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   https://doi.org/10.1137/1.9781611976700.75  

   Hsieh, Tsung-Yu; Wang, Suhang; Honavar, Vasant (January 2021, Proceedings of the SIAM International Conference on Data Mining)

   In many real-world applications, e.g., monitoring of individual health, climate, brain activity, environmental exposures, among others, the data of interest change smoothly over a continuum, e.g., time, yielding multi-dimensional functional data. Solving clustering, classification, and regression problems with functional data calls for effective methods for learning compact representations of functional data. Existing methods for representation learning from functional data, e.g., functional principal component analysis, are generally limited to learning linear mappings from the data space to the representation space. However, in many applications, such linear methods do not suffice. Hence, we study the novel problem of learning non-linear representations of functional data. Specifically, we propose functional autoencoders, which generalize neural network autoencoders so as to learn non-linear representations of functional data. We derive from first principles, a functional gradient based algorithm for training functional autoencoders. We present results of experiments which demonstrate that the functional autoencoders outperform the state-of-the-art baseline methods. 

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5. Functional Hypergraphs of Stock Markets

   https://doi.org/10.3390/e26100848  

   David, Jerry Jones; Sabhahit, Narayan G; Stramaglia, Sebastiano; Matteo, T Di; Boccaletti, Stefano; Jalan, Sarika (October 2024, Entropy)

   In stock markets, nonlinear interdependencies between various companies result in nontrivial time-varying patterns in stock prices. A network representation of these interdependencies has been successful in identifying and understanding hidden signals before major events like stock market crashes. However, these studies have revolved around the assumption that correlations are mediated in a pairwise manner, whereas, in a system as intricate as this, the interactions need not be limited to pairwise only. Here, we introduce a general methodology using information-theoretic tools to construct a higher-order representation of the stock market data, which we call functional hypergraphs. This framework enables us to examine stock market events by analyzing the following functional hypergraph quantities: Forman–Ricci curvature, von Neumann entropy, and eigenvector centrality. We compare the corresponding quantities of networks and hypergraphs to analyze the evolution of both structures and observe features like robustness towards events like crashes during the course of a time period. 

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