Convergence of Hyperbolic Neural Networks Under Riemannian Stochastic Gradient Descent

We prove,under mild conditions,the convergence of a Riemannian gradient descent method for a hyperbolic neural network regression model,both in batch gradient descent and stochastic gradient descent.We also discuss a Riemannian version of the Adam algorithm.We show numerical simulations of these algorithms on various benchmarks.

Inequalities for Scalar Curvature and Shape Operator of an R-Lightlike Submanifold in Semi-Riemannian Manifold

We establish the links between the lightlike geometry and basics invariants of the associated semi-Riemannian geometry on r-lightlike submanifold and semi-Riemannian constructed from a semi-Riemannian ambient. Then we establish some basic inequalities, involving the scalar curvature and shape operator on r-lightlike coisotropic submanifold in semi-Riemannian manifold. Equality cases are also discussed.

Geometry Flow-Based Deep Riemannian Metric Learning

Deep metric learning(DML)has achieved great results on visual understanding tasks by seamlessly integrating conventional metric learning with deep neural networks.Existing deep metric learning methods focus on designing pair-based distance loss to decrease intra-class distance while increasing interclass distance.However,these methods fail to preserve the geometric structure of data in the embedding space,which leads to the spatial structure shift across mini-batches and may slow down the convergence of embedding learning.To alleviate these issues,by assuming that the input data is embedded in a lower-dimensional sub-manifold,we propose a novel deep Riemannian metric learning(DRML)framework that exploits the non-Euclidean geometric structural information.Considering that the curvature information of data measures how much the Riemannian(nonEuclidean)metric deviates from the Euclidean metric,we leverage geometry flow,which is called a geometric evolution equation,to characterize the relation between the Riemannian metric and its curvature.Our DRML not only regularizes the local neighborhoods connection of the embeddings at the hidden layer but also adapts the embeddings to preserve the geometric structure of the data.On several benchmark datasets,the proposed DRML outperforms all existing methods and these results demonstrate its effectiveness.

A Discussion on the Establishment That a Fibre Metric on the Positive Definite Real Inner-Product of a Properly Embedded Smooth Submanifold Be Always Extended to a Riemannian Metric on the Positive Definite Real Inner-Product

Let M be a smooth manifold and S ⊆ M a properly embedded smooth submanifold. Suppose that we have a fibre metric on TM|s i.e. a positive definite real inner-product on TpM for all p ∈ S, which depends smoothly on p ∈ S. The purpose of this article is to figure out that the fibre metric on TM|s can always be extended to a Riemannian metric on TM from a special perspective.

几何-拓扑法建立的Riemannian空间弹性损伤连续性方程

损伤力学发展至今,涌现出了各种各样的损伤力学理论,但尚未出现比较公认的普遍的理论.并且在解决更复杂的损伤问题时碰到了更多的困难.用几何-拓扑的方法将弹性损伤缺陷与Riemannian空间建立了对应关系,把材料的这种物理缺陷转化为几何缺陷.由连续损伤变量定义了拟塑性损伤因子张量,再由拟塑性应变张量出发,推导出了Riemannian空间中的弹性损伤连续性方程.从而将一个非线性问题转化为一个线性问题和一个弯曲空间的叠加.

拟常曲率Riemannian流行的一个Pinching定理

本文证明了积分不等式∫M∑i=1β≠n+1hi^2βj[3-1/p-1+n^1/2)S-na-1/2(n+1)(b-│b│）]*1≥0从而得到如下Pinching定理：若S≤[na+1/2(n+1)(b-│b│）]/（3-1/p-1+n^1/2)则M落在N的一个全测地子流行S^n+1中或S=[na+1/2(n+1)(b-│b│)]/(3-1/p-1+n^1/2)所得积分不等式优于白正国教授的结果而Pinching定理是丘成桐教授相应定理的推广。

DYNAMICS IN NEWTONIAN-RIEMANNIAN SPACE-TIME (Ⅱ)

The relativity of motion and covariance of equation of motion in Newtonian_Riemannian space_time, some relationship between Newton's mechanics in N_R space_time and the general relativity, their difference and identity are discussed.

ON THE INVARIANT SUBMANIFOLDS OF RIEMANNIAN PRODUCT MANIFOLD

In this paper, the vertical and horizontal distributions of an invariant sub-manifold of a Riemannian product manifold are discussed. An invariant real space form in a Riemannian product manifold is researched. Finally, necessary and sufficient conditions are given on an invariant submanifold of a Riemannian product manifold to be a locally symmetric and real space form.

DELAUNAY TRIANGULATION METHOD OF CURVED SURFACES BASED ON RIEMANNIAN METRIC

A method for quality mesh generation of parametric curved surfaces isproposed. It is shown that the main difference between the proposed method and previous ones is thatour meshing process is done completely in the parametric domains with the guarantee of meshquality. To obtain this aim, the Delaunay method is extended to anisotropic context of 2D domains,and a Riemannian metric map is introduced to remedy the mapping distortion from object space toparametric domain. Compared with previous algorithms, the approach is much simpler, more robust andspeedy. The algorithm is implemented and examples for several geometries are presented todemonstrate the efficiency and validity of the method.

Fast Near-duplicate Image Detection in Riemannian Space by A Novel Hashing Scheme

There is a steep increase in data encoded as symmetric positive definite(SPD)matrix in the past decade.The set of SPD matrices forms a Riemannian manifold that constitutes a half convex cone in the vector space of matrices,which we sometimes call SPD manifold.One of the fundamental problems in the application of SPD manifold is to find the nearest neighbor of a queried SPD matrix.Hashing is a popular method that can be used for the nearest neighbor search.However,hashing cannot be directly applied to SPD manifold due to its non-Euclidean intrinsic geometry.Inspired by the idea of kernel trick,a new hashing scheme for SPD manifold by random projection and quantization in expanded data space is proposed in this paper.Experimental results in large scale nearduplicate image detection show the effectiveness and efficiency of the proposed method.

Cosmic Dark Energy Density from Classical Mechanics and Seemingly Redundant Riemannian Finitely Many Tensor Components of Einstein’s General Relativity

We determine the limit of the ratio formed by the independent components of the Riemann tensor to the non-zero component as space dimensionality tends to infinity and find it to be 12. Subsequently we use this result in conjunction with Newtonian classical mechanics to show that the ordinary measurable cosmic energy density is given by while the dark energy density is obviously the Legendre transformation dual energy E(D) = 1 -?E(O). The result is in complete agreement with the COBE, WMAP and type 1a supernova measurements.

SOME EXTENSIONS OF THE MEAN CURVATURE FLOW IN RIEMANNIAN MANIFOLDS

Given a family of smooth immersions of closed hypersurfaces in a locally symmetric Riemannian manifold with bounded geometry, moving by mean curvature flow, we show that at the first finite singular time of mean curvature flow, certain subcritical quantities concerning the second fundamental form blow up. This result not only generalizes a result of Le-Sesum and Xu-Ye-Zhao, but also extends the latest work of Le in the Euclidean case

Riemannian流形中DE算法算子最优特征量的量子渐进估计

该文主要分析和探讨了差分进化算法(Differential Eveolutionary Algorithm,DE)在Riemannian流形中的几何关系,对P-ε条件下Riemannian流形中的种群个体进行了收敛性分析,得到了迭代个体收敛精度与收敛速度的量子不确定渐进估计,如下式Δv^2 Δxβ^ε^2≥(√(λε)1+…+√(λε)n/2)^2,其中,Δv^2为种群个体的速度分辨率,Δxβ^ε^2为种群个体带有误差的位置分辨率,(λε)i,i=1,2,…,n.从本质上说明了Riemannian流形中迭代个体的局部特征量是不能从收敛精度和收敛速度同时达到算法高效.

GEOMETRIC INEQUALITIES FOR CERTAIN SUBMANIFOLDS IN A PINCHED RIEMANNIAN MANIFOLD

This article gives some geometric inequalities for a submanifold with parallel second fundamental form in a pinched Riemannian manifold and the distribution for the square norm of its second fundamental form.

Embeddings of Almost Hermitian Manifold in Almost Hyper Hermitian Manifold and Complex (Hypercomplex) Numbers in Riemannian Geometry

Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold. Further, an almost hyper Hermitian structure has been constructed on the tangent bundle TM with help of the Riemannian connection of an almost Hermitian structure on a manifold M then, we consider an embedding of the almost Hermitian manifold M in the corresponding normal tubular neighborhood of the null section in the tangent bundle TM equipped with the deformed almost hyper Hermitian structure of the special form. As a result, we have obtained that any Riemannian manifold M of dimension n can be embedded as a totally geodesic submanifold in a Kaehlerian manifold of dimension 2n (Theorem 6) and in a hyper Kaehlerian manifold of dimension 4n (Theorem 7). Such embeddings are “good” from the point of view of Riemannian geometry. They allow solving problems of Riemannian geometry by methods of Kaehlerian geometry (see Section 5 as an example). We can find similar situation in mathematical analysis (real and complex).

The Riemannian Structure of the Three-Parameter Gamma Distribution

In this paper, we will utilize the results already known in differential geometry and provide an intuitive understanding of the Gamma Distribution. This approach leads to the definition of new concepts to provide new results of statistical importance. These new results could explain Chen [1-3] experienced difficulty when he attempts to simulate the sampling distribution and power function of Cox’s [4,5] test statistics of separate families of hypotheses. It may also help simplify and clarify some known statistical proofs or results. These results may be of particular interest to mathematical physicists. In general, it has been shown that the parameter space is not of constant curvature. In addition, we calculated some invariant quantities, such as Sectional curvature, Ricci curvature, mean curvature and scalar curvature.

Notes on the Heat Diffusion Semigroup in a Riemannian Manifold

In this paper, the properties of the heat diffusion semigroup {e^(t△)}_(t≥0) generated by the Hodge-deRham operator in a Riemannian manifold are discussed.

A CHARACTERISTIC OF RIEMANNIAN SPACES ADMITTING QUASI-CONCIRCULAR TRANSFORMATION

In this paper, we have considered some properties of quasi-umbilical hypersurfaces of a Riemannian space and obtained a characteristic of Riemannian spaces admitting quasi-concircular transformation.

Riemannian Space-Time, de Donder Conditions and Gravitational Field in Flat Space-Time

Let the coordinate system xi of flat space-time to absorb a second rank tensor field Φij of the flat space-time deforming into a Riemannian space-time, namely, the tensor field Φuv is regarded as a metric tensor with respect to the coordinate system xu. After done this, xu is not the coordinate system of flat space-time anymore, but is the coordinate system of the new Riemannian space-time. The inverse operation also can be done. According to these notions, the concepts of the absorption operation and the desorption operation are proposed. These notions are actually compatible with Einstein’s equivalence principle. By using these concepts, the relationships of the Riemannian space-time, the de Donder conditions and the gravitational field in flat space-time are analyzed and elaborated. The essential significance of the de Donder conditions (the harmonic conditions or gauge) is to desorb the tensor field of gravitation from the Riemannian space-time to the Minkowski space-time with the Cartesian coordinates. Einstein equations with de Donder conditions can be solved in flat space-time. Base on Fock’s works, the equations of gravitational field in flat space-time are obtained, and the tensor expression of the energy-momentum of gravitational field is found. They all satisfy the global Lorentz covariance.

Compact Maximal Space-like Submanifolds in a Pseudo-Riemannian Spacetime S_p^(m+p)

Abstract： This paper concerns space-like submanifolds in a pseudo-Riemannianspace-time Sp^m＋p∪→Ep^m＋p＋1 （P ≥ 1）, and proves that connected compact maximalsuace-like submanifolds in a pseudo-Riemannian spacetime Sp^m＋p∪→Ep^m＋p＋1 （P ≥ 1） must be totally umbilical, and also totally geodesic. Particularly, when p = 1, our result is just Montiel＇s in case of H = 0.