Related papers: Orthogonal Persistence Revisited
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
Machine learning has achieved remarkable success in many applications. However, existing studies are largely based on the closed-world assumption, which assumes that the environment is stationary, and the model is fixed once deployed. In…
By general case we mean methods able to process simplicial sets and chain complexes not of finite type. A filtration of the object to be studied is the heart of both subjects persistent homology and spectral sequences. In this paper we…
A method to apply and visualize persistent homology of time series is proposed. The method captures persistent features in space and time, in contrast to the existing procedures, where one usually chooses one while keeping the other fixed.…
This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence barcodes, generalized persistence,…
Concurrency, the art of doing many things at the same time is slowly becoming a science. It is very difficult to master, yet it arises all over modern computing systems, both when the communication medium is shared memory and when it is by…
Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features,…
Software platforms often act as structure preserving systems. They provide consistent interfaces and behaviors that remain stable under specific transformations that we denote as symmetries. This paper explores the idea that architectural…
The study of complex systems has attracted widespread attention from researchers in the fields of natural sciences, social sciences, and engineering. Prediction is one of the central issues in this field. Although most related studies have…
Programming languages are engineered languages that allow to instruct a machine and share algorithmic information; they have a great influence on the society since they underlie almost every information technology artefact, and they are at…
Persistent homology was shown by Carlsson and Zomorodian to be homology of graded chain complexes with coefficients in the graded ring $\kk[t]$. As such, the behavior of persistence modules -- graded modules over $\kk[t]$ is an important…
Object-Oriented Programming (OOP) has become a crucial paradigm for managing the growing complexity of modern software systems, particularly in fields like machine learning, deep learning, large language models (LLM), and data analytics.…
Localized patterns are coherent structures embedded in a quiescent state and occur in both discrete and continuous media across a wide range of applications. While it is well-understood how domain covering patterns (for example stripes and…
Extended and zigzag persistence were introduced more than ten years ago, as generalizations of ordinary persistence. While overcoming certain limitations of ordinary persistence, they both enjoy nice computational properties, which make…
Recently, it was found that there is a remarkable intuitive similarity between studies in theoretical computer science dealing with large data sets on the one hand, and categorical methods of topology and geometry in pure mathematics, on…
Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to…
The current computer programmings encapsulate attributes and behaviours into objects, but miss the mechanism to support the connection among objects. A programming paradigm is presented to connect all objects. The connection supports…
In this paper we examine how concurrency has been embodied in mainstream programming languages. In particular, we rely on the evolutionary talking borrowed from biology to discuss major historical landmarks and crucial concepts that shaped…
While machine learning is traditionally a resource intensive task, embedded systems, autonomous navigation and the vision of the Internet-of-Things fuel the interest in resource efficient approaches. These approaches require a carefully…
Features in machine learning problems are often time-varying and may be related to outputs in an algebraic or dynamical manner. The dynamic nature of these machine learning problems renders current higher order accelerated gradient descent…