arXiv subjects

Emilie Purvine

Publications and source records attributed to Emilie Purvine.

At least 19 recordsLinked to original sources

arXiv2026

TAGTorch: A PyTorch Library for Geometry, Topology, and Symmetry-Aware Machine Learning

Over the last decade, neural networks have been applied to an increasingly diverse range of applications, including data with rich geometric, topological, or symmetry-related structure. As a result, researchers have increasingly drawn inspiration from topology, algebra, and geometry. Despite this rich algorithmic development, the supporting software ecosystem remains fragmented. Many important methods exist only as research prototypes in unmaintained repositories. We address this by introducing Topology, Algebra, and Geometry Torch (TAGTorch), an open-source, PyTorch-based library that unifies tools inspired by topology, algebra, and geometry, including data-preprocessing methods, architectures, training techniques, and model analysis tools. We describe the design philosophy of TAGTorch and then discuss its current architecture and capabilities, highlighting areas where it can fill gaps in the current software ecosystem. We conclude with a discussion of our future development priorities for the library.

cs.LG↗

arXiv2025

Large Language Model-Based Reward Design for Deep Reinforcement Learning-Driven Autonomous Cyber Defense

Designing rewards for autonomous cyber attack and defense learning agents in a complex, dynamic environment is a challenging task for subject matter experts. We propose a large language model (LLM)-based reward design approach to generate autonomous cyber defense policies in a deep reinforcement learning (DRL)-driven experimental simulation environment. Multiple attack and defense agent personas were crafted, reflecting heterogeneity in agent actions, to generate LLM-guided reward designs where the LLM was first provided with contextual cyber simulation environment information. These reward structures were then utilized within a DRL-driven attack-defense simulation environment to learn an ensemble of cyber defense policies. Our results suggest that LLM-guided reward designs can lead to effective defense strategies against diverse adversarial behaviors.

cs.LG↗

arXiv2024

A survey of simplicial, relative, and chain complex homology theories for hypergraphs

Hypergraphs have seen widespread application in network and data science communities in recent years. We present a survey of recent work to construct auxiliary structures from hypergraphs -- specifically simplicial, relative, and chain complexes -- that can be used to build homology theories for hypergraphs. We define and describe nine different constructions and their associated homology theories. We discuss some interesting properties of each homology theory to show how various hypergraph properties imply properties of the homology groups. We also include discussion of functoriality for several of the homology theories. Finally, we provide a series of illustrative examples by computing many of these homology theories for small hypergraphs to show the variability of the methods and build intuition.

math.AT↗

arXiv2024

Any Graph is a Mapper Graph

The Mapper algorithm is a popular tool for visualization and data exploration in topological data analysis. We investigate an inverse problem for the Mapper algorithm: Given a dataset $X$ and a graph $G$, does there exist a set of Mapper parameters such that the output Mapper graph of $X$ is isomorphic to $G$? We provide constructions that affirmatively answer this question. Our results demonstrate that it is possible to engineer Mapper parameters to generate a desired graph.

math.AT↗

arXiv2023

Hypergraph Topological Features for Autoencoder-Based Intrusion Detection for Cybersecurity Data

In this position paper, we argue that when hypergraphs are used to capture multi-way local relations of data, their resulting topological features describe global behaviour. Consequently, these features capture complex correlations that can then serve as high fidelity inputs to autoencoder-driven anomaly detection pipelines. We propose two such potential pipelines for cybersecurity data, one that uses an autoencoder directly to determine network intrusions, and one that de-noises input data for a persistent homology system, PHANTOM. We provide heuristic justification for the use of the methods described therein for an intrusion detection pipeline for cyber data. We conclude by showing a small example over synthetic cyber attack data.

cs.CR↗

arXiv2023

Stepping out of Flatland: Discovering Behavior Patterns as Topological Structures in Cyber Hypergraphs

Data breaches and ransomware attacks occur so often that they have become part of our daily news cycle. This is due to a myriad of factors, including the increasing number of internet-of-things devices, shift to remote work during the pandemic, and advancement in adversarial techniques, which all contribute to the increase in both the complexity of data captured and the challenge of protecting our networks. At the same time, cyber research has made strides, leveraging advances in machine learning and natural language processing to focus on identifying sophisticated attacks that are known to evade conventional measures. While successful, the shortcomings of these methods, particularly the lack of interpretability, are inherent and difficult to overcome. Consequently, there is an ever-increasing need to develop new tools for analyzing cyber data to enable more effective attack detection. In this paper, we present a novel framework based in the theory of hypergraphs and topology to understand data from cyber networks through topological signatures, which are both flexible and can be traced back to the log data. While our approach's mathematical grounding requires some technical development, this pays off in interpretability, which we will demonstrate with concrete examples in a large-scale cyber network dataset. These examples are an introduction to the broader possibilities that lie ahead; our goal is to demonstrate the value of applying methods from the burgeoning fields of hypernetwork science and applied topology to understand relationships among behaviors in cyber data.

cs.CR↗

arXiv2023

HyperNetX: A Python package for modeling complex network data as hypergraphs

HyperNetX (HNX) is an open source Python library for the analysis and visualization of complex network data modeled as hypergraphs. Initially released in 2019, HNX facilitates exploratory data analysis of complex networks using algebraic topology, combinatorics, and generalized hypergraph and graph theoretical methods on structured data inputs. With its 2023 release, the library supports attaching metadata, numerical and categorical, to nodes (vertices) and hyperedges, as well as to node-hyperedge pairings (incidences). HNX has a customizable Matplotlib-based visualization module as well as HypernetX-Widget, its JavaScript addon for interactive exploration and visualization of hypergraphs within Jupyter Notebooks. Both packages are available on GitHub and PyPI. With a growing community of users and collaborators, HNX has become a preeminent tool for hypergraph analysis.

cs.MS↗

arXiv2023

Malicious Cyber Activity Detection Using Zigzag Persistence

In this study we synthesize zigzag persistence from topological data analysis with autoencoder-based approaches to detect malicious cyber activity and derive analytic insights. Cybersecurity aims to safeguard computers, networks, and servers from various forms of malicious attacks, including network damage, data theft, and activity monitoring. Here we focus on the detection of malicious activity using log data. To do this we consider the dynamics of the data by exploring the changing topology of a hypergraph representation gaining insights into the underlying activity. Hypergraphs provide a natural representation of cyber log data by capturing complex interactions between processes. To study the changing topology we use zigzag persistence which captures how topological features persist at multiple dimensions over time. We observe that the resulting barcodes represent malicious activity differently than benign activity. To automate this detection we implement an autoencoder trained on a vectorization of the resulting zigzag persistence barcodes. Our experimental results demonstrate the effectiveness of the autoencoder in detecting malicious activity in comparison to standard summary statistics. Overall, this study highlights the potential of zigzag persistence and its combination with temporal hypergraphs for analyzing cybersecurity log data and detecting malicious behavior.

cs.CG↗

arXiv2023

G-Mapper: Learning a Cover in the Mapper Construction

The Mapper algorithm is a visualization technique in topological data analysis (TDA) that outputs a graph reflecting the structure of a given dataset. However, the Mapper algorithm requires tuning several parameters in order to generate a ``nice" Mapper graph. This paper focuses on selecting the cover parameter. We present an algorithm that optimizes the cover of a Mapper graph by splitting a cover repeatedly according to a statistical test for normality. Our algorithm is based on G-means clustering which searches for the optimal number of clusters in $k$-means by iteratively applying the Anderson-Darling test. Our splitting procedure employs a Gaussian mixture model to carefully choose the cover according to the distribution of the given data. Experiments for synthetic and real-world datasets demonstrate that our algorithm generates covers so that the Mapper graphs retain the essence of the datasets, while also running significantly faster than a previous iterative method.

cs.LG↗

arXiv2023

Topological Analysis of Temporal Hypergraphs

In this work we study the topological properties of temporal hypergraphs. Hypergraphs provide a higher dimensional generalization of a graph that is capable of capturing multi-way connections. As such, they have become an integral part of network science. A common use of hypergraphs is to model events as hyperedges in which the event can involve many elements as nodes. This provides a more complete picture of the event, which is not limited by the standard dyadic connections of a graph. However, a common attribution to events is temporal information as an interval for when the event occurred. Consequently, a temporal hypergraph is born, which accurately captures both the temporal information of events and their multi-way connections. Common tools for studying these temporal hypergraphs typically capture changes in the underlying dynamics with summary statistics of snapshots sampled in a sliding window procedure. However, these tools do not characterize the evolution of hypergraph structure over time, nor do they provide insight on persistent components which are influential to the underlying system. To alleviate this need, we leverage zigzag persistence from the field of Topological Data Analysis (TDA) to study the change in topological structure of time-evolving hypergraphs. We apply our pipeline to both a cyber security and social network dataset and show how the topological structure of their temporal hypergraphs change and can be used to understand the underlying dynamics.

cs.CG↗

arXiv2022

Experimental Observations of the Topology of Convolutional Neural Network Activations

Topological data analysis (TDA) is a branch of computational mathematics, bridging algebraic topology and data science, that provides compact, noise-robust representations of complex structures. Deep neural networks (DNNs) learn millions of parameters associated with a series of transformations defined by the model architecture, resulting in high-dimensional, difficult-to-interpret internal representations of input data. As DNNs become more ubiquitous across multiple sectors of our society, there is increasing recognition that mathematical methods are needed to aid analysts, researchers, and practitioners in understanding and interpreting how these models' internal representations relate to the final classification. In this paper, we apply cutting edge techniques from TDA with the goal of gaining insight into the interpretability of convolutional neural networks used for image classification. We use two common TDA approaches to explore several methods for modeling hidden-layer activations as high-dimensional point clouds, and provide experimental evidence that these point clouds capture valuable structural information about the model's process. First, we demonstrate that a distance metric based on persistent homology can be used to quantify meaningful differences between layers, and we discuss these distances in the broader context of existing representational similarity metrics for neural network interpretability. Second, we show that a mapper graph can provide semantic insight into how these models organize hierarchical class knowledge at each layer. These observations demonstrate that TDA is a useful tool to help deep learning practitioners unlock the hidden structures of their models.

cs.LG↗

arXiv2022

The SVD of Convolutional Weights: A CNN Interpretability Framework

Deep neural networks used for image classification often use convolutional filters to extract distinguishing features before passing them to a linear classifier. Most interpretability literature focuses on providing semantic meaning to convolutional filters to explain a model's reasoning process and confirm its use of relevant information from the input domain. Fully connected layers can be studied by decomposing their weight matrices using a singular value decomposition, in effect studying the correlations between the rows in each matrix to discover the dynamics of the map. In this work we define a singular value decomposition for the weight tensor of a convolutional layer, which provides an analogous understanding of the correlations between filters, exposing the dynamics of the convolutional map. We validate our definition using recent results in random matrix theory. By applying the decomposition across the linear layers of an image classification network we suggest a framework against which interpretability methods might be applied using hypergraphs to model class separation. Rather than looking to the activations to explain the network, we use the singular vectors with the greatest corresponding singular values for each linear layer to identify those features most important to the network. We illustrate our approach with examples and introduce the DeepDataProfiler library, the analysis tool used for this study.

cs.CV↗

arXiv2022

Proceedings of TDA: Applications of Topological Data Analysis to Data Science, Artificial Intelligence, and Machine Learning Workshop at SDM 2022

Topological Data Analysis (TDA) is a rigorous framework that borrows techniques from geometric and algebraic topology, category theory, and combinatorics in order to study the "shape" of such complex high-dimensional data. Research in this area has grown significantly over the last several years bringing a deeply rooted theory to bear on practical applications in areas such as genomics, natural language processing, medicine, cybersecurity, energy, and climate change. Within some of these areas, TDA has also been used to augment AI and ML techniques. We believe there is further utility to be gained in this space that can be facilitated by a workshop bringing together experts (both theorists and practitioners) and non-experts. Currently there is an active community of pure mathematicians with research interests in developing and exploring the theoretical and computational aspects of TDA. Applied mathematicians and other practitioners are also present in community but do not represent a majority. This speaks to the primary aim of this workshop which is to grow a wider community of interest in TDA. By fostering meaningful exchanges between these groups, from across the government, academia, and industry, we hope to create new synergies that can only come through building a mutual comprehensive awareness of the problem and solution spaces.

math.AT↗

arXiv2021

Sheaves as a Framework for Understanding and Interpreting Model Fit

As data grows in size and complexity, finding frameworks which aid in interpretation and analysis has become critical. This is particularly true when data comes from complex systems where extensive structure is available, but must be drawn from peripheral sources. In this paper we argue that in such situations, sheaves can provide a natural framework to analyze how well a statistical model fits at the local level (that is, on subsets of related datapoints) vs the global level (on all the data). The sheaf-based approach that we propose is suitably general enough to be useful in a range of applications, from analyzing sensor networks to understanding the feature space of a deep learning model.

cs.LG↗

arXiv2021

Topological Simplifications of Hypergraphs

We study hypergraph visualization via its topological simplification. We explore both vertex simplification and hyperedge simplification of hypergraphs using tools from topological data analysis. In particular, we transform a hypergraph to its graph representations known as the line graph and clique expansion. A topological simplification of such a graph representation induces a simplification of the hypergraph. In simplifying a hypergraph, we allow vertices to be combined if they belong to almost the same set of hyperedges, and hyperedges to be merged if they share almost the same set of vertices. Our proposed approaches are general, mathematically justifiable, and they put vertex simplification and hyperedge simplification in a unifying framework.

cs.HC↗

arXiv2021

Directional Laplacian Centrality for Cyber Situational Awareness

Cyber operations is drowning in diverse, high-volume, multi-source data. In order to get a full picture of current operations and identify malicious events and actors analysts must see through data generated by a mix of human activity and benign automated processes. Although many monitoring and alert systems exist, they typically use signature-based detection methods. We introduce a general method rooted in spectral graph theory to discover patterns and anomalies without a priori knowledge of signatures. We derive and propose a new graph-theoretic centrality measure based on the derivative of the graph Laplacian matrix in the direction of a vertex. To build intuition about our measure we show how it identifies the most central vertices in standard network data sets and compare to other graph centrality measures. Finally, we focus our attention on studying its effectiveness in identifying important IP addresses in network flow data. Using both real and synthetic network flow data, we conduct several experiments to test our measure's sensitivity to two types of injected attack profiles, and show that vertices participating in injected attack profiles exhibit noticeable changes in our centrality measures, even when the injected anomalies are relatively small, and in the presence of simulated network dynamics.

cs.SI↗

arXiv2020

Argumentative Topology: Finding Loop(holes) in Logic

Advances in natural language processing have resulted in increased capabilities with respect to multiple tasks. One of the possible causes of the observed performance gains is the introduction of increasingly sophisticated text representations. While many of the new word embedding techniques can be shown to capture particular notions of sentiment or associative structures, we explore the ability of two different word embeddings to uncover or capture the notion of logical shape in text. To this end we present a novel framework that we call Topological Word Embeddings which leverages mathematical techniques in dynamical system analysis and data driven shape extraction (i.e. topological data analysis). In this preliminary work we show that using a topological delay embedding we are able to capture and extract a different, shape-based notion of logic aimed at answering the question "Can we find a circle in a circular argument?"

cs.CL↗

arXiv2020

Hypergraph Models of Biological Networks to Identify Genes Critical to Pathogenic Viral Response

Background: Representing biological networks as graphs is a powerful approach to reveal underlying patterns, signatures, and critical components from high-throughput biomolecular data. However, graphs do not natively capture the multi-way relationships present among genes and proteins in biological systems. Hypergraphs are generalizations of graphs that naturally model multi-way relationships and have shown promise in modeling systems such as protein complexes and metabolic reactions. In this paper we seek to understand how hypergraphs can more faithfully identify, and potentially predict, important genes based on complex relationships inferred from genomic expression data sets. Results: We compiled a novel data set of transcriptional host response to pathogenic viral infections and formulated relationships between genes as a hypergraph where hyperedges represent significantly perturbed genes, and vertices represent individual biological samples with specific experimental conditions. We find that hypergraph betweenness centrality is a superior method for identification of genes important to viral response when compared with graph centrality. Conclusions: Our results demonstrate the utility of using hypergraphs to represent complex biological systems and highlight central important responses in common to a variety of highly pathogenic viruses.

q-bio.QM↗