Related papers: Data, geometry and homology
Most Machine Learning (ML) methods, from clustering to classification, rely on a distance function to describe relationships between datapoints. For complex datasets it is hard to avoid making some arbitrary choices when defining a distance…
Given a set of data points sampled from some underlying space, there are two important challenges in geometric and topological data analysis when dealing with sampled data: reconstruction -- how to assemble discrete samples into global…
The input data features set for many data driven tasks is high-dimensional while the intrinsic dimension of the data is low. Data analysis methods aim to uncover the underlying low dimensional structure imposed by the low dimensional hidden…
Evaluating the quality of learned representations without relying on a downstream task remains one of the challenges in representation learning. In this work, we present Geometric Component Analysis (GeomCA) algorithm that evaluates…
Statistical analysis on object data presents many challenges. Basic summaries such as means and variances are difficult to compute. We apply ideas from topology to study object data. We present a framework for using persistence landscapes…
Continual learning aims to efficiently learn from a non-stationary stream of data while avoiding forgetting the knowledge of old data. In many practical applications, data complies with non-Euclidean geometry. As such, the commonly used…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…
The aim of this article is to describe a new perspective on functoriality of persistent homology and explain its intrinsic symmetry that is often overlooked. A data set for us is a finite collection of functions, called measurements, with a…
Change in viewpoint is one of the major factors for variation in object appearance across different images. Thus, view-invariant object recognition is a challenging and important image understanding task. In this paper, we propose a method…
We use the method of maximum entropy to model physical space as a curved statistical manifold. It is then natural to use information geometry to explain the geometry of space. We find that the resultant information metric does not describe…
Big data often has emergent structure that exists at multiple levels of abstraction, which are useful for characterizing complex interactions and dynamics of the observations. Here, we consider multiple levels of abstraction via a…
A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded C1-submanifold without boundary in…
Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including…
Neural representations of 3D data have been widely adopted across various applications, particularly in recent work leveraging coordinate-based networks to model scalar or vector fields. However, these approaches face inherent challenges,…
A crucial step in the analysis of persistent homology is the transformation of data into an appropriate topological object (in our case, a simplicial complex). Modern packages for persistent homology often construct Vietoris--Rips or other…
This research uses deep learning to estimate the topology of manifolds represented by sparse, unordered point cloud scenes in 3D. A new labelled dataset was synthesised to train neural networks and evaluate their ability to estimate the…
Object data analysis is concerned with statistical methodology for datasets whose elements reside in an arbitrary, unspecified metric space. In this work we propose the object shape, a novel measure of shape/symmetry for object data. The…
Hypergraph data appear and are hidden in many places in the modern age. They are data structure that can be used to model many real data examples since their structures contain information about higher order relations among data points. One…
This paper presents the geometric aspect of the autoencoder framework, which, despite its importance, has been relatively less recognized. Given a set of high-dimensional data points that approximately lie on some lower-dimensional…