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A Riemannian cone (C,gC) is by definition a warped product C=R+×L with metric gC=dr2⊕r2gL, where (L,gL) is a compact Riemannian manifold without boundary. We say that C is a Calabi-Yau cone if gC is a Ricci-flat Kähler metric and if C admits a gC-parallel holomorphic volume form; this is equivalent to the cross-section (L,gL) being a Sasaki-Einstein manifold. In this paper, we give a complete classification of all smooth complete Calabi-Yau manifolds asymptotic to some given Calabi-Yau cone at a polynomial rate at infinity. As a special case, this includes a proof of Kronheimer's classification of ALE hyper-Kähler 4-manifolds without twistor theory.
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Rethinking Maximum Flow Problem and Beamforming Design through Brain-inspired Geometric Lens
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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- Award ID(s):
- 1816112
- PAR ID:
- 10212931
- Date Published:
- Journal Name:
- GLOBECOM 2020 - 2020 IEEE Global Communications Conference
- Page Range / eLocation ID:
- 1 to 6
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/GLOBECOM42002.2020.9321983
