Related papers: Topological Singularity Detection at Multiple Scal…

Recent observations seem to indicate that we live in a universe whose spatial sections are nearly or exactly flat. Motivated by this we study the problem of observational detection of the topology of universes with flat spatial sections. We…

Nonlinear dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. However, many popular methods can fail dramatically, even on simple two-dimensional manifolds, due to problems such as…

The topological information of a network can be retrieved equivalently from its complement consisting of the same nodes but complementary edges. Hence the partition of a network into certain substructures based on given criteria should be…

Deep learning models have seen significant successes in numerous applications, but their inner workings remain elusive. The purpose of this work is to quantify the learning process of deep neural networks through the lens of a novel…

Complex networks are a powerful modeling tool, allowing the study of countless real-world systems. They have been used in very different domains such as computer science, biology, sociology, management, etc. Authors have been trying to…

Deep neural networks progressively transform their inputs across multiple processing layers. What are the geometrical properties of the representations learned by these networks? Here we study the intrinsic dimensionality (ID) of…

Density estimation is an important technique for characterizing distributions given observations. Much existing research on density estimation has focused on cases wherein the data lies in a Euclidean space. However, some kinds of data are…

Surface extraction from implicit neural representations modelling a single class surface is a well-known task. However, there exist no surface extraction methods from an implicit representation of multiple classes that guarantee topological…

The hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold is the basis of manifold learning. The goal of this paper is to develop an algorithm (with accompanying complexity guarantees) for fitting a…

Clustering on the data with multiple aspects, such as multi-view or multi-type relational data, has become popular in recent years due to their wide applicability. The approach using manifold learning with the Non-negative Matrix…

Latent space geometry provides a rigorous and empirically valuable framework for interacting with the latent variables of deep generative models. This approach reinterprets Euclidean latent spaces as Riemannian through a pull-back metric,…

Topological data analysis has emerged as a powerful tool for extracting the metric, geometric and topological features underlying the data as a multi-resolution summary statistic, and has found applications in several areas where data…

With the growing adoption of AI-based systems across everyday life, the need to understand their decision-making mechanisms is correspondingly increasing. The level at which we can trust the statistical inferences made from AI-based…

Topological Data Analysis (TDA) gives practioners the ability to analyse the global structure of cybersecurity data. We use TDA for anomaly detection in host-based logs collected with the open-source Logging Made Easy (LME) project. We…

We present a generalised Hanson-Wright inequality and use it to establish new statistical insights into the geometry of data point-clouds. In the setting of a general random function model of data, we clarify the roles played by three…

Subspace clustering is an unsupervised clustering technique designed to cluster data that is supported on a union of linear subspaces, with each subspace defining a cluster with dimension lower than the ambient space. Many existing…

Despite the great performance of deep learning models in many areas, they still make mistakes and underperform on certain subsets of data, i.e. error slices. Given a trained model, it is important to identify its semantically coherent error…

In this paper, we present a method of embedding physics data manifolds with metric structure into lower dimensional spaces with simpler metrics, such as Euclidean and Hyperbolic spaces. We then demonstrate that it can be a powerful step in…

The space-like hypersurface of the Universe at the present cosmological time is a three-dimensional manifold. A non-trivial global topology of this space-like hypersurface would imply that the apparently observable universe (the sphere of…

We present a new manifold learning algorithm that takes a set of data points lying on or near a lower dimensional manifold as input, possibly with noise, and outputs a simplicial complex that fits the data and the manifold. We have…