Related papers: DTM-based Filtrations
Outlier detection is an important topic in machine learning and has been used in a wide range of applications. Outliers are objects that are few in number and deviate from the majority of objects. As a result of these two properties, we…
In this work, we present a generalization of extended persistent homology to filtrations of graded sub-groups by defining relative homology in this setting. Our work provides a more comprehensive and flexible approach to get an algebraic…
The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions…
We give bounds for dimension 0 persistent homology and codimension 1 homology of Vietoris--Rips, alpha, and cubical complex filtrations from finite sets related by enrichment (adding new elements), sparsification (removing elements), and…
Topological data analysis, as a tool for extracting topological features and characterizing geometric shapes, has experienced significant development across diverse fields. Its key mathematical techniques include persistent homology and the…
A common machine learning task is to discriminate between normal and anomalous data points. In practice, it is not always sufficient to reach high accuracy at this task, one also would like to understand why a given data point has been…
Deep neural networks (DNNs) are known to produce incorrect predictions with very high confidence on out-of-distribution (OOD) inputs. This limitation is one of the key challenges in the adoption of deep learning models in high-assurance…
Persistent Homology (PH) offers stable, multi-scale descriptors of intrinsic shape structure by capturing connected components, loops, and voids that persist across scales, providing invariants that complement purely geometric…
Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically -- as long as the algorithm does not…
We introduce a new flavor of functor cocalculus, called \emph{poset cocalculus}, as a tool for studying approximations in topological data analysis. Given a functor from a distributive lattice to a model category, poset cocalculus produces…
Geometric graphs form an important family of hidden structures behind data. In this paper, we develop an efficient and robust algorithm to infer a graph skeleton of a high-dimensional point cloud dataset (PCD). Previously, there has been…
We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence…
In recent years, persistent homology (PH) has been successfully applied to real-world data in many different settings. Despite significant computational advances, PH algorithms do not yet scale to large datasets preventing interesting…
Clustering is a fundamental tool in unsupervised learning, used to group objects by distinguishing between similar and dissimilar features of a given data set. One of the most common clustering algorithms is k-means. Unfortunately, when…
The scores of distance-based outlier detection methods are difficult to interpret, making it challenging to determine a cut-off threshold between normal and outlier data points without additional context. We describe a generic…
The \v{C}ech complex is one of the most widely used tools in applied algebraic topology. Unfortunately, due to the inclusive nature of the \v{C}ech filtration, the number of simplices grows exponentially in the number of input points. A…
Many complex networks, ranging from social to biological systems, exhibit structural patterns consistent with an underlying hyperbolic geometry. Revealing the dimensionality of this latent space can disentangle the structural complexity of…
For the importance of differentiation theorems in metric spaces (starting with Pansu Rademacher type theorem in Carnot groups) and relations with rigidity of embeddings see the section 1.2 in Cheeger and Kleiner paper arXiv:math/0611954 and…
A large body of work in the statistics and computer science communities dating back to Huber (Huber, 1960) has led to statistically and computationally efficient outlier-robust estimators. Two particular outlier models have received…
Information networks are becoming increasingly popular to capture complex relationships across various disciplines, such as social networks, citation networks, and biological networks. The primary challenge in this domain is measuring…