The extension of SGGMC from previous work (SGRHMC)[1] are in two folds. First, the proposed method use Geodesic flow rather than Riemmannian manifold. Second, the proposed method leverage a symmetric splitting integrator (ABOBA) scheme. However, unfortunately none of extensions have a clear and convincing novelty as far as I can see. The drop-in replacement of D and Q are not surprising.

Automated driving has become more and more popular due to its potential to eliminate road accidents by taking over driving tasks from humans. One of the remaining challenges is to follow a planned path autonomously, especially when uncertainties in self-localizing or understanding the surroundings can influence the decisions made by autonomous vehicles, such as calculating how much they need to steer to minimize tracking errors. In this paper, a modified geometric pure pursuit path-tracking algorithm is proposed, taking into consideration such uncertainties using the unscented transform. The algorithm is tested through simulations for typical road geometries, such as straight and circular lines.

Physics-informed neural networks (PINNs) effectively embed physical principles into machine learning, but often struggle with complex or alternating geometries. We propose a novel method for integrating geometric transformations within PINNs to robustly accommodate geometric variations. Our method incorporates a diffeomorphism as a mapping of a reference domain and adapts the derivative computation of the physics-informed loss function. This generalizes the applicability of PINNs not only to smoothly deformed domains, but also to lower-dimensional manifolds and allows for direct shape optimization while training the network. We demonstrate the effectivity of our approach on several problems: (i) Eikonal equation on Archimedean spiral, (ii) Poisson problem on surface manifold, (iii) Incompressible Stokes flow in deformed tube, and (iv) Shape optimization with Laplace operator. Through these examples, we demonstrate the enhanced flexibility over traditional PINNs, especially under geometric variations. The proposed framework presents an outlook for training deep neural operators over parametrized geometries, paving the way for advanced modeling with PDEs on complex geometries in science and engineering.

For a manifold embedded in an inner product space, we express geometric quantities such as {\it Hamilton vector fields, affine and Levi-Civita connections, curvature} in global coordinates. Instead of coordinate indices, the global formulas for most quantities are expressed as {\it operator-valued} expressions, using an {\it affine projection} to the tangent bundle. For a submersion image of an embedded manifold, we introduce {\it liftings} of Hamilton vector fields, allowing us to use embedded coordinates on horizontal bundles. We derive a {\it Gauss-Codazzi equation} for affine connections on vector bundles. This approach allows us to evaluate geometric expressions globally, and could be used effectively with modern numerical frameworks in applications. Examples considered include rigid body mechanics and Hamilton mechanics on Grassmann manifolds. We show explicitly the cross-curvature (MTW-tensor) for the {\it Kim-McCann} metric with a reflector antenna-type cost function on the space of positive-semidefinite matrices of fixed rank has nonnegative cross-curvature, while the corresponding cost could have negative cross-curvature on Grassmann manifolds, except for projective spaces.

This paper presents a novel approach for localising a GPS (Global Positioning System)-denied Unmanned Aerial Vehicle (UAV) with the aid of a GPS-equipped UAV in three-dimensional space. The GPS-equipped UAV makes discrete-time broadcasts of its global coordinates. The GPS-denied UAV simultaneously receives the broadcast and takes direction of arrival (DOA) measurements towards the origin of the broadcast in its local coordinate frame (obtained via an inertial navigation system (INS)). The aim is to determine the difference between the local and global frames, described by a rotation and a translation. In the noiseless case, global coordinates were recovered exactly by solving a system of linear equations. When DOA measurements are contaminated with noise, rank relaxed semidefinite programming (SDP) and the Orthogonal Procrustes algorithm are employed. Simulations are provided and factors affecting accuracy, such as noise levels and number of measurements, are explored.

We consider an alignment algorithm for reconstructing global coordinates from local coordinates constructed for sections of manifolds. We show that, under certain conditions, the align- ment algorithm can successfully recover global coordinates even when local neighborhoods have different dimensions. Our results generalize an earlier analysis to allow alignment of sections of different dimensions. We also apply our result to a semisupervised learning problem.

We present an automatic alignment procedure which maps the disparate internal representations learned by several local dimensionality reduction experts into a single, coherent global coordinate system for the original data space. Our algorithm can be applied to any set of experts, each of which produces a low-dimensional local representation of a highdimensional input. Unlike recent efforts to coordinate such models by modifying their objective functions [1, 2], our algorithm is invoked after training and applies an efficient eigensolver to post-process the trained models. The post-processing has no local optima and the size of the system it must solve scales with the number of local models rather than the number of original data points, making it more efficient than model-free algorithms such as Isomap [3] or LLE [4].

We present an automatic alignment procedure which maps the disparate internal representations learned by several local dimensionality reduction experts into a single, coherent global coordinate system for the original data space. Our algorithm can be applied to any set of experts, each of which produces a low-dimensional local representation of a highdimensional input. Unlike recent efforts to coordinate such models by modifying their objective functions [1, 2], our algorithm is invoked after training and applies an efficient eigensolver to post-process the trained models. The post-processing has no local optima and the size of the system it must solve scales with the number of local models rather than the number of original data points, making it more efficient than model-free algorithms such as Isomap [3] or LLE [4].

We present an automatic alignment procedure which maps the disparate internal representations learned by several local dimensionality reduction experts into a single, coherent global coordinate system for the original data space. Our algorithm can be applied to any set of experts, each of which produces a low-dimensional local representation of a highdimensional input.Unlike recent efforts to coordinate such models by modifying their objective functions [1, 2], our algorithm is invoked after training and applies an efficient eigensolver to post-process the trained models. The post-processing has no local optima and the size of the system itmust solve scales with the number of local models rather than the number of original data points, making it more efficient than model-free algorithms such as Isomap [3] or LLE [4].

High dimensional data that lies on or near a low dimensional manifold can be described by a collection of local linear models. Such a description, however, does not provide a global parameterization of the manifold--arguably an important goal of unsupervised learning. In this paper, we show how to learn a collection of local linear models that solves this more difficult problem. Our local linear models are represented by a mixture of factor analyzers, and the "global coordination" of these models is achieved by adding a regularizing term to the standard maximum likelihood objective function. The regularizer breaks a degeneracy in the mixture model's parameter space, favoring models whose internal coordinate systems are aligned in a consistent way. As a result, the internal coordinates change smoothly and continuously as one traverses a connected path on the manifold--even when the path crosses the domains of many different local models. The regularizer takes the form of a Kullback-Leibler divergence and illustrates an unexpected application of variational methods: not to perform approximate inference in intractable probabilistic models, but to learn more useful internal representations in tractable ones.