In this paper, we develop a consensus algorithm for distributed computation of the Riemannian center of mass (RCM) on Lie Groups. The algorithm is built upon a distributed optimization reformulation that allows developing an intrinsic, distributed (without relying on a consensus subroutine), and a computationally efficient protocol for the RCM computation. The novel idea for developing this fast distributed algorithm is to utilize a Riemannian version of distributed gradient flow combined with a gradient tracking technique. We first guarantee that, under certain conditions, the limit point of our algorithm is the RCM point of interest. We then provide a proof of global convergence in the Euclidean setting, that can be viewed as a "geometric" dynamic consensus that converges to the average from arbitrary initial points. Finally, we proceed to showcase the superior convergence properties of the proposed approach as compared with other classes of consensus optimization-based algorithms for the RCM computation.
Texas A&M University（德克萨斯农工大学）；University of Minnesota（明尼苏达大学）
We study the convergence properties of Riemannian gradient method for solving the consensus problem (for an undirected connected graph) over the Stiefel manifold. The Stiefel manifold is a non-convex set and the standard notion of averaging in the Euclidean space does not work for this problem. We propose Distributed Riemannian Consensus on Stiefel Manifold (DRCS) and prove that it enjoys a local linear convergence rate to global consensus. More importantly, this local rate asymptotically scales with the second largest singular value of the communication matrix, which is on par with the well-k…
In the context of sensor networks, gossip algorithms are a popular, well esthablished technique for achieving consensus when sensor data is encoded in linear spaces. Gossip algorithms also have several extensions to non linear data spaces. Most of these extensions deal with Riemannian manifolds and use Riemannian gradient descent. This paper, instead, exhibits a very simple metric property that do not rely on any differential structure. This property strongly suggests that gossip algorithms could be studied on a broader family than Riemannian manifolds. And it turns out that, indeed, (local) c…
This work addresses the problem of distributed formation tracking for a group of follower holonomic mobile robots around a reference signal. The reference signal is comprised of the geometric center of the positions of multiple leaders. This work's main contribution is a novel Modulated Distributed Virtual Observer (MDVO) for the reference signal. Moreover, the proposed MDVO is based on an exact dynamic consensus algorithm with a prescribed convergence time. In addition, we provide simulation examples showcasing two different application scenarios for the proposal.
We study the problem of finding the global Riemannian center of mass of a set of data points on a Riemannian manifold. Specifically, we investigate the convergence of constant step-size gradient descent algorithms for solving this problem. The challenge is that often the underlying cost function is neither globally differentiable nor convex, and despite this one would like to have guaranteed convergence to the global minimizer. After some necessary preparations we state a conjecture which we argue is the best (in a sense described) convergence condition one can hope for. The conjecture specifi…
We study the convergence issue for the gradient algorithm (employing general step sizes) for optimization problems on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity (resp. weak sharp minima), local/global convergence (resp. linear convergence) results are established. As an application, the linear convergence properties of the gradient algorithm employing the constant step sizes and the Armijo step sizes for finding the Riemannian $L^p$ ($p\in[1,+\infty)$) centers of mass are explored, respectively, which in particular …
Decentralized Optimization on Compact Submanifolds by Quantized Riemannian Gradient Tracking
National Special Education Resource Center for Children with Autism, Zhejiang Normal University（浙江师范大学生自闭症教育资源中心）；School of Computer Science and Technology, Zhejiang Normal University（浙江师范大学计算机科学与技术学院）；China Mobile Research Institute（中国移动研究院）；Institute of Cyber-Systems and Control, Zhejiang University（浙江大学控制系统研究所）；SGIT AI Lab, State Grid Corporation of China（国网SGIT人工智能实验室）；Zhejiang Key Laboratory of Intelligent Education Technology and Application, Zhejiang Normal University（浙江智能教育技术与应用重点实验室）；School of Information Engineering, Huzhou University（湖州大学信息工程学院）；Centre for Frontier Artificial Intelligence Research, A*STAR（A*STAR前沿人工智能研究中心）
The gradient method for minimize a differentiable convex function on Riemannian manifolds with lower bounded sectional curvature is analyzed in this paper. The analysis of the method is presented with three different finite procedures for determining the stepsize, namely, Lipschitz stepsize, adaptive stepsize and Armijo's stepsize. The first procedure requires that the objective function has Lipschitz continuous gradient, which is not necessary for the other approaches. Convergence of the whole sequence to a minimizer, without any level set boundedness assumption, is proved. Iteration-complexi…
Recently, there have been growing interests in solving consensus optimization problems in a multi-agent network. In this paper, we develop a decentralized algorithm for the consensus optimization problem $$\min\limits_{x\in\mathbb{R}^p}~\bar{f}(x)=\frac{1}{n}\sum\limits_{i=1}^n f_i(x),$$ which is defined over a connected network of $n$ agents, where each function $f_i$ is held privately by agent $i$ and encodes the agent's data and objective. All the agents shall collaboratively find the minimizer while each agent can only communicate with its neighbors. Such a computation scheme avoids a data…
We investigate the number of geodesics between two points $p$ and $q$ on a contact sub-Riemannian manifold M. We show that the count of geodesics on $M$ is controlled by the count on its nilpotent approximation at $p$ (a contact Carnot group). For contact Carnot groups we make the count explicit in exponential coordinates $(x,z) \in \mathbb{R}^{2n} \times \mathbb{R}$ centered at $p$. In this case we prove that for the generic $q$ the number of geodesics $ν(q)$ between $p$ and $q=(x,z)$ satisfies: \[ C_1\frac{|z|}{\|x\|^2} + R_1 \leq ν(q) \leq C_2\frac{|z|}{\|x\|^2} + R_2\] for some constants $…
Department of Mathematics, University of Oslo（奥斯陆大学数学系）
In this paper, we give explicit descriptions of versions of (Local-) Backtracking Gradient Descent and New Q-Newton's method to the Riemannian setting.Here are some easy to state consequences of results in this paper, where X is a general Riemannian manifold of finite dimension and $f:X\rightarrow \mathbb{R}$ a $C^2$ function which is Morse (that is, all its critical points are non-degenerate). {\bf Theorem.} For random choices of the hyperparameters in the Riemanian Local Backtracking Gradient Descent algorithm and for random choices of the initial point $x_0$, the sequence $\{x_n\}$ construc…
We consider the problem of recovering a complete (i.e., square and invertible) matrix $\mathbf A_0$, from $\mathbf Y \in \mathbb{R}^{n \times p}$ with $\mathbf Y = \mathbf A_0 \mathbf X_0$, provided $\mathbf X_0$ is sufficiently sparse. This recovery problem is central to theoretical understanding of dictionary learning, which seeks a sparse representation for a collection of input signals and finds numerous applications in modern signal processing and machine learning. We give the first efficient algorithm that provably recovers $\mathbf A_0$ when $\mathbf X_0$ has $O(n)$ nonzeros per column,…
