🍩 Database of Original & Non-Theoretical Uses of Topology

(found 4 matches in 0.001039s)

  1. Cell Complex Neural Networks (2020)

    Mustafa Hajij, Kyle Istvan, Ghada Zamzami
    Abstract Cell complexes are topological spaces constructed from simple blocks called cells. They generalize graphs, simplicial complexes, and polyhedral complexes that form important domains for practical applications. We propose a general, combinatorial, and unifying construction for performing neural network-type computations on cell complexes. Furthermore, we introduce inter-cellular message passing schemes, message passing schemes on cell complexes that take the topology of the underlying space into account. In particular, our method generalizes many of the most popular types of graph neural networks.

  2. A Topological Framework for Deep Learning (2020)

    Mustafa Hajij, Kyle Istvan
    Abstract We utilize classical facts from topology to show that the classification problem in machine learning is always solvable under very mild conditions. Furthermore, we show that a softmax classification network acts on an input topological space by a finite sequence of topological moves to achieve the classification task. Moreover, given a training dataset, we show how topological formalism can be used to suggest the appropriate architectural choices for neural networks designed to be trained as classifiers on the data. Finally, we show how the architecture of a neural network cannot be chosen independently from the shape of the underlying data. To demonstrate these results, we provide example datasets and show how they are acted upon by neural nets from this topological perspective.

  3. Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology (2018)

    Mustafa Hajij, Bei Wang, Carlos Scheidegger, Paul Rosen
    Abstract Topological data analysis is an emerging area in exploratory data analysis and data mining. Its main tool, persistent homology, has become a popular technique to study the structure of complex, high-dimensional data. In this paper, we propose a novel method using persistent homology to quantify structural changes in time-varying graphs. Specifically, we transform each instance of the time-varying graph into a metric space, extract topological features using persistent homology, and compare those features over time. We provide a visualization that assists in time-varying graph exploration and helps to identify patterns of behavior within the data. To validate our approach, we conduct several case studies on real-world datasets and show how our method can find cyclic patterns, deviations from those patterns, and one-time events in time-varying graphs. We also examine whether a persistence-based similarity measure satisfies a set of well-established, desirable properties for graph metrics.

  4. TDA-Net: Fusion of Persistent Homology and Deep Learning Features for COVID-19 Detection From Chest X-Ray Images (2021)

    Mustafa Hajij, Ghada Zamzmi, Fawwaz Batayneh
    Abstract Topological Data Analysis (TDA) has emerged recently as a robust tool to extract and compare the structure of datasets. TDA identifies features in data (e.g., connected components and holes) and assigns a quantitative measure to these features. Several studies reported that topological features extracted by TDA tools provide unique information about the data, discover new insights, and determine which feature is more related to the outcome. On the other hand, the overwhelming success of deep neural networks in learning patterns and relationships has been proven on various data applications including images. To capture the characteristics of both worlds, we propose TDA-Net, a novel ensemble network that fuses topological and deep features for the purpose of enhancing model generalizability and accuracy. We apply the proposed TDA-Net to a critical application, which is the automated detection of COVID-19 from CXR images. Experimental results showed that the proposed network achieved excellent performance and suggested the applicability of our method in practice.