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1. A Riemannian Geometric Approach to Blind SignalRecovery for Grant Free Radio Network Access

   Feres, Carlos J.; Ding, Zhi (January 2021, IEEE transactions on signal processing)

   We propose a new nonconvex framework for blind multiple signal demixing and recovery. The proposed Riemann geometric approach extends the well known constant modulus algorithm to facilitate grant-free wireless access. For multiple signal demixing and recovery, we formulate the problem as non-convex problem optimization problem with signal orthogonality constraint in the form of Riemannian Orthogonal CMA(ROCMA). Unlike traditional stochastic gradient solutions that require large data samples, parameter tuning, and careful initialization, we leverage Riemannian geometry and transform the orthogonality requirement of recovered signals into a Riemannian manifold optimization. Our solution demonstrates full recovery of multiple access signals without large data sample size or special initialization with high probability of success. 

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2. Riemannian Optimization for Euclidean Distance Geometry

   Smith, Chandler; Lichtenberg, Samuel; Cai, HanQin; Tasissa, Abiy (December 2023, 15th Annual Workshop on Optimization for Machine Learning)

   The Euclidean distance geometry (EDG) problem is a crucial machine learning task that appears in many applications. Utilizing the pairwise Euclidean distance information of a given point set, EDG reconstructs the configuration of the point system. When only partial distance information is available, matrix completion techniques can be incorporated to fill in the missing pairwise distances. In this paper, we propose a novel dual basis Riemannian gradient descent algorithm, coined RieEDG, for the EDG completion problem. The numerical experiments verify the effectiveness of the proposed algorithm. In particular, we show that RieEDG can precisely reconstruct various datasets consisting of 2- and 3-dimensional points by accessing a small fraction of pairwise distance information. 

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3. Riemannian Optimization for Euclidean Distance Geometry

   Smith, Chander; Lichtenberg, Samuel; Cai, HanQin; Tasissa, Abiy (October 2023, OPT2023: 15th Annual Workshop on Optimization for Machine Learning)

   The Euclidean distance geometry (EDG) problem is a crucial machine learning task that appears in many applications. Utilizing the pairwise Euclidean distance information of a given point set, EDG reconstructs the configuration of the point system. When only partial distance information is available, matrix completion techniques can be incorporated to fill in the missing pairwise distances. In this paper, we propose a novel dual basis Riemannian gradient descent algorithm, coined RieEDG, for the EDG completion problem. The numerical experiments verify the effectiveness of the proposed algorithm. In particular, we show that RieEDG can precisely reconstruct various datasets consisting of 2- and 3-dimensional points by accessing a small fraction of pairwise distance information. 

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4. Policy Optimization Over Submanifolds for Linearly Constrained Feedback Synthesis

   https://doi.org/10.1109/TAC.2023.3306384  

   Talebi, Shahriar; Mesbahi, Mehran (May 2024, IEEE Transactions on Automatic Control)

   In this article, we study linearly constrained policy optimization over the manifold of Schur stabilizing controllers, equipped with a Riemannian metric that emerges naturally in the context of optimal control problems. We provide extrinsic analysis of a generic constrained smooth cost function that subsequently facilitates subsuming any such constrained problem into this framework. By studying the second-order geometry of this manifold, we provide a Newton-type algorithm that does not rely on the exponential mapping nor a retraction, while ensuring local convergence guarantees. The algorithm hinges instead upon the developed stability certificate and the linear structure of the constraints. We then apply our methodology to two well-known constrained optimal control problems. Finally, several numerical examples showcase the performance of the proposed algorithm. 

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5. Distributed Consensus on Manifolds using the Riemannian Center of Mass

   Spencer Kraisler, Shahriar Talebi (July 2023, Control Technology and Applications)

   In this paper, we develop a distributed consensus algorithm for agents whose states evolve on a manifold. This algorithm is complementary to traditional consensus, predominantly developed for systems with dynamics on vector spaces. We provide theoretical convergence guarantees for the proposed manifold consensus provided that agents are initialized within a geodesically convex (g-convex) set. This required condition on initialization is not restrictive as g-convex sets may be comparatively “large” for relevant Riemannian manifolds. Our approach to manifold consensus builds upon the notion of Riemannian Center of Mass (RCM) and the intrinsic structure of the manifold to avoid projections in the ambient space. We first show that on a g-convex ball, all states coincide if and only if each agent’s state is the RCM of its neighbors’ states. This observation facilitates our convergence guarantee to the consensus submanifold. Finally, we provide simulation results that exemplify the linear convergence rate of the proposed algorithm and illustrates its statistical properties over randomly generated problem instances. 

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