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Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological…
Persistent homology is one of the most popular methods in topological data analysis. An initial step in its use involves constructing a nested sequence of simplicial complexes. There is an abundance of different complexes to choose from,…
Hash-based sampling and estimation are common themes in computing. Using hashing for sampling gives us the coordination needed to compare samples from different sets. Hashing is also used when we want to count distinct elements. The quality…
In this paper, we show how to construct a factor graph from a network code. This provides a systematic framework for decoding using message passing algorithms. The proposed message passing decoder exploits knowledge of the underlying…
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…
In this paper, we propose a new method to compute the barycenter of large weighted graphs endowed with probability measures on their nodes. We suppose that the edge weights are distances between the nodes and that the probability measure on…
Topological Data Analysis (TDA) is a rising field of computational topology in which the topological structure of a data set can be observed by persistent homology. By considering a sequence of sublevel sets, one obtains a filtration that…
We explored the statistics of filtering of simple patterns on a number of deterministic and random graphs as a tractable simple example of information processing in complex systems. In this problem, multiple inputs map to the same output,…
Speculative decoding is a technique to leverage hardware concurrency in order to enable multiple steps of token generation in a single forward pass, thus improving the efficiency of large-scale autoregressive (AR) Transformer models.…
We propose a novel exact multi-parameter persistent homology method for analyzing time-series data utilizing the Liouville torus. In the field of topological data analysis (TDA), the conventional approach to analyzing time-series data often…
Persistent homology barcodes and diagrams are a cornerstone of topological data analysis that capture the "shape" of a wide range of complex data structures, such as point clouds, networks, and functions. However, their use in statistical…
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…
We propose a statistical framework to identify topological differences in two populations of random geometric objects. The proposed framework involves first associating a topological signature with random geometric objects and then…
Previous work has suggested that the structural restrictions of graphs from classes of bounded expansion--locally dense pockets in a globally sparse graph--naturally coincide with common properties of real-world networks such as clustering…
Topological Data Analysis (TDA) refers to an approach that uses concepts from algebraic topology to study the "shapes" of datasets. The main focus of this paper is persistent homology, a ubiquitous tool in TDA. Basing our study on this, we…
We introduce "quantum barcodes," a theoretical framework that applies persistent homology to classify topological phases in quantum many-body systems. By mapping quantum states to classical data points through strategic observable…
The contour of a family of filters along a filter is a set-theoretic lower limit. Topologicity and regularity of convergences can be characterized with the aid of the contour operation. Contour inversion is studied, in particular, for…
This paper develops the idea of homology for 1-parameter families of topological spaces. We express parametrized homology as a collection of real intervals with each corresponding to a homological feature supported over that interval or,…
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field.…
0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the…