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1. Rethinking Maximum Flow Problem and Beamforming Design through Brain-inspired Geometric Lens

   https://doi.org/10.1109/GLOBECOM42002.2020.9321983  

   Ibrahim, Ahmed S. ( December 2020 , GLOBECOM 2020 - 2020 IEEE Global Communications Conference)

   null (Ed.)

   Increasing data rate in wireless networks (e.g., vehicular ones) can be accomplished through a two-pronged approach, which are 1) increasing the network flow rate through parallel independent routes and 2) increasing the user's link rate through beamforming codebook adaptation. Mobile relays (e.g., mobile road side units) are utilized to enable achieving these goals given their flexible positioning. First at the network level, we model regularized Laplacian matrices, which are symmetric positive definite (SPD) ones representing relay-dependent network graphs, as points over Riemannian manifolds. Inspired by the geometric classification of different tasks in the brain network, Riemannian metrics, such as Log- Euclidean metric (LEM), are utilized to choose relay positions that result in maximum LEM. Simulation results show that the proposed LEM- based relay positioning algorithm enables parallel routes and achieves maximum network flow rate, as opposed to other conventional metrics (e.g., algebraic connectivity). Second at the link level, we propose an unsupervised geometric machine learning (G-ML) approach to learn the unique channel characteristics of each relay-dependent environment. Given that spatially-correlated fading channels have SPD covariance matrices, they can be represented over Riemannian manifolds. Consequently, LEM-based Riemannian metric is utilized for unsupervised learning of the environment channels, and a matched beamforming codebook is constructed accordingly. Simulation results show that the proposed G-ML model increases the link rate after a short training period. 

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2. A topological study of functional data and Frechet functions of metric measure spaces

   https://doi.org/10.1007/s41468-019-00037-8  

   Hang, H ; Memoli, F ; Mio, W. ( August 2019 , Journal of applied and computational topology)

   We study the persistent homology of both functional data on compact topological spaces and structural data presented as compact metric measure spaces. One of our goals is to define persistent homology so as to capture primarily properties of the shape of a signal, eliminating otherwise highly persistent homology classes that may exist simply because of the nature of the domain on which the signal is defined. We investigate the stability of these invariants using metrics that downplay regions where signals are weak. The distance between two signals is small if they exhibit high similarity in regions where they are strong, regardless of the nature of their full domains, in particular allowing different homotopy types. Consistency and estimation of persistent homology of metric measure spaces from data are studied within this framework. We also apply the methodology to the construction of multi-scale topological descriptors for data on compact Riemannian manifolds via metric relaxations derived from the heat kernel. 

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3. On the Space of Locally Sobolev-Slobodeckij Functions

   https://doi.org/10.1155/2022/9094502  

   Behzadan, A. ; Holst, M. ( July 2022 , Journal of Function Spaces)

   Lu, Guozhen (Ed.)

   The study of certain differential operators between Sobolev spaces of sections of vector bundles on compact manifolds equipped with rough metric is closely related to the study of locally Sobolev functions on domains in the Euclidean space. In this paper, we present a coherent rigorous study of some of the properties of locally Sobolev-Slobodeckij functions that are especially useful in the study of differential operators between sections of vector bundles on compact manifolds with rough metric. The results of this type in published literature generally can be found only for integer order Sobolev spaces W m , p or Bessel potential spaces H s . Here, we have presented the relevant results and their detailed proofs for Sobolev-Slobodeckij spaces W s , p where s does not need to be an integer. We also develop a number of results needed in the study of differential operators on manifolds that do not appear to be in the literature. 

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4. Distance distributions and inverse problems for metric measure spaces

   https://doi.org/10.1111/sapm.12526  

   Mémoli, Facundo ; Needham, Tom ( August 2022 , Studies in Applied Mathematics)

   Applications in data science, shape analysis, and object classification frequently require comparison of probability distributions defined on different ambient spaces. To accomplish this, one requires a notion of distance on a given class of metric measure spaces—that is, compact metric spaces endowed with probability measures. Such distances are typically defined as comparisons between metric measure space invariants, such as distance distributions (also referred to as shape distributions, distance histograms, or shape contexts in the literature). Generally, distances defined in terms of distance distributions are actually pseudometrics, in that they may vanish when comparing nonisomorphic spaces. The goal of this paper is to set up a formal framework for assessing the discrimininative power of distance distributions, that is, the extent to which these pseudometrics fail to define proper metrics. We formulate several precise inverse problems in terms of these invariants and answer them in several categories of metric measure spaces, including the category of plane curves, where we give a counterexample to the curve histogram conjecture of Brinkman and Olver, the categories of embedded and Riemannian manifolds, where we obtain sphere rigidity results, and the category of metric graphs, where we obtain a local injectivity result along the lines of classical work of Boutin and Kemper on point cloud configurations. The inverse problems are further contextualized by the introduction of a variant of the Gromov–Wasserstein distance on the space of metric measure spaces, which is inspired by the original Monge formulation of optimal transport.

    

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5. Composable Geometric Motion Policies using Multi-Task Pullback Bundle Dynamical Systems

   Bylard, A. ; Bonalli, R. ; Pavone, M. ( January 2021 , IEEE International Conference on Robotics and Automation)

   null (Ed.)

   Despite decades of work in fast reactive planning and control, challenges remain in developing reactive motion policies on non-Euclidean manifolds and enforcing constraints while avoiding undesirable potential function local minima. This work presents a principled method for designing and fusing desired robot task behaviors into a stable robot motion policy, leveraging the geometric structure of non-Euclidean manifolds, which are prevalent in robot configuration and task spaces. Our Pullback Bundle Dynamical Systems (PBDS) framework drives desired task behaviors and prioritizes tasks using separate position-dependent and position/velocity-dependent Riemannian metrics, respectively, thus simplifying individual task design and modular composition of tasks. For enforcing constraints, we provide a class of metric-based tasks, eliminating local minima by imposing non-conflicting potential functions only for goal region attraction. We also provide a geometric optimization problem for combining tasks inspired by Riemannian Motion Policies (RMPs) that reduces to a simple least-squares problem, and we show that our approach is geometrically well-defined. We demonstrate the PBDS framework on the sphere S2 and at 300-500 Hz on a manipulator arm, and we provide task design guidance and an open-source Julia library implementation. Overall, this work presents a fast, easy-to-use framework for generating motion policies without unwanted potential function local minima on general manifolds. 

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