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1. 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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2. Convergence of graph Laplacian with kNN self-tuned kernels

   https://doi.org/10.1093/imaiai/iaab019  

   Cheng, Xiuyuan; Wu, Hau-Tieng (September 2021, Information and Inference: A Journal of the IMA)

   Abstract Kernelized Gram matrix $$W$$ constructed from data points $$\{x_i\}_{i=1}^N$$ as $$W_{ij}= k_0( \frac{ \| x_i - x_j \|^2} {\sigma ^2} ) $$ is widely used in graph-based geometric data analysis and unsupervised learning. An important question is how to choose the kernel bandwidth $$\sigma $$, and a common practice called self-tuned kernel adaptively sets a $$\sigma _i$$ at each point $$x_i$$ by the $$k$$-nearest neighbor (kNN) distance. When $$x_i$$s are sampled from a $$d$$-dimensional manifold embedded in a possibly high-dimensional space, unlike with fixed-bandwidth kernels, theoretical results of graph Laplacian convergence with self-tuned kernels have been incomplete. This paper proves the convergence of graph Laplacian operator $$L_N$$ to manifold (weighted-)Laplacian for a new family of kNN self-tuned kernels $$W^{(\alpha )}_{ij} = k_0( \frac{ \| x_i - x_j \|^2}{ \epsilon \hat{\rho }(x_i) \hat{\rho }(x_j)})/\hat{\rho }(x_i)^\alpha \hat{\rho }(x_j)^\alpha $$, where $$\hat{\rho }$$ is the estimated bandwidth function by kNN and the limiting operator is also parametrized by $$\alpha $$. When $$\alpha = 1$$, the limiting operator is the weighted manifold Laplacian $$\varDelta _p$$. Specifically, we prove the point-wise convergence of $$L_N f $$ and convergence of the graph Dirichlet form with rates. Our analysis is based on first establishing a $C^0$ consistency for $$\hat{\rho }$$ which bounds the relative estimation error $$|\hat{\rho } - \bar{\rho }|/\bar{\rho }$$ uniformly with high probability, where $$\bar{\rho } = p^{-1/d}$$ and $$p$$ is the data density function. Our theoretical results reveal the advantage of the self-tuned kernel over the fixed-bandwidth kernel via smaller variance error in low-density regions. In the algorithm, no prior knowledge of $$d$$ or data density is needed. The theoretical results are supported by numerical experiments on simulated data and hand-written digit image data. 

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3. Global Attitude Control via Contraction on Manifolds with Reference Trajectory and Optimization

   https://doi.org/10.1109/CDC42340.2020.9303862  

   Vang, Bee; Tron, Roberto (December 2020, IEEE Conference on Decision and Control)

   null (Ed.)

   In this paper, we present a simple geometric attitude controller that is globally, exponentially stable. To overcome the topological restriction, the controller is designed to follow a reference trajectory that in turn converges to the desired equilibrium (making it discontinuous in the initial conditions, but continuous in time). The system and reference dynamics are studied as a single augmented system that can be analyzed and tuned simultaneously. The controller's stability is proved using contraction analysis (on the manifold), and the bounds on the convergence rate can be found via a semi-definite program with linear matrix inequalities. Additionally, our approach allows the use of the Nelder-Mead algorithm to automatically select controller gains and reference trajectory parameters by optimizing the aforementioned bounds. The resulting controller is verified through simulations. 

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4. Boomerang: Local Sampling on Image Manifolds using Diffusion Models

   Luzi, L; Siahkoohi, A; Mayer, P M; Casco-Rodriguez, J; Baraniuk, R G (November 2023, Transactions on machine learning research)

   The inference stage of diffusion models involves running a reverse-time diffusion stochastic differential equation, transforming samples from a Gaussian latent distribution into samples from a target distribution on a low-dimensional manifold. The intermediate values can be interpreted as noisy images, with the amount of noise determined by the forward diffusion process noise schedule. Boomerang is an approach for local sampling of image manifolds, which involves adding noise to an input image, moving it closer to the latent space, and mapping it back to the image manifold through a partial reverse diffusion process. Boomerang can be used with any pretrained diffusion model without adjustments to the reverse diffusion process, and we present three applications: constructing privacy-preserving datasets with controllable anonymity, increasing generalization performance with Boomerang for data augmentation, and enhancing resolution with a perceptual image enhancement framework. 

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5. Weighted L ² -contractivity of Langevin dynamics with singular potentials

   https://doi.org/10.1088/1361-6544/ac4152  

   Camrud, Evan; Herzog, David P; Stoltz, Gabriel; Gordina, Maria (December 2021, Nonlinearity)

   Abstract Convergence to equilibrium of underdamped Langevin dynamics is studied under general assumptions on the potential U allowing for singularities. By modifying the direct approach to convergence in L 2 pioneered by Hérau and developed by Dolbeault et al , we show that the dynamics converges exponentially fast to equilibrium in the topologies L 2 (d μ ) and L 2 ( W * d μ ), where μ denotes the invariant probability measure and W * is a suitable Lyapunov weight. In both norms, we make precise how the exponential convergence rate depends on the friction parameter γ in Langevin dynamics, by providing a lower bound scaling as min( γ , γ −1 ). The results hold for usual polynomial-type potentials as well as potentials with singularities such as those arising from pairwise Lennard-Jones interactions between particles. 

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