High-dimensional Euclidean Space Research Articles

Learning on manifolds without manifold learning

Function approximation based on data drawn randomly from an unknown distribution is an important problem in machine learning. The manifold hypothesis assumes that the data is sampled from an unknown submanifold of a high dimensional Euclidean space. A great deal of research deals with obtaining information about this manifold, such as the eigendecomposition of the Laplace–Beltrami operator or coordinate charts, and using this information for function approximation. This two-step approach implies some extra errors in the approximation stemming from estimating the basic quantities of the data manifold in addition to the errors inherent in function approximation. In this paper, we project the unknown manifold as a submanifold of an ambient hypersphere and study the question of constructing a one-shot approximation using a specially designed sequence of localized spherical polynomial kernels on the hypersphere. Our approach does not require preprocessing of the data to obtain information about the manifold other than its dimension. We give optimal rates of approximation for relatively “rough” functions.

G[formula omitted]BFNN: Generalized geodesic basis function neural network

Real-world data is typically distributed on low-dimensional manifolds embedded in high-dimensional Euclidean spaces. Accurately extracting spatial distribution features on general manifolds that reflect the intrinsic characteristics of data is crucial for effective feature representation. Therefore, we propose a generalized geodesic basis function neural network (G2BFNN) architecture. The generalized geodesic basis functions (G2BF) are defined based on generalized geodesic distances. The generalized geodesic distance metric (G2DM) is obtained by learning the manifold structure. To implement this architecture, a specific G2BFNN, named discriminative local preserving projection-based G2BFNN (DLPP-G2BFNN) is proposed. DLPP-G2BFNN mainly contains two modules, namely the manifold structure learning module (MSLM) and the network mapping module (NMM). In the MSLM module, a supervised adjacency graph matrix is constructed to constrain the learning of the manifold structure. This enables the learned features in the embedding subspace to maintain the manifold structure while enhancing the discriminability. The features and G2DM learned in the MSLM are fed into the NMM. Through the G2BF in the NMM, the spatial distribution features on manifold are obtained. Finally, the output of the network is obtained through the fully connected layer. Compared with the local response neural network based on Euclidean distance, the proposed network can reveal more essential spatial structure characteristics of the data. Meanwhile, the proposed G2BFNN is a generalized network architecture that can be combined with any manifold learning method, showcasing high scalability. The experimental results demonstrate that the proposed DLPP-G2BFNN outperforms existing methods by utilizing fewer kernels while achieving higher recognition performance.

Some Aspects of Maxwell’s Equations, Klein-Gordon Equations, and Heat and Mass Transfer Equations in an n-Dimensional Maximally Symmetric Space-Time from the Classical and Quantum Mechanical Standpoints

This manuscript examines Maxwell’s equations, Klein-Gordon equations, and heat and mass transfer equations in n-dimensional maximally symmetric space-time. It investigates these equations in spherical and hyperbolic spaces embedded in higher-dimensional Euclidean and Minkowski spaces. The study focuses on the implications of these geometries and symmetries on the behaviour of the equations, highlighting how specific transformations and parametrizations impact their solutions. The findings reveal the underlying connections between geometric symmetries and physical laws, providing insights into their possible applications in theoretical physics. We touch upon both classical and quantum mechanical aspects of density and velocity evolutions with time in the universe. Quantum mechanical aspects of single and two-particle state evolution and statistical moments of the matter four-current are derived from the quantum Boltzmann equation and Feynman’s path integral method for fields applied to gravity interacting with electrons and positrons.

Sparse Signal Recovery via Rescaled Matching Pursuit

We propose the Rescaled Matching Pursuit (RMP) algorithm to recover sparse signals in high-dimensional Euclidean spaces. The RMP algorithm has less computational complexity than other greedy-type algorithms, such as Orthogonal Matching Pursuit (OMP). We show that if the restricted isometry property is satisfied, then the upper bound of the error between the original signal and its approximation can be derived. Furthermore, we prove that the RMP algorithm can find the correct support of sparse signals from random measurements with a high probability. Our numerical experiments also verify this conclusion and show that RMP is stable with the noise. So, the RMP algorithm is a suitable method for recovering sparse signals.

On the Budgeted Priority p-Median Problem in High-Dimensional Euclidean Spaces

On the Budgeted Priority p-Median Problem in High-Dimensional Euclidean Spaces

How do kernel-based sensor fusion algorithms behave under high-dimensional noise?

Abstract We study the behavior of two kernel based sensor fusion algorithms, nonparametric canonical correlation analysis (NCCA) and alternating diffusion (AD), under the nonnull setting that the clean datasets collected from two sensors are modeled by a common low-dimensional manifold embedded in a high-dimensional Euclidean space and the datasets are corrupted by high-dimensional noise. We establish the asymptotic limits and convergence rates for the eigenvalues of the associated kernel matrices assuming that the sample dimension and sample size are comparably large, where NCCA and AD are conducted using the Gaussian kernel. It turns out that both the asymptotic limits and convergence rates depend on the signal-to-noise ratio (SNR) of each sensor and selected bandwidths. On one hand, we show that if NCCA and AD are directly applied to the noisy point clouds without any sanity check, it may generate artificial information that misleads scientists’ interpretation. On the other hand, we prove that if the bandwidths are selected adequately, both NCCA and AD can be made robust to high-dimensional noise when the SNRs are relatively large.

Visual cue-aided human supervised robot navigation guidance in photometrically challenging environments using adaptive spatial-feature kernel-guided bilateral LPP

The objective of an effective human-robot collaborative (HRC) task is the maximization of human-robot competencies while ensuring the user’s convenience. In photometrically challenging and unstructured HRC environments, data obtained from vision sensors often tend to get degraded due to illumination irregularities and spatio-temporal complexities. To extract useful and discriminative features from the data under such situations, locality-sensitive methods like locality preserving projections (LPPs) become quite useful as it captures the local geometric structure of the high-dimensional data. In LPP, the local structural information is encoded in the form of weight values between two samples in the higher-dimensional Euclidean space. The weight values are learned in a regular and continuous manner which only depends on the spatial distribution of the data. Moreover, because of its weight dependency solely on the Euclidean distance, improper weight values can occur frequently, as the Euclidean distance is susceptible to noise, outliers, and different types of geometrical transformations. This paper proposes an adaptive weight learning method to be utilized in the weight computation of LPP, which allows it to adaptively select and extract more discriminative features from the higher-dimensional input data while preserving the intrinsic structural information of the data. Additionally, to alleviate the issues with spatial dependency, the concept of bilateral filtering that incorporates the range weights from the feature space along with the similarity weight in the Euclidean space has been utilized here. This paper proposes an augmented version of adaptive spatial-feature kernel-guided bilateral filtering inspired LPP which addresses two of these basic and fundamental issues of the conventional LPP.

Subsets of rectifiable curves in Banach spaces I: Sharp exponents in traveling salesman theorems

The analyst’s traveling salesman problem (TSP) is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jones in 1990 and subsequently solved in higher-dimensional Euclidean spaces by Okikiolu in 1992 and in the infinite-dimensional Hilbert space ℓ2 by Schul in 2007. In this paper, we establish sharp extensions of Schul’s necessary and sufficient conditions for a bounded set E⊂ℓp to be contained in a rectifiable curve from p=2 to 1<p<∞. While the necessary and sufficient conditions coincide when p=2, we demonstrate that there is a strict gap between the necessary condition and sufficient condition when p≠2. We also identify and correct technical errors in the proof by Schul. This investigation is partly motivated by recent work of Edelen, Naber, and Valtorta on Reifenberg-type theorems in Banach spaces and complements work of Hahlomaa and recent work of David and Schul on the analyst’s TSP in general metric spaces.

Local approximation of operators

Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator between metric spaces X and Y. We study the problem of determining the degree of approximation of such operators on a compact subset KX⊂X using a finite amount of information. If F:KX→KY, a well established strategy to approximate F(F) for some F∈KX is to encode F (respectively, F(F)) in terms of a finite number d (respectively m) of real numbers. Together with appropriate reconstruction algorithms (decoders), the problem reduces to the approximation of m functions on a compact subset of a high dimensional Euclidean space Rd, equivalently, the unit sphere Sd embedded in Rd+1. The problem is challenging because d, m, as well as the complexity of the approximation on Sd are all large, and it is necessary to estimate the accuracy keeping track of the inter-dependence of all the approximations involved. In this paper, we establish constructive methods to do this efficiently; i.e., with the constants involved in the estimates on the approximation on Sd being O(d1/6). We study different smoothness classes for the operators, and also propose a method for approximation of F(F) using only information in a small neighborhood of F, resulting in an effective reduction in the number of parameters involved. To further mitigate the problem of large number of parameters, we propose prefabricated networks, resulting in a substantially smaller number of effective parameters. The problem is studied in both deterministic and probabilistic settings.

A Logic of East and West

We propose a logic of east and west (LEW ) for points in 1D Euclidean space. It formalises primitive direction relations: east (E), west (W) and indeterminate east/west (Iew). It has a parameter τ ∈ N&gt;1, which is referred to as the level of indeterminacy in directions. For every τ ∈ N&gt;1, we provide a sound and complete axiomatisation of LEW , and prove that its satisfiability problem is NP-complete. In addition, we show that the finite axiomatisability of LEW depends on τ : if τ = 2 or τ = 3, then there exists a finite sound and complete axiomatisation; if τ &gt; 3, then the logic is not finitely axiomatisable. LEW can be easily extended to higher-dimensional Euclidean spaces. Extending LEW to 2D Euclidean space makes it suitable for reasoning about not perfectly aligned representations of the same spatial objects in different datasets, for example, in crowd-sourced digital maps.

Side effects of learning from low-dimensional data embedded in a Euclidean space

The low-dimensional manifold hypothesis posits that the data found in many applications, such as those involving natural images, lie (approximately) on low-dimensional manifolds embedded in a high-dimensional Euclidean space. In this setting, a typical neural network defines a function that takes a finite number of vectors in the embedding space as input. However, one often needs to consider evaluating the optimized network at points outside the training distribution. This paper considers the case in which the training data are distributed in a linear subspace of $${\mathbb {R}}^d$$ . We derive estimates on the variation of the learning function, defined by a neural network, in the direction transversal to the subspace. We study the potential regularization effects associated with the network’s depth and noise in the codimension of the data manifold. We also present additional side effects in training due to the presence of noise.

Extrinsic Bayesian Optimization on Manifolds

We propose an extrinsic Bayesian optimization (eBO) framework for general optimization problems on manifolds. Bayesian optimization algorithms build a surrogate of the objective function by employing Gaussian processes and utilizing the uncertainty in that surrogate by deriving an acquisition function. This acquisition function represents the probability of improvement based on the kernel of the Gaussian process, which guides the search in the optimization process. The critical challenge for designing Bayesian optimization algorithms on manifolds lies in the difficulty of constructing valid covariance kernels for Gaussian processes on general manifolds. Our approach is to employ extrinsic Gaussian processes by first embedding the manifold onto some higher dimensional Euclidean space via equivariant embeddings and then constructing a valid covariance kernel on the image manifold after the embedding. This leads to efficient and scalable algorithms for optimization over complex manifolds. Simulation study and real data analyses are carried out to demonstrate the utilities of our eBO framework by applying the eBO to various optimization problems over manifolds such as the sphere, the Grassmannian, and the manifold of positive definite matrices.

A High-Gain Observer for Embedded Polynomial Dynamical Systems

This article deals with the construction of high-gain observers for autonomous polynomial dynamical systems. In contrast to the usual approach, the system’s state is embedded into a higher dimensional Euclidean space. The observer state will be contained in said Euclidean space, which has usually higher dimension than the system’s state space. Due to this embedding it is possible to avoid singularities in the observation matrix. For some systems this even allows constructing global observers in a structured way, which would not be possible in the lower-dimensional case. Finally, the state estimate in the original coordinates can be obtained by a projection. The proposed method is applied on some example systems.

Generalized n-Dimensional Rigid Registration: Theory and Applications.

The generalized rigid registration problem in high-dimensional Euclidean spaces is studied. The loss function is minimized with an equivalent error formulation by the Cayley formula. The closed-form linear least-square solution to such a problem is derived which generates the registration covariances, i.e., uncertainty information of rotation and translation, providing quite accurate probabilistic descriptions. Simulation results indicate the correctness of the proposed method and also present its efficiency on computation-time consumption, compared with previous algorithms using singular value decomposition (SVD) and linear matrix inequality (LMI). The proposed scheme is then applied to an interpolation problem on the special Euclidean group SE(n) with covariance-preserving functionality. Finally, experiments on covariance-aided Lidar mapping show practical superiority in robotic navigation.

Different types of multifractal measures in separable metric spaces and their applications

&lt;abstract&gt;&lt;p&gt;The properties of various fractal and multifractal measures and dimensions have been under extensive study in the real-line and higher-dimensional Euclidean spaces. In non-Euclidean spaces, it is often impossible to construct non-trivial self-similar or self-conformal sets, etc. We consider in the present paper the proper way to phrase the definitions for use in general metric spaces. We investigate the relative Hausdorff measures $ {\mathscr H}_{ {\boldsymbol{\mu}}}^{q, t} $ and the relative packing measures $ {\mathscr P}_{ {\boldsymbol{\mu}}}^{q, t} $ defined in a separable metric space. We give some product inequalities which are a consequence of a new version of density theorems for these measures. Moreover, we prove that $ {\mathscr H}_{ {\boldsymbol{\mu}}}^{q, t} $ and $ {\mathscr P}_{ {\boldsymbol{\mu}}}^{q, t} $ can be expressed as Henstock-Thomson variation measures. The question of the weak-Vitali property arises in this context.&lt;/p&gt;&lt;/abstract&gt;

Byzantine Resilient Distributed Learning in Multirobot Systems

Distributed machine learning algorithms are increasingly used in multirobot systems and are prone to Byzantine attacks. In this article, we consider a distributed implementation of the stochastic gradient descent (SGD) algorithm in a cooperative network, where networked agents optimize a global loss function using SGD on the local data and aggregation of the estimates of immediate neighbors. Byzantine agents can send arbitrary estimates to their neighbors, which may disrupt the convergence of normal agents to the optimum state. We show that if every normal agent combines its neighbors’ estimates (states) such that the aggregated state is in the convex hull of its normal neighbors’ states, then the resilient convergence is guaranteed. To assure this sufficient condition, we propose a resilient aggregation rule based on the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">centerpoint</i> , which is a generalization of the median in the higher-dimensional Euclidean space. We evaluate our results using examples of target pursuit and pattern recognition in multirobot systems. The evaluation results demonstrate that distributed learning with average, coordinate-wise median, and geometric median-based aggregation rules fail to converge to the optimum state, whereas the centerpoint-based aggregation rule is resilient in the same scenario.

Geometric characterizations of Gromov hyperbolic Hölder domains

Abstract In this paper, we investigate Hölder continuity of quasiconformal mappings in ℝ n {\mathbb{R}^{n}} from the points of view of quasihyperbolic geometry and the theory of Gromov hyperbolic spaces. We establish several characterizations of Gromov hyperbolic domains satisfying the Gehring–Martio-type quasihyperbolic boundary conditions. As applications, we generalize certain results concerning Hölder continuity of conformal mappings, establishing counterparts of results of Becker and Pommerenke, Smith and Stegenga, and Näkki and Palka in higher-dimensional Euclidean spaces.

Vlasov equations on digraph measures

Many science phenomena are described as interacting particle systems (IPS). The mean field limit (MFL) of large all-to-all coupled deterministic IPS is given by the solution of a PDE, the Vlasov Equation (VE). Yet, many applications demand IPS coupled on networks/graphs. In this paper, we are interested in IPS on a sequence of directed graphs, or digraphs for short. It is interesting to know, how the limit of a sequence of digraphs associated with the IPS influences the macroscopic MFL. This paper studies VEs on a generalized digraph, regarded as limit of a sequence of digraphs, which we refer to as a digraph measure (DGM) to emphasize that we work with its limit via measures. We provide (i) unique existence of solutions of the VE on continuous DGMs, and (ii) discretization of the solution of the VE by empirical distributions supported on solutions of an IPS via ODEs coupled on a sequence of digraphs converging to the given DGM. Our result extends existing results on one-dimensional Kuramoto-type networks coupled on dense graphs. Here we allow the underlying digraphs to be not necessarily dense which include many interesting graphical structures such that stars, trees and rings, which have been frequently used in many sparse network models in finance, telecommunications, physics, genetics, neuroscience, and social sciences. A key contribution of this paper is a nontrivial generalization of Neunzert's in-cell-particle approach for all-to-all coupled indistinguishable IPS with global Lipschitz continuity in Euclidean spaces to distinguishable IPS on heterogeneous digraphs with local Lipschitz continuity, via a measure-theoretic viewpoint. The approach together with the metrics is different from the known techniques in Lp-functions using graphons and their generalization by means of harmonic analysis of locally compact Abelian groups. Finally, to demonstrate the wide applicability, we apply our results to various models in higher-dimensional Euclidean spaces in epidemiology, ecology, and social sciences.

Reflections on Equidistant Sets

We show that reflections on curves that can be obtained as the equidistant set to convex focal sets have exactly the same focal property as reflections on perpendicular bisectors when viewed on the level of first-order linear approximations. Hence, we obtain a suitable focal property for reflections on equidistant sets with convex focal sets. This focal property generalizes the classical focal properties of conic sections. All these results can be extended to higher-dimensional Euclidean spaces.

Regular Solutions of the Stationary Navier–Stokes Equations on High Dimensional Euclidean Space

We study the existence of regular solutions of the incompressible stationary Navier–Stokes equations in n-dimensional Euclidean space with a given bounded external force of compact support. In dimensions \(n\le 5\), the existence of such solutions was known. In this paper, we extend it to dimensions \(n\le 15\).