Tangent Manifold Research Articles

On pseudo-Riemannian quartics in Finsler geometry

Finsler geometry usually describes an extension of Riemannian geometry into a direction-dependent geometric structure. Historically, the well-known Riemann quartic length element example served as the inspiration for this construction. Surprisingly, the same quartic expression emerges as a fundamental dispersion relationcovariant Fresnel equationin solid-state electrodynamics. As a result, it is possible to conceive of the Riemann quartic length expression as a mathematical representation of a well-known physical phenomenon. This paper provides a number of Riemann quartic examples that show Finsler geometry to be overly constrictive for many applications, even when the signature space is positive definite in the Euclidean sense. The strong axioms of Finsler geometry are broken down on many more singular hypersurfaces for the spaces having an indefinite (Minkowski) signature. We suggest a more flexible definition of a Finsler structure that only has to hold for open subsets of a manifold’s tangent bundle. We demonstrate the distinctive singular hypersurfaces connected to the Riemann quartic and discuss the potential physics explanations for them. As an illustration of the pseudo-Riemannian quartic, we took into consideration the dispersion relation that appears in electromagnetic wave propagation in uniaxial crystal. Our analysis suggests that the signature of the Finsler measure may be altered for large anisotropy factors.

Unification of Gravity and Internal Interactions

AbstractIn the gauge theoretic approach of gravity, general relativity is described by gauging the symmetry of the tangent manifold in four dimensions. Usually the dimension of the tangent space is considered to be equal to the dimension of the curved manifold. However, the tangent group of a manifold of dimension d is not necessarily . It has been suggested earlier that by gauging an enlarged symmetry of the tangent space in four dimensions one could unify gravity with internal interactions. Here, such a unified model is considered by gauging the as the extended Lorentz group overcoming in this way some difficulties of the previous attempts of similar unification and eventually obtained the GUT, supplemented by an global symmetry.

A linear transportation [formula omitted] distance for pattern recognition

The transportation Lp distance, denoted TLp, has been proposed as a generalisation of Wasserstein Wp distances motivated by the property that it can be applied directly to colour or multi-channelled images, as well as multivariate time-series without normalisation or mass constraints. Both TLp and Wp assign a cost based on the transport distance (i.e. the “Lagrangian” model), the key difference between the distances is that TLp interprets the signal as a function whilst Wp interprets the signal as a measure. Both distances are powerful tools in modelling data with spatial or temporal perturbations. However, their computational cost can make them infeasible to apply to even moderate pattern recognition tasks. The linear Wasserstein distance was proposed as a method for projecting signals into a Euclidean space where the Euclidean distance is approximately the Wasserstein distance (more formally, this is a projection on to the tangent manifold). We propose linear versions of the TLp distance (LTLp) and we show significant improvement over the linear Wp distance on signal processing tasks, whilst being several orders of magnitude faster to compute than the TLp distance.

Model reduction for nonlinearizable dynamics via delay-embedded spectral submanifolds

Delay embedding is a commonly employed technique in a wide range of data-driven model reduction methods for dynamical systems, including the dynamic mode decomposition, the Hankel alternative view of the Koopman decomposition (HAVOK), nearest-neighbor predictions and the reduction to spectral submanifolds (SSMs). In developing these applications, multiple authors have observed that delay embedding appears to separate the data into modes, whose orientations depend only on the spectrum of the sampled system. In this work, we make this observation precise by proving that the eigenvectors of the delay-embedded linearized system at a fixed point are determined solely by the corresponding eigenvalues, even for multi-dimensional observables. This implies that the tangent space of a delay-embedded invariant manifold can be predicted a priori using an estimate of the eigenvalues. We apply our results to three datasets to identify multimodal SSMs and analyse their nonlinear modal interactions. While SSMs are the focus of our study, these results generalize to any delay-embedded invariant manifold tangent to a set of eigenvectors at a fixed point. Therefore, we expect this theory to be applicable to a number of data-driven model reduction methods.

Finsler metrics and semi-symmetric compatible linear connections

Finsler metrics are direct generalizations of Riemannian metrics such that the quadratic Riemannian indicatrices in the tangent spaces of a manifold are replaced by more general convex bodies as unit spheres. A linear connection on the base manifold is called compatible with the Finsler metric if the induced parallel transports preserve the Finslerian length of tangent vectors. Finsler manifolds admitting compatible linear connections are called generalized Berwald manifolds Wagner (Dokl Acad Sci USSR (N.S.) 39:3–5, 1943). Compatible linear connections are the solutions of the so-called compatibility equations containing the components of the torsion tensor as unknown quantities. Although there are some theoretical results for the solvability of the compatibility equations (monochromatic Finsler metrics Bartelmeß and Matveev (J Diff Geom Appl 58:264–271, 2018), extremal compatible linear connections and algorithmic solutions Vincze (Aequat Math 96:53–70, 2022)), it is very hard to solve them in general because compatible linear connections may or may not exist on a Finsler manifold and may or may not be unique. Therefore special cases are of special interest. One of them is the case of the so-called semi-symmetric compatible linear connection with decomposable torsion tensor. It is proved Vincze (Publ Math Debrecen 83(4):741–755, 2013 (see also Vincze (Euro J Math 3:1098–1171, 2017))) that such a compatible linear connection must be uniquely determined. The original proof is based on averaging in the sense that the 1-form in the decomposition of the torsion tensor can be expressed by integrating differential forms on the tangent manifold over the Finslerian indicatrices. The integral formulas are very difficult to compute in practice. In what follows we present a new proof for the uniqueness by using linear algebra and some basic facts about convex bodies. We present an explicit formula for the solution without integration. The method has a new contribution to the problem as well: necessary conditions of the solvability are formulated in terms of intrinsic equations without unknown quantities.

Loop Closure Detection Based on Differentiable Manifold

Loop closure detection is an important part of SLAM (simultaneous location and mapping), which can effectively reduce the cumulative error of the system after long period of exploration. The existing loop closure detection methods are mainly to evenly distribute the accumulated error in the robot trajectory, but the motion error of the actual robot is also related to its motion speed and rotation angle, while the corrected motion trajectory of the robot is difficult to match the real trajectory. Based on the analysis of the mechanism of robot motion error, this paper proposes a novel loop closure detection method based on differentiable manifold, which mainly includes real-time pose based on manifold tangent space and smooth motion trajectory model of robot based on differential geometry. Firstly, we introduce the Frenet framework structure and establish the corresponding manifold tangent space theory for the keyframe pose nodes. The real-time problem of robot motion is equivalent to the problem of finding the optimal angle tangent vector. Secondly, the motion speed between keyframes is used to determine the characteristics of the robot motion trajectory. We calculate the curvature and torsion of the curve composed of several nodes based on the manifold tangent space and then combine the curve interpolation and fitting of the keyframe nodes to achieve the approximation of the robot motion trajectory, and the smooth curve of the robot trajectory is obtained. Finally, the experiment verifies that the method in this paper can effectively ensure the continuity and smoothness of the robot’s trajectory, thereby reducing the cumulative error of the system and improving the accuracy of loop closure detection.

Dimensionality reduction of SPD data based on Riemannian manifold tangent spaces and local affinity

Dimensionality reduction of SPD data based on Riemannian manifold tangent spaces and local affinity

A Hybrid Projection Algorithm for Stochastic Differential Equations on Manifolds

Stochastic differential equations projected onto manifolds occur widely in physics, chemistry, biology, engineering, nanotechnology and optimization theory. In some problems one can use an intrinsic coordinate system on the manifold, but this is often computationally impractical. Numerical projections are preferable in many cases. We derive an algorithm to solve these, using adiabatic elimination and a constraining potential. We also review earlier proposed algorithms. Our hybrid midpoint projection algorithm uses a midpoint projection on a tangent manifold, combined with a normal projection to satisfy the constraints. We show from numerical examples on spheroidal and hyperboloidal surfaces that this has greatly reduced errors compared to earlier methods using either a hybrid Euler with tangential and normal projections or purely tangential derivative methods. Our technique can handle multiple constraints. This allows, for example, the treatment of manifolds that embody several conserved quantities. The resulting algorithm is accurate, relatively simple to implement and efficient.

Dictionary Learning of Symmetric Positive Definite Data Based on Riemannian Manifold Tangent Spaces and Local Homeomorphism

Symmetric Positive Definite (SPD) data are increasingly prevalent in dictionary learning recently. SPD data are the typical non-Euclidean data and cannot constitute a Euclidean space. Therefore, many dictionary learning algorithms cannot be directly adopted on SPD data. Reproducing Kernel Hilbert Spaces (RKHS) is now commonly used to deal with this problem. However, RKHS is an infinite-dimensional Hilbert space, rather than a Euclidean space, resulting in the inability of the dictionary learning to be directly used on SPD data. In this paper, we propose a novel dictionary learning algorithm for SPD data, which is based on the Riemannian Manifold Tangent Space (RMTS). Since RMTS is based on a finite-dimensional Hilbert space, i.e., Euclidean space, most machine learning algorithms developed on Euclidean space can be directly applied to RMTS. The introduction of RMTS provides a better mathematical platform for machine learning of non-Euclidean data. The proposed RMTS method first selects a point on a Riemannian manifold as an anchor point. It transforms the other points on the Riemannian manifold into tangent vectors of the geodesics between these points and the anchor point, with the tangent vectors at the anchor point. We set the length of the tangent vector equal to the length of the geodesic. As a result, such tangent vectors have an explicit geometric meaning, such as direction information, while the RKHS method may cause some geometric meaning to be lost in the original data during the mapping process. In addition, the proposed algorithm adds a regular term of local homomorphism between SPD data and its RMTS dictionary coding coefficients, so that the similarity of SPD data and its RMTS dictionary coding coefficients are as close as possible. Experimental results on four public datasets demonstrate that the proposed algorithm significantly outperforms five state-of-the-art algorithms. This work opens a new pathway towards SPD data dictionary learning methods.

Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry.

Symmetric positive definite (SPD) data have become a hot topic in machine learning. Instead of a linear Euclidean space, SPD data generally lie on a nonlinear Riemannian manifold. To get over the problems caused by the high data dimensionality, dimensionality reduction (DR) is a key subject for SPD data, where bilinear transformation plays a vital role. Because linear operations are not supported in nonlinear spaces such as Riemannian manifolds, directly performing Euclidean DR methods on SPD matrices is inadequate and difficult in complex models and optimization. An SPD data DR method based on Riemannian manifold tangent spaces and global isometry (RMTSISOM-SPDDR) is proposed in this research. The main contributions are listed: (1) Any Riemannian manifold tangent space is a Hilbert space isomorphic to a Euclidean space. Particularly for SPD manifolds, tangent spaces consist of symmetric matrices, which can greatly preserve the form and attributes of original SPD data. For this reason, RMTSISOM-SPDDR transfers the bilinear transformation from manifolds to tangent spaces. (2) By log transformation, original SPD data are mapped to the tangent space at the identity matrix under the affine invariant Riemannian metric (AIRM). In this way, the geodesic distance between original data and the identity matrix is equal to the Euclidean distance between corresponding tangent vector and the origin. (3) The bilinear transformation is further determined by the isometric criterion guaranteeing the geodesic distance on high-dimensional SPD manifold as close as possible to the Euclidean distance in the tangent space of low-dimensional SPD manifold. Then, we use it for the DR of original SPD data. Experiments on five commonly used datasets show that RMTSISOM-SPDDR is superior to five advanced SPD data DR algorithms.

Statistical mirror symmetry

In this paper, we investigate a duality between Hermitian and almost Kähler structures on the tangent manifold TM induced by pairs of conjugate connections on its base, affine Riemannian manifold M. In the context of information geometry, the classical theory of statistical manifold (which we call S-geometry) prescribes a parametrized family of probability distributions with a Fisher-Rao metric g and, using the Amari-Chensov tensor, a family of dualistic, torsion-free connections ∇(α), known as α-connections on M. Here we prescribe an alternative geometric framework (which we call P-geometry or partially flat geometry) by treating such parametrization as affine coordinates with respect to a flat connection ∇, and considering its g-conjugate connection ∇⁎ which is curvature-free but generally carries torsion. Under P-geometry, the triplet (g,∇,∇⁎) on M leads to a pair of complex and almost Kähler structures on TM, in “mirror correspondence” to each other. Such complex-to-symplectic correspondence is reminiscent of mirror symmetry in string theory. We discuss the statistical meaning of mirror correspondence in terms of reference duality and representation duality in (various generalizations of) contrast/divergence functions characterizing proximity of probability distributions within a parametric statistical model.

On the connections of sub-Finslerian geometry

A sub-Finslerian manifold is, roughly speaking, a manifold endowed with a Finsler type metric which is defined on a [Formula: see text]-dimensional smooth distribution only, not on the whole tangent manifold. Our purpose is to construct a generalized nonlinear connection for a sub-Finslerian manifold, called [Formula: see text]-connection by the Legendre transformation which characterizes normal extremals of a sub-Finsler structure as geodesics of this connection. We also wish to investigate some of its properties like normal, adapted, partial and metrical.

Determination of the critical manifold tangent space and curvature with Monte Carlo renormalization group.

We show that the critical manifold of a statistical mechanical system in the vicinity of a critical point is locally accessible through correlation functions at that point. A practical numerical method is presented to determine the tangent space and the curvature to the critical manifold with variational Monte Carlo renormalization group. Because of the use of a variational bias potential of the coarse-grained variables, critical slowing down is greatly alleviated in the Monte Carlo simulation. In addition, this method is free of truncation error. We study the isotropic Ising model on square and cubic lattices, the anisotropic Ising model, and the tricritical Ising model on square lattices to illustrate the method.

Notes on the Sasaki metric

We survey on the geometry of the tangent bundle of a Riemannian manifold, endowed with the classical metric established by S. Sasaki 60 years ago. Following the results of Sasaki, we try to write and deduce them by different means. Questions of vector fields, mainly those arising from the base, are related as invariants of the classical metric, contact and Hermitian structures. Attention is given to the natural notion of extension or complete lift of a vector field, from the base to the tangent manifold. Few results are original, but finally new equations of the mirror map are considered.

The shape space of discrete orthogonal geodesic nets

Discrete orthogonal geodesic nets (DOGs) are a quad mesh analogue of developable surfaces. In this work we study continuous deformations on these discrete objects. Our main theoretical contribution is the characterization of the shape space of DOGs for a given net connectivity. We show that generally, this space is locally a manifold of a fixed dimension, apart from a set of singularities, implying that DOGs are continuously deformable. Smooth flows can be constructed by a smooth choice of vectors on the manifold's tangent spaces, selected to minimize a desired objective function under a given metric. We show how to compute such vectors by solving a linear system, and we use our findings to devise a geometrically meaningful way to handle singular points. We base our shape space metric on a novel DOG Laplacian operator, which is proved to converge under sampling of an analytical orthogonal geodesic net. We further show how to extend the shape space of DOGs by supporting creases and curved folds and apply the developed tools in an editing system for developable surfaces that supports arbitrary bending, stretching, cutting, (curved) folds, as well as smoothing and subdivision operations.

TEC Forecasting Based on Manifold Trajectories

In this paper, we present a method for forecasting the ionospheric Total Electron Content (TEC) distribution from the International GNSS Service’s Global Ionospheric Maps. The forecasting system gives an estimation of the value of the TEC distribution based on linear combination of previous TEC maps (i.e., a set of 2D arrays indexed by time), and the computation of a tangent subspace in a manifold associated to each map. The use of the tangent space to each map is justified because it allows modeling the possible distortions from one observation to the next as a trajectory on the tangent manifold of the map. The coefficients of the linear combination of the last observations along with the tangent space are estimated at each time stamp to minimize the mean square forecasting error with a regularization term. The estimation is made at each time stamp to adapt the forecast to short-term variations in solar activity.

Maximum Likelihood Estimation for Multiple Camera Target Tracking on Grassmann Tangent Subspace.

In this paper, we introduce a likelihood model for tracking the location of object in multiple view systems. Our proposed model transforms conventional nonlinear Euclidean estimation model to an estimation model based on the manifold tangent subspace. In this paper, we show that by decomposition of input noise into two parts and description of model by exponential map, real observations in the Euclidean geometry can be transformed to the manifold tangent subspace. Moreover, by obtained tangent subspace likelihood function, we propose two iterative and noniterative maximum likelihood estimation approaches which numerical results show their good performance.

Extended disformal approach in the scenario of rainbow gravity

We investigate all feasible mathematical representations of disformal transformations on a space-time metric according to the action of a linear operator upon the manifold's tangent and cotangent bundles. The geometric, algebraic and group structures of this operator and their interfaces are analyzed in detail. Then, we scrutinize a possible physical application, providing a new covariant formalism for a phenomenological approach to quantum gravity known as Rainbow Gravity.

An Efficient Dynamic Formulation for Solving Rigid and Flexible Multibody Systems Based on Semirecursive Method and Implicit Integration

This paper presents a method for solving the dynamic equations of multibody systems containing both rigid and flexible bodies. The proposed method uses independent coordinates and projects the dynamic equations on the constraint tangent manifold by means of a velocity transformation matrix. It can be used with a wide variety of integration formulae, considering both fixed and variable stepsizes. Topological semirecursive methods are used to take advantage of the relatively small number of parameters needed. An in depth implementation analysis is performed in order to evaluate the terms involved in the integration process. Numerical and stability issues are also discussed.

Semisupervised Support Vector Machines With Tangent Space Intrinsic Manifold Regularization.

Semisupervised learning has been an active research topic in machine learning and data mining. One main reason is that labeling examples is expensive and time-consuming, while there are large numbers of unlabeled examples available in many practical problems. So far, Laplacian regularization has been widely used in semisupervised learning. In this paper, we propose a new regularization method called tangent space intrinsic manifold regularization. It is intrinsic to data manifold and favors linear functions on the manifold. Fundamental elements involved in the formulation of the regularization are local tangent space representations, which are estimated by local principal component analysis, and the connections that relate adjacent tangent spaces. Simultaneously, we explore its application to semisupervised classification and propose two new learning algorithms called tangent space intrinsic manifold regularized support vector machines (TiSVMs) and tangent space intrinsic manifold regularized twin SVMs (TiTSVMs). They effectively integrate the tangent space intrinsic manifold regularization consideration. The optimization of TiSVMs can be solved by a standard quadratic programming, while the optimization of TiTSVMs can be solved by a pair of standard quadratic programmings. The experimental results of semisupervised classification problems show the effectiveness of the proposed semisupervised learning algorithms.