Geodesic Function Research Articles

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.

9D Rotation Representation-SVD Fusion with Deep Learning for Unconstrained Head Pose Estimation

Abstract Accurately estimating human head pose poses a significant challenge across various application domains. To address the inherent limitations of previous approaches, this research proposes an unconstrained head pose estimation strategy. The method combines deep learning with rotation matrices, utilizing nine-dimensional vectors output by the neural network, which are projected back to rotation matrices in SO (3) space through singular value decomposition. This ensures both the smoothness and uniqueness of the rotation representation. The approach demonstrates distinct advantages in handling the rotation estimation task, particularly when the rotated representation is used as the model output. It not only avoids the discontinuity and double-coverage issues associated with prior methods but also enhances the stability of the representation in high-dimensional space, thereby improving the learning process. Additionally, the geodesic loss function is incorporated to train the network. The proposed strategy surpasses previous state-of-the-art methods, as evidenced by experiments conducted on the AFLW2000 and BIWI datasets.

Geodesic Basis Function Neural Network.

In the learning of existing radial basis function neural networks-based methods, it is difficult to propagate errors back. This leads to an inconsistency between the learning and recognition task. This article proposes a geodesic basis function neural network with subclass extension learning (GBFNN-ScE). The geodesic basis function (GBF), which is defined here for the first time, uses the geodetic distance in the manifold as a measure to obtain the response of the sample with respect to the local center. To learn network parameters by back-propagating errors for the purpose of correct classification, a specific GBF based on a pruned gamma encoding cosine function is constructed. This function has a concise and explicit expression on the hyperspherical manifold, which is conducive to the realization of error back propagation. In the preprocessing layer, a sample unitization method with no loss of information, nonnegative unit hyperspherical crown (NUHC) mapping, is proposed. The sample can be mapped to the support set of the GBF. To alleviate the problem that one-hot encoding is not effective enough in the differential expression of data labels within a class, a subclass extension (ScE) learning strategy is proposed. The ScE learning strategy can help the learned network be more robust. For the working of GBFNN-ScE, the original sample is projected onto the support set of GBF through the NUHC mapping. Then the mapped samples are sent to the nonlinear computation units composed of GBFs in the hidden layer. Finally, the response obtained in the hidden layer is weighted by the learned weight to obtain the network output. This article theoretically proves that the separability of the data with ScE learning is stronger. The experimental results show that the proposed GBFNN-ScE has a better performance in recognition tasks than existing methods. The ablation experiments show that the ideas of the GBFNN-ScE contribute to the algorithm performance.

Riemannian Maps From Kaehler Manifold With Generic Fibers

We study Riemannian maps from almost Hermitian manifolds to Riemannian manifolds for the case when the fibers are genericsubmanifold of the total space. We obtain the integrability conditionsfor the distributions while vertical distribution is always integrable. Wealso study the geometry of the leaves of the distribution which arisefrom such maps, and obtain the necessary and sufficient conditions forthe fibers as well as the total manifold to be generic product manifolds. We, further, obtain the necessary and sufficient condition for aRiemannian maps with generic fibers to be totally geodesic map.

On the almost h-conformal semi-slant Riemannian maps

As a generalization of conformal semi-slant submersions, semi-slant Riemannian maps, almost h-semi-slant submersions and almost h-semi-slant Riemannian maps, we introduce almost h-conformal semi-slant submersions and almost h-conformal semi-slant Riemannian maps. We give some examples of such maps and also introduce some types of pluriharmonic maps, invariant maps and geodesic maps. We study the geometry of foliations, the integrability of distributions, the properties of pluriharmonic maps, invariant maps and geodesic maps. We also investigate the condition for such maps to be totally geodesic and the harmonicity of such maps.

Features of the geometry of the five-dimensional pseudo-Euclidean space of index two

The article is devoted to the study of the geometry of subspaces of a five-dimensional pseudo-Euclidean space. This space is attractive because all kinds of semi-Euclidean, semi-pseudo-Euclidean, hyperbolic three-dimensional spaces with projective metrics are realized in its subspaces. In the sphere of the imaginary radius of space, de Sitter space is realized. Here there is a space with projective metrics in the sense of Cayley-Klein. It is a three-dimensional space with a metric that preserves space on itself when mapped linearly. The corresponding linear transformation is called the motion of this space. An interpretation of de Sitter space in a four-dimensional pseudo-Euclidean space is proved. Studies have confirmed that in subspaces of space , in addition to elliptic spaces, there is a geometry of three-dimensional spaces with projective metrics. De Sitter space of the second kind is also realized in the sphere of imaginary radius. De Sitter space is a geodesic mapping in four-dimensional Minkowski space.

Digitization of Precipitation

In addressing the challenges posed by outdated and insufficiently documented urban drainage, irrigation, and storm systems, this study aims to revolutionize urban infrastructure through digitalization. Leveraging insights from 25 invited experts representing various organizations, including highway committees, city administration, and city improvement departments, the research focuses on the city of Tashkent. Employing geodesic methodologies, the team meticulously documented the locations and conditions of underground facilities, utilizing city maps and visual aids provided by specialists. The resulting digital database, curated by geodesists, identifies responsible personnel, object types, and areas requiring attention. This innovative initiative not only fills crucial knowledge gaps but also lays the foundation for enhanced maintenance strategies and resource optimization, offering a paradigm shift in urban planning and management. Highlights: Comprehensive Digital Mapping: The study employs advanced geodesic techniques to create a detailed digital map of aging drainage, irrigation, and storm systems, offering a comprehensive overview of their current conditions. Knowledge Integration from Diverse Experts: Gathering insights from 25 experts across various organizations, the research consolidates valuable information on underground facilities, enabling a holistic understanding of the urban infrastructure challenges. Strategic Resource Optimization: The resulting digital database not only identifies critical maintenance areas but also assigns responsibility, paving the way for strategic resource allocation and efficient urban infrastructure management. Keyword: Urban Infrastructure, Digitalization, Drainage Systems, Geodesic Mapping, Resource Optimization

Critical Analyzing on Sand, Gravel, and Stone Mining on the Ongka River for Materials which are Not Metalic

Historically, mining rivers for building materials has harmed both the river and the surrounding landscape. To ensure that mining in the river won't cause irreparable damage, we need to conduct in-depth scientific research. They plan to excavate the Osaka Demos River bed for building materials. The researchers employed geodesic mapping to figure out the contours of the river at various points. They examined the river's flow over a distance of 600 metres and over three distinct time spans (25, 50, and 100 years). Sand, gravel, and stones can settle to the river floor as sediment as a river alters its course. For a 100-year return rate flood, the HSS-Snyder technique calculated a flow rate of 1249.1692 cubic metres per second, 1178.8064 cubic metres per second for a 50-year return rate flood, and 1104.2372 cubic metres per second for a 25-year return rate flood. The river is shown to overflow in the simulation when large floods occur at regular intervals of 25, 50, or 100 years. The trapezoid shape will be used in the new technological river design. We use data on river form changes to calculate the volume of dirt that needs to be excavated: 88,347.42 cubic metres.

Properties of Anti-Invariant Submersions and Some Applications to Number Theory

In this article, we investigate anti-invariant Riemannian and Lagrangian submersions onto Riemannian manifolds from the Lorentzian para-Sasakian manifold. We demonstrate that, for these submersions, horizontal distributions are not integrable and their fibers are not totally geodesic. As a result, they are not totally geodesic maps. The harmonicity of such submersions is also examined. We specifically prove that they are not harmonic when the Reeb vector field is horizontal. Finally, we provide an illustration of our findings and mention some number-theoretic applications for the same submersions.

Karush-Kuhn-Tucker optimality conditions for non-smooth geodesic quasi-convex optimization on Riemannian manifolds

This paper is committed to studying Karush-Kuhn-Tucker (in short, KKT) type necessary and sufficient optimality conditions for non-smooth quasi-convex (geodesic sense) optimization problems on Riemannian manifolds. Recently, Ansari et al. [Ansari QH, Babu F, Zeeshan M. Incremental quasi-subgradient method for minimizing geodesic quasi-convex function on Riemannian manifolds with applications. Numer Funct Anal Optim. 2022;42(13):1492–1521. doi: 10.1080/01630563.2021.2001823] defined the quasi-subdifferential on Riemannian manifolds and established the existence results of the quasi-subdifferential. We provide several auxiliary results for the quasi-subdifferential in the current study. We offer the KKT optimality conditions for the quasi-convex optimization problems on Riemannian manifolds with or without the Slater-constraint qualifications. To verify the suggested outcomes, we formulate numerical examples. In addition, we also provide our results in the Euclidean spaces, which are original and distinct from earlier findings in the Euclidean spaces.

English

Mining nonmetallic sand, gravel, and river stones in a river channel has previously caused damage to the river channel itself and the surrounding environment. Technical studies are needed to minimize the damage in mining in the river channel. Ongkag Dumoga River channel is planned to be the location of mining non-metal material for sand, gravel and river stone. The research method was carried out by geodesic mapping to determine the profile of the river cross-section, hydrological analysis with a return period of 25, 50, and 100 years and hydraulic analysis on the river flow along the 600 m. The design of the post-mining river technical profile is carried out to improve the capacity of the existing river cross-section. Changes in river profile are the potential volume of nonmetallic material in the form of sand, gravel and river stones that become river sediment. The results of the analysis of flood discharge HSS-Snyder method get a discharge of 1249.1692 m³ / s at the 100-year return, 1178.8064 m³ / s at the 50-year return period, 1104.2372 m³ / s at the 25-year return period. The simulated flood discharge shows a flood overflow in the existing river during the 25, 50 and 100-year floods. The new technical river cross-section will use the economic cross-section of the trapezium cross-section. Changes in river profiles are calculated to get a potential excavation volume of 88,347.42 m³.

A note on conformal bi-slant submersion from Kenmotsu manifold

In this article, we examine conformal bi-slant submersion from Kenmotsu manifold onto a Riemannian manifold which generalizes the conformal semi-invariant, conformal semi-slant and conformal hemi-slant submersions and non-trivial examples are provided. We have also covered integrability requirements and address the necessary and sufficient conditions for the totally geodesicness of distributions. Moreover, the sufficient conditions for a conformal bi-slant submersion to be a totally geodesic map is investigated. The condition for a total manifold of the submersion to be twisted product is also studied, followed by other decomposition theorems.

Hessian-based semi-supervised feature selection using generalized uncorrelated constraint

Feature selection (FS) aims to eliminate redundant features and choose the informative ones. Since labeled data are not always easily available and abundant unlabeled data are often accessible, the importance of semi-supervised FS (SSFS), which selects the informative features utilizing the labeled and unlabeled data, becomes apparent. Semi-supervised sparse FS based on graph Laplacian (GL) regularization has become a popular technique. The GL regularization biases the solution towards a constant geodesic function, has a poor extrapolating ability, and cannot preserve well the topological structure of data. Traditional ridge regression is widely utilized by the GL-based semi-supervised sparse FS, but the results cannot gain the closed-form solution and carry out the manifold structure. To tackle these problems, we propose a Hessian-based SSFS framework using the generalized uncorrelated constraint (HSFSGU) and the mixed ℓ2,p-norm (0<p≤1) regularization. We also propose a Hessian–Laplacian-based SSFS framework using the generalized uncorrelated constraint (HLSFSGU) which utilizes the combination of GL and Hessian matrices. Our frameworks use the Hessian regularization for maintaining the topological structure of data well, the ℓ2,p-norm regularization for making the projection matrix appropriate for FS, and the generalized uncorrelated constraint for preventing exceeding row sparsity of the projection matrix and providing the elegant closed-form solution for our frameworks. We also present a unified algorithm to solve the frameworks for convex and non-convex cases, and prove its convergence. The results of the experiments depict the ability of the Hessian-based frameworks under generalized uncorrelated constraint in selecting the informative features.

New Generalization of Geodesic Convex Function

As a generalization of a geodesic function, this paper introduces the notion of the geodesic φE-convex function. Some properties of the φE-convex function and geodesic φE-convex function are established. The concepts of a geodesic φE-convex set and φE-epigraph are also given. The characterization of geodesic φE-convex functions in terms of their φE-epigraphs, are also obtained.

Functional Additive Models on Manifolds of Planar Shapes and Forms

The “shape” of a planar curve and/or landmark configuration is considered its equivalence class under translation, rotation, and scaling, its “form” its equivalence class under translation and rotation while scale is preserved. We extend generalized additive regression to models for such shapes/forms as responses respecting the resulting quotient geometry by employing the squared geodesic distance as loss function and a geodesic response function to map the additive predictor to the shape/form space. For fitting the model, we propose a Riemannian L 2-Boosting algorithm well suited for a potentially large number of possibly parameter-intensive model terms, which also yields automated model selection. We provide novel intuitively interpretable visualizations for (even nonlinear) covariate effects in the shape/form space via suitable tensor-product factorization. The usefulness of the proposed framework is illustrated in an analysis of (a) astragalus shapes of wild and domesticated sheep and (b) cell forms generated in a biophysical model, as well as (c) in a realistic simulation study with response shapes and forms motivated from a dataset on bottle outlines. Supplementary materials for this article are available online.

A local spline regression-based framework for semi-supervised sparse feature selection

Feature selection (FS) is extensively applied in many machine learning applications for the selection of relevant features from data sets. A lot of unlabeled data are available in a variety of applications that can be exploited for semi-supervised FS to address the lack of labeled data and improve learning performance. Recently, semi-supervised sparse FS based on graph Laplacian has obtained considerable research interest, which uses the correlation between features in the process of FS. However, the Laplacian regularization has a weak extrapolating power and a bias towards the constant geodesic function, and cannot retain the local topology well. In this paper, a spline regression-based framework for semi-supervised sparse FS (SRS3FS) is proposed, which uses the mixed convex and non-convex ℓ2,p-norm (0 <p≤1) regularization to select the relevant features and consider the correlation between features. The framework exploits local spline regression to retain the geometry structure of labeled and unlabeled data and encodes the data distribution. A unified iterative algorithm is presented to solve the proposed framework for the convex and non-convex cases, and its convergence is theoretically and experimentally proved. Experiments on several data sets illustrate the effectiveness of our framework in the selection of the most relevant and discriminative features.

Three-halves variation of geodesics in the directed landscape

We show that geodesics in the directed landscape have $3/2$-variation and that weight functions along the geodesics have cubic variation. We show that the geodesic and its landscape environment around an interior point has a small-scale limit. This limit is given in terms of the directed landscape with Brownian-Bessel boundary conditions. The environments around different interior points are asymptotically independent. We give tail bounds with optimal exponents for geodesic and weight function increments. As an application of our results, we show that geodesics are not H\"older-$2/3$ and that weight functions are not H\"older-$1/3$, although these objects are known to be H\"older with all lower exponents.

New Hadamard-type inequality for new class of geodesic convex functions

This paper aims to introduce the concept of (\( E,\mu,\kappa \))-convex function by using special inequality. Hadamard integral inequality for this new class of geodesic convex function in the case of Lebesgue and Sugeno integrals is given.

Pluriharmonic conformal bi-slant Riemannian maps

In this study, notion of pluriharmonic map applied onto conformal bi-slant Riemannian maps from a Kaehler manifold to a Riemannian manifold to examine its geometric properties. Such that, relations between pluriharmonic map, horizontally homothetic map and totally geodesic map were obtained.

Anti-Invariant Lorentzian Submersions From Lorentzian Concircular Structure Manifolds

This research article attempts to investigate anti-invariant Lorentzian submersions and the Lagrangian Lorentzian submersions (LLS) from the Lorentzian concircular structure [in short (LCS)n] manifolds onto semi-Riemannian manifolds with relevant non-trivial examples. It is shown that the horizontal distributions of such submersions are not integrable and their fibers are not totally geodesic. As a result, they can not be totally geodesic maps. Anti-invariant and Lagrangian submersions are also explored for their harmonicity. We illustrate that if the Reeb vector field is horizontal, the anti-invariant and LLS can not be harmonic.