algorithms or models, especially deep neural networks (DNNs), have shown significant promise in several areas. However, researchers have recently demonstrated that ML algorithms, especially DNNs, are vulnerable to adversarial examples (slightly perturbed samples that cause misclassification). The existence of adversarial examples has hindered the deployment of ML algorithms in safety-critical sectors, such as security. Several defenses for adversarial ex...

Natural data observed in $\mathbb{R}^n$ is often constrained to an $m$-dimensional manifold $\mathcal{M}$, where $m < n$. This work focuses on the task of building theoretically principled generative models for such data. Current generative models learn $\mathcal{M}$ by mapping an $m$-dimensional latent variable through a neural network $f_\theta: \mathbb{R}^m \to \mathbb{R}^n$. These procedures, which we call pushforward models, incur a straightforward limitation: manifolds ...

We develop a framework for comparing data manifolds, aimed, in particular, towards the evaluation of deep generative models. We describe a novel tool, Cross-Barcode(P,Q), that, given a pair of distributions in a high-dimensional space, tracks multiscale topology spacial discrepancies between manifolds on which the distributions are concentrated. Based on the Cross-Barcode, we introduce the Manifold Topology Divergence score (MTop-Divergence) and apply it to assess the perform...

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 simple feedforward neural network's layer representations of a modified torus with a Klein bottle-like twist were computed. The network appeared to approximate homeomorphisms in early layers, before significantly changing the topology of t...

The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework...

Topological data analysis (TDA) uncovers crucial properties of objects in medical imaging. Methods based on persistent homology have demonstrated their advantages in capturing topological features that traditional deep learning methods cannot detect in both radiology and pathology. However, previous research primarily focused on 2D image analysis, neglecting the comprehensive 3D context. In this paper, we propose an innovative 3D TDA approach that incorporates the concept of ...

Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms an...

We show how, given a sufficiently large point cloud sampled from an embedded 2-manifold in $\mathbb{R}^n$, we may obtain a global representation as a cell complex with vertices given by a representative subset of the point cloud. The vertex spacing is based on obtaining an approximation of the tangent plane which insures that the vertex accurately summarizes the local data. Using results from topological graph theory, we couple our cell complex representation with the known C...