An official website of the United States government
The study of Einstein constraint equations in general relativity naturally leads to considering Riemannian manifolds equipped with nonsmooth metrics. There are several important differential operators on Riemannian manifolds whose definitions depend on the metric: gradient, divergence, Laplacian, covariant derivative, conformal Killing operator, and vector Laplacian, among others. In this article, we study the approximation of such operators, defined using a rough metric, by the corresponding operators defined using a smooth metric. This paves the road to understanding to what extent the nice properties such operators possess, when defined with smooth metric, will transfer over to the corresponding operators defined using a nonsmooth metric. These properties are often assumed to hold when working with rough metrics, but to date the supporting literature is slim.
more »
« less
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.
more »
« less
- 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
More Like this
-
-
The deep linear network (DLN) is a model for implicit regularization in gradient based optimization of overparametrized learning architectures. Training the DLN corresponds to a Riemannian gradient flow, where the Riemannian metric is defined by the architecture of the network and the loss function is defined by the learning task. We extend this geometric framework, obtaining explicit expressions for the volume form, including the case when the network has infinite depth. We investigate the link between the Riemannian geometry and the training asymptotics for matrix completion with rigorous analysis and numerics. We propose that under small initialization, implicit regularization is a result of bias towards high state space volume.more » « less
-
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.more » « less
-
Wei, Guofang (Ed.)We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank, giving a classification of closed, smooth, simply-connected 4-manifolds of almost non-negative curvature under the assumption of torus symmetry.more » « less
-
The relay channel, consisting of a source-destination pair and a relay, is a fundamental component of cooperative communications. While the capacity of a general relay channel remains unknown, various relaying strategies, including compress-and-forward (CF), have been proposed. For CF, given the correlated signals at the relay and destination, distributed compression techniques, such as Wyner–Ziv coding, can be harnessed to utilize the relay-to-destination link more efficiently. In light of the recent advancements in neural network-based distributed compression, we revisit the relay channel problem, where we integrate a learned one-shot Wyner–Ziv compressor into a primitive relay channel with a finite-capacity and orthogonal (or out-of-band) relay-to-destination link. The resulting neural CF scheme demonstrates that our task-oriented compressor recovers binning of the quantized indices at the relay, mimicking the optimal asymptotic CF strategy, although no structure exploiting the knowledge of source statistics was imposed into the design. We show that the proposed neural CF scheme, employing finite order modulation, operates closely to the capacity of a primitive relay channel that assumes a Gaussian codebook. Our learned compressor provides the first proof-of-concept work toward a practical neural CF relaying scheme. Published in: 2024 IEEE 25th Internmore » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/GLOBECOM42002.2020.9321983
