Showing papers on "Distribution (differential geometry) published in 2021"

Poisson Generated Family of Distributions: A Review

Abstract: The present article represents a survey on Poisson generated family of distributions. Based on this family of distribution, several transformations and distributions have been proposed. Out of which, some of them are proposed by referencing it, and some are independent. The family can be proposed by using the compounding concept of zero truncated Poisson distribution with any other model or family of distributions. Here, we provide a complete survey on this family of distributions and list the contributory related research works. We also address 12 power series distributions, 77 distributions based on the Poisson family of distribution, and 23 distributions, based on different ten transformation methods based on this family of distribution. These numbers show the importance of the Poisson family of distribution.

Reduction by λ –symmetries and σ –symmetries: a Frobenius approach

Abstract: Different kinds of reduction for ordinary differential equations, such as λ –symmetry and σ –symmetry reductions, are recovered as particular cases of Frobenius reduction theorem for distribution of vector fields. This general approach provides some hints to tackle the reconstruction problem and to solve it under suitable assumptions on the distribution involved in the reduction process.

Local Manifold Embedding Cross-Domain Subspace Learning for Drift Compensation of Electronic Nose Data

Abstract: The gas sensor drift problem arises from the bias of data, which is known as a significant problem in the artificial olfactory community. Traditionally, hardware calibration methods are laborious and ineffective due to frequent recalibration actions involving different gases, and some calibration transfer and baseline calibration methods are not effective enough. In this work, a local manifold embedding cross-domain subspace learning (LME-CDSL) model is proposed based on domain distribution alignment. It is a unified subspace learning model combined with manifold learning and domain adaptation, which tends to explore a latent transform matrix that not only enforces the drifted target domain data to learn the manifold of nondrifted source domain data but also adopts the domain adaptation method to align the domain data distribution. In general, the LME-CDSL model has three features: 1) the unsupervised and adaptation distribution subspace projection can be easily computed through eigenvector decomposition; 2) the local linear manifold learns to achieve the compact representations of high-dimensional data and is capable of preserving the local features of nondrifted samples; and 3) the domain adaptation part utilizes the maximum mean discrepancy (MMD) and variance maximization to make the sample distributions of different domains more similar and preserve the intrinsic properties. For long-term and short-term drift compensation on a single E-nose system, the local manifold embedding cross-domain subspace learning (LME-CDSL) model obtains the average recognition accuracy of 70.95% and 74.09%, respectively, while 71.71% and 73.96%, respectively for multiple identical E-nose systems with both long-term and interplate drift, which are higher than several comparative methods and proves the its effectiveness and superiority on anti-drift and gas recognition.

Codimension Two Spacelike Submanifolds Through a Null Hypersurface in a Lorentzian Manifold

Abstract: Most important examples of null hypersurfaces in a Lorentzian manifold admit an integrable screen distribution, which determines a spacelike foliation of the null hypersurface. In this paper, we obtain conditions for a codimension two spacelike submanifold contained in a null hypersurface to be a leaf of the (integrable) screen distribution. For this, we use the rigging technique to endow the null hypersurface with a Riemannian metric, which allows us to apply the classical Eschenburg maximum principle. We apply the obtained results to classical examples as generalized Robertson–Walker spaces and Kruskal space.

Sliding Mode Control for a Generalization of the Caginalp Phase-Field System

Abstract: In the present paper, we present and solve the sliding mode control (SMC) problem for a second-order generalization of the Caginalp phase-field system. This generalization, inspired by the theories developed by Green and Naghdi on one side, and Podio-Guidugli on the other, deals with the concept of thermal displacement, i.e., a primitive with respect to the time of the temperature. Two control laws are considered: the former forces the solution to reach a sliding manifold described by a linear constraint between the temperature and the phase variable; the latter forces the phase variable to reach a prescribed distribution $$\varphi ^*$$ . We prove existence, uniqueness as well as continuous dependence of the solutions for both problems; two regularity results are also given. We also prove that, under suitable conditions, the solutions reach the sliding manifold within finite time.

Spinors of real type as polyforms and the generalized Killing equation

Abstract: We develop a new framework for the study of generalized Killing spinors, where every generalized Killing spinor equation, possibly with constraints, can be formulated equivalently as a system of partial differential equations for a polyform satisfying algebraic relations in the Kahler–Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module $$\Sigma $$ of real type as a real algebraic variety in the Kahler–Atiyah algebra, which gives necessary and sufficient conditions for a polyform to be the square of a real spinor. We apply these results to Lorentzian four-manifolds, obtaining a new description of a real spinor on such a manifold through a certain distribution of parabolic 2-planes in its cotangent bundle. We use this result to give global characterizations of real Killing spinors on Lorentzian four-manifolds and of four-dimensional supersymmetric configurations of heterotic supergravity. In particular, we find new families of Einstein and non-Einstein four-dimensional Lorentzian metrics admitting real Killing spinors, some of which are deformations of the metric of $$\text {AdS}_4$$ space-time.

Manifold Matching via Deep Metric Learning for Generative Modeling.

Abstract: We propose a manifold matching approach to generative models which includes a distribution generator (or data generator) and a metric generator. In our framework, we view the real data set as some manifold embedded in a high-dimensional Euclidean space. The distribution generator aims at generating samples that follow some distribution condensed around the real data manifold. It is achieved by matching two sets of points using their geometric shape descriptors, such as centroid and $p$-diameter, with learned distance metric; the metric generator utilizes both real data and generated samples to learn a distance metric which is close to some intrinsic geodesic distance on the real data manifold. The produced distance metric is further used for manifold matching. The two networks are learned simultaneously during the training process. We apply the approach on both unsupervised and supervised learning tasks: in unconditional image generation task, the proposed method obtains competitive results compared with existing generative models; in super-resolution task, we incorporate the framework in perception-based models and improve visual qualities by producing samples with more natural textures. Experiments and analysis demonstrate the feasibility and effectiveness of the proposed framework.

Varieties of minimal rational tangents of unbendable rational curves subordinate to contact structures

Abstract: A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $${\mathcal O}_{{\mathbb P}^1}(1)^{\oplus p} \oplus {\mathcal O}_{{\mathbb P}^1}^{\oplus q}$$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x \in C,$ which is the germ of submanifolds ${\mathcal C}^C_x \subset {\mathbb P} T_x X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D \subset TX$ such that all small deformations of $C$ are tangent to $D$, one asks what kind of submanifolds of projective space can be realized as the VMRT ${\mathcal C}^C_x \subset {\mathbb P} D_x$. When $D \subset TX$ is a contact distribution, a well-known necessary condition is that ${\mathcal C}_x^C$ should be Legendrian with respect to the induced contact structure on ${\mathbb P} D_x$. We prove that this is also a sufficient condition: we construct a complex manifold $X$ with a contact structure $D \subset TX$ and an unbendable rational curve $C \subset X$ such that all small deformations of $C$ are tangent to $D$ and the VMRT ${\mathcal C}^C_x \subset {\mathbb P} D_x$ at some point $x\in C$ is projectively isomorphic to an arbitrarily given Legendrian submanifold. Our construction uses the geometry of contact lines on the Heisenberg group and a technical ingredient is the symplectic geometry of distributions the study of which has originated from geometric control theory.

Sturm theory with applications in geometry and classical mechanics

Abstract: Classical Sturm non-oscillation and comparison theorems as well as the Sturm theorem on zeros for solutions of second order differential equations have a natural symplectic version, since they describe the rotation of a line in the phase plane of the equation. In the higher dimensional symplectic version of these theorems, lines are replaced by Lagrangian subspaces and intersections with a given line are replaced by non-transversality instants with a distinguished Lagrangian subspace. Thus the symplectic Sturm theorems describe some properties of the Maslov index. Starting from the celebrated paper of Arnol’d on symplectic Sturm theory for optical Hamiltonians, we provide a generalization of his results to general Hamiltonians. We finally apply these results for detecting some geometrical information about the distribution of conjugate and focal points on semi-Riemannian manifolds and for studying the geometrical properties of the solutions space of singular Lagrangian systems arising in Celestial Mechanics.

Stability and Hopf Bifurcation Analysis of an Epidemic Model with Time Delay.

Abstract: Epidemic models are normally used to describe the spread of infectious diseases. In this paper, we will discuss an epidemic model with time delay. Firstly, the existence of the positive fixed point is proven; and then, the stability and Hopf bifurcation are investigated by analyzing the distribution of the roots of the associated characteristic equations. Thirdly, the theory of normal form and manifold is used to drive an explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions. Finally, some simulation results are carried out to validate our theoretic analysis.

Lightlike manifolds and Cartan geometries

Abstract: Lightlike Cartan geometries are introduced as Cartan geometries modelled on the future lightlike cone in Lorentz-Minkowski spacetime. Then, we provide an approach to the study of lightlike manifolds from this point of view. It is stated that every lightlike Cartan geometry on a manifold N provides a lightlike metric h with radical distribution globally spanned by a vector field Z. For lightlike hypersurfaces of a Lorentz manifold, we give the condition that characterizes when the pull-back of the Levi-Civita connection form of the ambient manifold is a lightlike Cartan connection on such hypersurface. In the particular case that a lightlike hypersurface is properly totally umbilical, this construction essentially returns the original lightlike metric. From the intrinsic point of view, starting from a given lightlike manifold (N, h), we show a method to construct a family of ambient Lorentzian manifolds that realize (N, h) as a hypersurface. This method is inspired on the Feffermann-Graham ambient metric construction in conformal geometry and provides a lightlike Cartan geometry on the original manifold when (N, h) is generic.

Geodesic-based distance reveals nonlinear topological features in neural activity from mouse visual cortex.

Abstract: An increasingly popular approach to the analysis of neural data is to treat activity patterns as being constrained to and sampled from a manifold, which can be characterized by its topology. The persistent homology method identifies the type and number of holes in the manifold, thereby yielding functional information about the coding and dynamic properties of the underlying neural network. In this work, we give examples of highly nonlinear manifolds in which the persistent homology algorithm fails when it uses the Euclidean distance because it does not always yield a good approximation to the true distance distribution of a point cloud sampled from a manifold. To deal with this issue, we instead estimate the geodesic distance which is a better approximation of the true distance distribution and can therefore be used to successfully identify highly nonlinear features with persistent homology. To document the utility of the method, we utilize a toy model comprised of a circular manifold, built from orthogonal sinusoidal coordinate functions and show how the choice of metric determines the performance of the persistent homology algorithm. Furthermore, we explore the robustness of the method across different manifold properties, like the number of samples, curvature and amount of added noise. We point out strategies for interpreting its results as well as some possible pitfalls of its application. Subsequently, we apply this analysis to neural data coming from the Visual Coding-Neuropixels dataset recorded at the Allen Institute in mouse visual cortex in response to stimulation with drifting gratings. We find that different manifolds with a non-trivial topology can be seen across regions and stimulus properties. Finally, we interpret how these changes in manifold topology along with stimulus parameters and cortical region inform how the brain performs visual computation.

Spectral geometry on manifolds with fibred boundary metrics II: heat kernel asymptotics

Abstract: In this paper we continue the analysis of spectral problems in the setting of complete manifolds with fibred boundary metrics, also referred to as $\phi$-metrics, as initiated in our previous work. We consider the Hodge Laplacian for a $\phi$-metric and construct the corresponding heat kernel as a polyhomogeneous conormal distribution on an appropriate manifold with corners. Our discussion is a generalization of an earlier work by Albin and provides a fundamental first step towards analysis of Ray-Singer torsion, eta-invariants and index theorems in the setting.

Manifold fitting algorithm of noisy manifold data based on variable-scale spectral graph

Abstract: Manifold fitting is a manifold verification technique for data with noise and manifold structures. By extracting the expected manifold structure, the reliability of the data manifold hypothesis can be determined, and the true structure of the data without noise can conform to a manifold. This paper proposes a manifold fitting algorithm for the variable-scale spectral graph theory and estimates the deviation of the manifold structure caused by noise. Considering the scale variations in frequency-domain analysis based on spectral graph theory, details of the data under the effect of small scale and characteristics of data shape under the effect of large scale are highlighted. This study uses the average calculation and meanshift method to obtain two types of mean vectors from each neighborhood, which are essential in suppressing noise and maintaining shape, respectively. Therefore, manifold fitting is carried out from two aspects, specifically weakening noise and characterizing the manifold shape. To obtain a closer estimate of the deviation caused by noise to the manifold structure, this study also estimates the neighborhood distribution of data under the effect of medium scale, obtains the covariance information of each neighborhood, and uses the variance information to estimate the manifold structure deviations.

Frobenius trace distributions for K3 surfaces

Abstract: We study the distribution of the Frobenius traces on $K3$ surfaces. We compare experimental data with the predictions made by the Sato--Tate conjecture, i.e. with the theoretical distributions derived from the theory of Lie groups assuming equidistribution. Our sample consists of generic $K3$ surfaces, as well as of such having real and complex multiplication. We report evidence for the Sato--Tate conjecture for the surfaces considered.

Rectangular Flows for Manifold Learning

Abstract: Normalizing flows are invertible neural networks with tractable change-of-volume terms, which allows optimization of their parameters to be efficiently performed via maximum likelihood. However, data of interest is typically assumed to live in some (often unknown) low-dimensional manifold embedded in high-dimensional ambient space. The result is a modelling mismatch since -- by construction -- the invertibility requirement implies high-dimensional support of the learned distribution. Injective flows, mapping from low- to high-dimensional space, aim to fix this discrepancy by learning distributions on manifolds, but the resulting volume-change term becomes more challenging to evaluate. Current approaches either avoid computing this term entirely using various heuristics, or assume the manifold is known beforehand and therefore are not widely applicable. Instead, we propose two methods to tractably calculate the gradient of this term with respect to the parameters of the model, relying on careful use of automatic differentiation and techniques from numerical linear algebra. Both approaches perform end-to-end nonlinear manifold learning and density estimation for data projected onto this manifold. We study the trade-offs between our proposed methods, empirically verify that we outperform approaches ignoring the volume-change term by more accurately learning manifolds and the corresponding distributions on them, and show promising results on out-of-distribution detection.

Kernel-based methods for Solving Time-Dependent Advection-Diffusion Equations on Manifolds.

Abstract: In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM) and ghost point diffusion maps (GPDM), to solve the time-dependent advection-diffusion PDE on unknown smooth manifolds without and with boundaries. The core idea is to directly approximate the spatial components of the differential operator on the manifold with a local integral operator and combine it with the standard implicit time difference scheme. When the manifold has a boundary, a simplified version of the GPDM approach is used to overcome the bias of the integral approximation near the boundary. The Monte-Carlo discretization of the integral operator over the point cloud data gives rise to a mesh-free formulation that is natural for randomly distributed points, even when the manifold is embedded in high-dimensional ambient space. Here, we establish the convergence of the proposed solver on appropriate topologies, depending on the distribution of point cloud data and boundary type. We provide numerical results to validate the convergence results on various examples that involve simple geometry and an unknown manifold. Additionally, we also found positive results in solving the one-dimensional viscous Burger's equation where GPDM is adopted with a pseudo-spectral Galerkin framework to approximate nonlinear advection term.

Partial Differential Equations and Quantum States in Curved Spacetimes

Abstract: In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction of physically admissible quantum states—the so-called Hadamard states—on globally hyperbolic spacetimes. We will review the notion of a propagator and discuss how it can be constructed in an explicit and invariant fashion, first on a Riemannian manifold and then on a Lorentzian spacetime. Finally, we will recall the notion of Hadamard state and relate the latter to hyperbolic propagators via the wavefront set, a subset of the cotangent bundle capturing the information about the singularities of a distribution.

Joint Discriminative Distribution Adaptation and Manifold Regularization for Unsupervised Domain Adaptation

Abstract: Unsupervised Domain Adaptation (UDA) has become a basic technology for cross-domain recognition and has received extensive attention in recent years. UDA aims to obtain a classifier for the target domain by learning source instances with different data distributions. However, traditional domain adaptation algorithms cannot effectively explore the manifold structure of data while reducing the distribution differences between domains. To address this problem, this paper proposes a new UDA framework called Joint Discriminative Distribution Adaptation and Manifold Regularization (DDAMR). DDAMR makes full use of the category information and geometric structure of samples in the Grassmann manifold to learn the domain-invariant classifier. Specifically, DDAMR performs discriminative distribution adaptation during dynamic distribution calibration to enhance the discrimination ability of the feature space. In addition, DDAMR introduces manifold regularization that can maintain the proximity relationship of the samples. It can maximize effectively the consistency between the prediction structure of the domain-invariant classifier $f$ and the inherent manifold structure of the sample. A large number of results from cross-domain experiments have demonstrated the effectiveness of our DDAMR algorithm.

Normal-Bundle Bootstrap

Abstract: Probabilistic models of data sets often exhibit salient geometric structure. Such a phenomenon is summed up in the manifold distribution hypothesis and can be exploited in probabilistic learning. H...

Studying partial hyperbolicity inside regimes of motion in Hamiltonian systems

Abstract: A chaotic trajectory in weakly chaotic higher-dimensional Hamiltonian systems may locally present distinct regimes of motion, namely, chaotic, semiordered, or ordered. Such regimes, which are consequences of dynamical traps, are defined by the values of the Finite-Time Lyapunov Exponents (FTLEs) calculated during specific time windows. The Covariant Lyapunov Vectors (CLVs) contain the information about the local geometrical structure of the manifolds, and the distribution of the angles between them has been used to quantify deviations from hyperbolicity. In this work, we propose to study the deviation of partial hyperbolicity using the distribution of the local and mean angles during each of the mentioned regimes of motion. A system composed of two coupled standard maps and the Henon–Heiles system are used as examples. Both are paradigmatic models to study the dynamics of mixed phase-space of conservative systems in discrete and continuous dynamical systems, respectively. Hyperbolic orthogonality is a general tendency in strong chaotic regimes. However, this is not true anymore for weakly chaotic systems and we must look separately at the regimes of motion. Furthermore, the distribution of angles between the manifolds in a given regime of motion allows us to obtain geometrical information about manifold structures in the tangent spaces. The description proposed here helps to explain important characteristics between invariant manifolds that occur inside the regimes of motion and furnishes a kind of visualization tool to perceive what happens in phase and tangent space of dynamical systems. This is crucial for higher-dimensional systems and to discuss distinct degrees of (non)hyperbolicity.

Information Geometry of the Exponential Family of Distributions with Progressive Type-II Censoring.

Abstract: In geometry and topology, a family of probability distributions can be analyzed as the points on a manifold, known as statistical manifold, with intrinsic coordinates corresponding to the parameters of the distribution. Consider the exponential family of distributions with progressive Type-II censoring as the manifold of a statistical model, we use the information geometry methods to investigate the geometric quantities such as the tangent space, the Fisher metric tensors, the affine connection and the α-connection of the manifold. As an application of the geometric quantities, the asymptotic expansions of the posterior density function and the posterior Bayesian predictive density function of the manifold are discussed. The results show that the asymptotic expansions are related to the coefficients of the α-connections and metric tensors, and the predictive density function is the estimated density function in an asymptotic sense. The main results are illustrated by considering the Rayleigh distribution.

PARALLELISM OF DISTRIBUTIONS AND GEODESICS ON F(±a2; ±b2)-STRUCTURE LAGRANGIAN MANIFOLD

Abstract: This paper deals with the Lagrange vertical structure on the vertical space T V ( E ) endowed with a non null (1,1) tensor field F V satisfying ( F v 2 - a 2 )( F v 2 + a 2 )( F v 2 - b 2 )( F v 2 + b 2 ) = 0. In this paper, the authors have proved that if an almost product structure P on the tangent space of a 2 n -dimensional Lagrange manifold E is defined and the F ( ± a 2 ; ± b 2 )-structure on the vertical tangent space T V ( E ) is given, then it is possible to define the similar structure on the horizontal subspace T H ( E ) and also on T ( E ). In the next section, we have proved some theorems and have obtained conditions under which the distribution L and M are r -parallel, r ¯ anti half parallel when r = r ¯ . The last section is devoted to proving theorems on geodesics on the Lagrange manifold

Characteristic foliations of material evolution: from remodeling to aging

Abstract: For any body-time manifold $\mathbb{R} \times \mathcal{B}$ there exists a groupoid, called material groupoid, encoding all the material properties of the evolution material. A smooth distribution, the material distribution, is constructed to deal with the case in which the material groupoid is not a Lie groupoid. This new tool provides a unified framework to deal with general non-uniform evolution materials.

Helicity, linking and the distribution of null-homologous periodic orbits for Anosov flows

Abstract: This paper concerns connections between dynamical systems, knots and helicity of vector fields. For a divergence-free vector field on a closed $3$-manifold that generates an Anosov flow, we show that the helicity of the vector field may be recovered as the limit of appropriately weighted averages of linking numbers of periodic orbits, regarded as knots. This complements a classical result of Arnold and Vogel that, when the manifold is a real homology $3$-sphere, the helicity may be obtained as the limit of the normalised linking numbers of typical pairs of long trajectories. We also obtain results on the asymptotic distribution of weighted averages of null-homologous periodic orbits.