Symbolic analysis of linear and nonlinear circuits utilizing digital computers
Topological symbolic analysis of circuits defined by topological constraints, utilizing digital computers
Kobylarz, T. J.↗
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Topological formulation of active network functions
Active network analysis by topological formulation, eliminating k-trees
Chan, S. P.↗
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 towards 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 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 as to 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.
topological data analysis, deep learning↗
Landscaper v1
Understanding the inner workings of machine learning models through their loss landscapes offers crucial insights into model properties, optimization dynamics, and generalizability. However, accessing these insights has traditionally required specialized mathematical expertise, limiting broader adoption. Landscaper is an open-source Python package designed to bridge this gap. Landscaper seamlessly integrates a suite of multi-dimensional loss landscape analyses with cutting-edge topological data analysis (TDA) methods. This powerful combination makes both fundamental loss landscape analysis and advanced TDA techniques accessible to the broader scientific ML community, without requiring deep pre-existing mathematical knowledge. Landscaper offers three key functionalities: * Construction: Builds detailed loss landscape representations through versatile low and high-dimensional sampling techniques. * Quantification: Applies advanced metrics, including a novel topological data analysis (TDA) based smoothness metric, enabling new perspectives on model behavior. * Visualization: Offers intuitive tools to visualize and interpret loss landscapes, providing actionable insights beyond traditional performance metrics.
Weber, Gunther [Lawrence Berkeley National Laborat↗
Topological graph-based analysis of solid-state ion migration
To accelerate the development of ion conducting materials, we present a general graph-theoretic analysis framework for ion migration in any crystalline structure. The nodes of the graph represent metastable sites of the migrating ion and the edges represent discrete migration events between adjacent sites. Starting from a collection of possible metastable migration sites, the framework assigns a weight to the edges by calculating the individual migration energy barriers between those sites. Connected pathways in the periodic simulation cell corresponding to macroscopic ion migration are identified by searching for the lowest-cost cycle in the periodic migration graph. To exemplify the utility of the framework, we present the automatic analyses of Li migration in different polymorphs of VO(PO 4 ), with the resulting identification of two distinct crystal structures with simple migration pathways demonstrating overall <300 meV migration barriers.
36 MATERIALS SCIENCE↗
Automated analysis for fluid flow topology
A new approach for visualizing vector data sets was developed by reducing the original vector field to a set of critical points and their connections, and was applied to fluid flow data sets. The critical point representation allows for considerable reduction in the data complexity. The representations are displayed as surfaces which are much simpler than the original data set, yet retain all the pertinent flow topology information. It is suggested that topological representations may be useful for database comparison.
Helman, James↗
Operational Analysis of Vertiport Surface Topology
Urban Air Mobility (UAM) concepts and technologies are being developed to safely enable operations of small, electric-powered or hybrid, pilot-optional, vertical-takeoff-and-landing (VTOL) passenger and cargo aircraft at vertiport facilities in urban and suburban environments. It is likely that many of the highest demand locations for vertiports will be in space constrained urban environments, requiring vertiport designs to maximize throughput within a compact surface footprint. This paper presents several generic vertiport topology design approaches and evaluates their relative surface area utilization and operational efficiency under different wind constrained configurations while meeting safety driven spacing constraints derived from heliport design standards and subject matter expert interviews.
Urban air mobility↗
Anomalies in the topology of the temperature fluctuations in the cosmic microwave background: An analysis of the NPIPE and FFP10 data releases
We present a topological analysis of the temperature fluctuation maps from the Planck 2020 Data Release 4 NPIPE dataset and the Planck 2018 Data Release 3 FFP10 dataset. We performed a multiscale analysis in terms of the homology characteristics of the maps, invoking relative homology to account for the analysis in the presence of masks. We performed our analysis for a range of smoothing scales spanning sub- and super-horizon scales corresponding to a full width at half maximum (FWHM) of 5',10',20',40',80',160',320', and 640', and employed simulations based on the standard model for comparison, which assumes the initial fluctuation field to be an isotropic and homogeneous Gaussian random field. Examining the behavior of topological components, represented by the 0D homology group, we find the observations to be approximately 2σ or less deviant from the simulations for all resolutions and scales for the NPIPE dataset. For the FFP10 dataset, we detect a 2.96σ deviation between the observations and simulations at N = 128, FWHM = 80'. For the topological loops, represented by the first homology group, the simulations and observations are consistent within 2σ for most resolutions and scales for both the datasets. However, for the NPIPE dataset, we observe a high deviation between the observation and simulations in the number of loops at FWHM = 320', but at a low dimensionless threshold ν = –2.5. Under a Gaussian assumption, this would amount to a deviation of ~4σ. However, the distribution in this bin is manifestly non-Gaussian and does not obey Poisson statistics either. In the absence of a true theoretical understanding, we simply note that the significance is higher than what may be resolved by 600 simulations, yielding an empirical p-value of at most 0.0016. Specifically in this case, our tests indicate that the numbers arise from a statistically stable regime, despite being based on small numbers. For the FFP10 dataset, the differences are not as strong as for the NPIPE dataset, indicating a 2.77σ deviation at this resolution and threshold. The Euler characteristic, which is the alternating sum of the ranks of relative homology groups, reflects the deviations in the components and loops. To assess the significance of combined levels for a given scale, we employed the empirical and theoretical versions of the χ 2 test as well as the nonparametric Tukey depth test. Although all statistics exhibit a stable distribution, we favor the empirical version of the χ 2 test in the final interpretation, as it indicates the most conservative differences. For the NPIPE dataset, we find that the components and loops differ at more than 95%, but agree within the 99% confidence level with respect to the base model at N = 32, FWHM = 320'. The Euler characteristic at this resolution displays a per mil deviation. In contrast, the FFP10 dataset shows that the observations are consistent with the base model within the 95% confidence level, at this and smaller scales. This is consistent with the observations of the Planck analysis pipeline via Minkowski functionals. For the largest smoothing scale, N = 16, FWHM = 640', both datasets exhibit an anomalous behavior of the loops, where FFP10 data exhibit a deviation that is larger by an order of magnitude than that of the NPIPE dataset. In contrast, the values for the topological components and the Euler characteristic agree between observations and model to within a confidence level of 99%. However, for the largest scales, the statistics are based on low numbers and may have to be regarded with caution. Even though both datasets exhibit mild to significant discrepancies, they also exhibit contrasting behaviors at various instances. Therefore, we do not find it feasible to convincingly accept or reject the null hypothesis. Disregarding the large-scale anomalies that persist at similar scales in WMAP and Planck, observations of the cosmic microwave background are largely consistent with the standard cosmological model within 2σ.
79 ASTRONOMY AND ASTROPHYSICS↗
Topological Simplifications of Hypergraphs
Here 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 into 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 and mathematically justifiable, and put vertex simplification and hyperedge simplification in a unifying framework.
97 MATHEMATICS AND COMPUTING↗
Topological formulation of network functions without generation of k-trees.
Linear network analysis by topological formulas for derivation of k-tree terms without generation of k-trees of graph
Chan, S. P.↗
Do Neural Networks Trained with Topological Features Learn Different Internal Representations?
There is a growing body of work that leverages features extracted via topological data analysis to train machine learning models. While this field, sometimes known as topological machine learning (TML), has seen some notable successes, an understanding of how the process of learning from topological features differs from the process of learning from raw data is still limited. In this work, we begin to address one component of this larger issue by asking whether a model trained with topological features learns internal representations of data that are fundamentally different than those learned by a model trained with the original raw data. To quantify ``different'', we exploit two popular metrics that can be used to measure the similarity of the hidden representations of data within neural networks, neural stitching and centered kernel alignment. From these we draw a range of conclusions about how training with topological features does and does not change the representations that a model learns. Perhaps unsurprisingly, we find that structurally, the hidden representations of models trained and evaluated on topological features differ substantially compared to those trained and evaluated on the corresponding raw data. On the other hand, our experiments show that in some cases, these representations can be reconciled (at least to the degree required to solve the corresponding task) using a simple affine transformation. We conjecture that this means that neural networks trained on raw data may extract some limited topological features in the process of making predictions.
McGuire, Sarah L.↗
Materials Fingerprinting Classification
Significant progress in many classes of materials could be made with the availability of experimentally-derived large datasets composed of atomic identities and three-dimensional coordinates. Methods for visualizing the local atomic structure, such as atom probe tomography (APT), which routinely generate datasets comprised of millions of atoms, are an important step in realizing this goal. However, state-of-the-art APT instruments generate noisy and sparse datasets that provide information about elemental type, but obscure atomic structures, thus limiting their subsequent value for materials discovery. The application of a materials fingerprinting process, a machine learning algorithm coupled with topological data analysis, provides an avenue by which here-to-fore unprecedented structural information can be extracted from an APT dataset. As a proof of concept, the material fingerprint is applied to high-entropy alloy APT datasets containing body-centered cubic (BCC) and face-centered cubic (FCC) crystal structures. A local atomic configuration centered on an arbitrary atom is assigned a topological descriptor, with which it can be characterized as a BCC or FCC lattice with near perfect accuracy, despite the inherent noise in the dataset. Finally, this successful identification of a fingerprint is a crucial first step in the development of algorithms which can extract more nuanced information, such as chemical ordering, from existing datasets of complex materials.
36 MATERIALS SCIENCE↗
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. We focus on the cybersecurity domain and investigate the detection of malicious activity using log data. We consider the dynamics of the log data and explore the changing topology of a hypergraph representation of this data to gain insights into the underlying activity. These hypergraphs capture complex interactions between processes, together with their temporal information. To study the changing topology we use zigzag persistence, which captures how topological features persist at multiple dimensions over time. We observe that this detects malicious activity in a cyber data set. 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. Overall, this study highlights the potential of zigzag persistence and its combination with temporal hypergraphs for analyzing cybersecurity log data and detecting malicious behavior.
hypergraphs, temporal hypergraph, topological data↗
Topological Machine Learning Methods for Power System Responses to Contingencies: Preprint
While deep learning tools, coupled with the emerging machinery of topological data analysis, are proven to deliver various performance gains in a broad range of applications, from image classification to biosurveillance to blockchain fraud detection, their utility in areas of high societal importance such as power system modeling and, particularly, resilience quantification in the energy sector yet remain untapped. To provide fast acting synthetic regulation and contingency reserve services to the grid while having minimal disruptions on customer quality of service, we propose a new topologybased system that depends on neural network architecture for impact metrics classification and prediction in power systems. This novel topology-based system allows one to evaluate the impact of three power system contingency types, namely, in conjunction with transmission lines, transformers, and transmission lines combined with transformers. We show that the proposed new neural network architecture equipped with local topological measures facilitates both more accurate classification of unserved load as well as the amount of unserved load. In addition, we are able to learn complex relationships between electrical properties and local topological measurements on the simulated response to contingencies for NREL-SIIP power system.
graph theory↗
Seven open problems in applied combinatorics
We present and discuss seven different open problems in applied combinatorics. Additionally, the application areas relevant to this compilation include quantum computing, algorithmic differentiation, topological data analysis, iterative methods, hypergraph cut algorithms, and power systems.
97 MATHEMATICS AND COMPUTING↗
Distributed Augmentation, Hypersweeps, and Branch Decomposition of Contour Trees for Scientific Exploration
Contour trees describe the topology of level sets in scalar fields and are widely used in topological data analysis and visualization. A main challenge of utilizing contour trees for large-scale scientific data is their computation at scale using highperformance computing. To address this challenge, recent work has introduced distributed hierarchical contour trees for distributed computation and storage of contour trees. However, effective use of these distributed structures in analysis and visualization requires subsequent computation of geometric properties and branch decomposition to support contour extraction and exploration. In this work, we introduce distributed algorithms for augmentation, hypersweeps, and branch decomposition that enable parallel computation of geometric properties, and support the use of distributed contour trees as query structures for scientific exploration. Finally, we evaluate the parallel performance of these algorithms and apply them to identify and extract important contours for scientific visualization.
97 MATHEMATICS AND COMPUTING↗
A Survey of Mathematical Structures for Lunar Networks
To sustain the current and increasing accessibility of space, a scalable communications infrastructure (i.e. the Solar System Internet, SSI) is necessary. The goal of this paper is to begin the discovery of the fundamental underlying mathematical structure of space networks to help the research community harness these structures for algorithm development and optimization. To ensure the applicability of the research, the approaches are considered through the lens of simulated scenarios inspired by the Artemis Back-to-the-Moon mission set for 2024. We note that any approach to an SSI must fit under the umbrella of Delay Tolerant Networking (DTN), due to celestial mobility, high link latencies, high variance in link latencies, disconnections, lack of end-to-end paths, and so on. These difficulties are exacerbated by the fact that the underlying structure of a space network is a time-evolving network and may experience multiple discontinuities in its topology. In this paper we propose several novel approaches to a mathematical foundation for Delay Tolerant Networking Theory that fall outside the traditional scope of temporal network theory. These techniques include methods from Topological Data Analysis, Dynamic Graph Analysis, Applied Algebraic Geometry, Probability Theory, and Game Theory. Some of these methods include tools adapted to the study of dynamic metric spaces, such as zigzag persistent homology and their higher parameter analogs. We find that several of these methods target desired engineering outcomes such as discovery and automatic sub-netting. While each approach is theoretical, they are also algorithmic in nature and offer immediate practical applications. The paper concludes with comparisons of the various methods along with suggestions for future work.
Delay tolerant networking↗
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. In conclusion, 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.
G-means clustering↗