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Visualization in the emerging field of topological data analysis has progressed from persistence barcodes and persistence diagrams to display of two-parameter persistent homology. Although persistence barcodes and diagrams have permitted…
Persistent homology is a branch of computational algebraic topology that studies shapes and extracts features over multiple scales. In this paper, we present an unsupervised approach that uses persistent homology to study divergent behavior…
Multi-parameter persistent homology naturally arises in applications of persistent topology to data that come with extra information depending on additional parameters, like for example time series data. We introduce the concept of a…
Persistence modules that decompose into interval modules are important in topological data analysis because we can interpret such intervals as the lifetime of topological features in the data. We can classify the settings in which…
A class of two-bit bit flipping algorithms for decoding low-density parity-check codes over the binary symmetric channel was proposed in [1]. Initial results showed that decoders which employ a group of these algorithms operating in…
Distributed storage systems provide reliable access to data through redundancy spread over individually unreliable nodes. Application scenarios include data centers, peer-to-peer storage systems, and storage in wireless networks. Storing…
Peer-to-peer distributed storage systems provide reliable access to data through redundancy spread over nodes across the Internet. A key goal is to minimize the amount of bandwidth used to maintain that redundancy. Storing a file using an…
The Betti tables of a multigraded module encode the grades at which there is an algebraic change in the module. Multigraded modules show up in many areas of pure and applied mathematics, and in particular in topological data analysis, where…
The long computational time and large memory requirements for computing Vietoris Rips persistent homology from point clouds remains a significant deterrent to its application to big data. This paper aims to reduce the memory footprint of…
The machinery of topological data analysis becomes increasingly popular in a broad range of machine learning tasks, ranging from anomaly detection and manifold learning to graph classification. Persistent homology is one of the key…
This article introduces a novel binary representation of the canonical genetic code based on both the structural similarities of the nucleotides, as well as the physicochemical properties of the encoded amino acids. Each of the four mRNA…
The intrinsic connection between lattice theory and topology is fairly well established, For instance, the collection of open subsets of a topological subspace always forms a distributive lattice. Persistent homology has been one of the…
Reservoir Computing is a machine learning approach that uses the rich repertoire of complex system dynamics for function approximation. Current approaches to reservoir computing use a network of coupled integrating neurons that require a…
How do vision transformers (ViTs) represent and process the world? This paper addresses this long-standing question through the first systematic analysis of 6.6K features across all layers, extracted via sparse autoencoders, and by…
We present an algorithm for the computation of Vietoris-Rips persistence barcodes and describe its implementation in the software Ripser. The method relies on implicit representations of the coboundary operator and the filtration order of…
Recent advances in convolutional neural networks have considered model complexity and hardware efficiency to enable deployment onto embedded systems and mobile devices. For example, it is now well-known that the arithmetic operations of…
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
We review some recent work in fast, efficient and accurate methods to compute viscosity solutions and non-viscosity solutions to static Hamilton-Jacobi equations which arise in optimal control, anisotropic front propagation, and multiple…
The field of mathematical morphology offers well-studied techniques for image processing. In this work, we view morphological operations through the lens of persistent homology, a tool at the heart of the field of topological data analysis.…
The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of $\mathbb{R}$-valued functions, the result was later cast in a…