Newton Method on Riemannian Manifolds: Covariant Alpha-Theory
Jean-Pierre Dedieu, Pierre Priouret, Gregorio Malajovich
本文研究黎曼流形上牛顿法求映射和向量场零点的定量理论，提出协变Alpha理论。
本文研究了牛顿法在寻找映射f: M_n -> R^n和向量场X: M_x -> TM_n的零点时的定量方面。
In this paper we study quantitative aspects of Newton method for finding zeros of mappings f: M_n -> R^n and vector fields X: M_x -> TM_n
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This paper is concerned with the inverse problem of constructing a symmetric nonnegative matrix from realizable spectrum. We reformulate the inverse problem as an underdetermined nonlinear matrix equation over a Riemannian product manifold. To solve it, we develop a Riemannian underdetermined inexact Newton dogleg method for solving a general underdetermined nonlinear equation defined between Riemannian manifolds and Euclidean spaces. The global and quadratic convergence of the proposed method is established under some mild assumptions. Then we solve the inverse problem by applying the propose…
Local convergence analysis of Inexact Newton method with relative residual error tolerance under majorant condition in Riemannian Manifolds
本文基于主导原理，研究了在完全Riemann流形上求解可微向量场奇点的不精确牛顿法，证明其在局部条件下具有Q-线性收敛性，并展示了在无误差情况下恢复Riemann流形中牛顿法的经典局部定理。
Convergence analysis of Riemannian Gauss-Newton methods and its connection with the geometric condition number
本文分析了Riemannian Gauss-Newton方法在流形上求解最小二乘问题的局部收敛性，并将所得的乘法常数估计与几何条件数联系起来。
Effective Hamiltonians arise in several problems, including homogenization of Hamilton--Jacobi equations, nonlinear control systems, Hamiltonian dynamics, and Aubry--Mather theory. In Aubry--Mather theory, related objects, Mather measures, are also of great importance. Here, we combine ideas from mean-field games with the Hessian Riemannian flow to compute effective Hamiltonians and Mather measures simultaneously. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flo…
A Preconditioned Riemannian Gauss-Newton Method for Least Squares Inverse Eigenvalue Problems
本文提出了一种Riemannian近似广义牛顿法用于最小二乘反特征值问题，分析了其全局和局部收敛性，并提出预条件共轭梯度法求解Riemannian广义牛顿方程，通过数值实验验证了方法的有效性。
In this paper, we consider the inverse eigenvalue problem for the positive doubly stochastic matrices, which aims to construct a positive doubly stochastic matrix from the prescribed realizable spectral data. By using the real Schur decomposition, the inverse problem is written as a nonlinear matrix equation on a matrix product manifold. We propose monotone and nonmonotone Riemannian inexact Newton-CG methods for solving the nonlinear matrix equation. The global and quadratic convergence of the proposed methods is established under some assumptions. We also provide invariant subspaces of the c…
A Riemannian Inexact Newton-CG Method for Nonnegative Inverse Eigenvalue Problems: Nonsymmetric Case
本文提出一种非负逆特征值问题的黎曼近似牛顿-共轭梯度方法，用于求解非对称情况下的非负矩阵，其谱为给定的自共轭复数集。
We describe how to approximate the Riemann curvature tensor as well as sectional curvatures on possibly infinite-dimensional shape spaces that can be thought of as Riemannian manifolds. To this end, we extend the variational time discretization of geodesic calculus presented in Rumpf and Wirth (2015), which just requires an approximation of the squared Riemannian distance that is typically easy to compute. First we obtain first order discrete covariant derivatives via a Schild's ladder type discretization of parallel transport. Second order discrete covariant derivatives are then computed as n…
The Symmetric Tensor Approximation problem (STA) consists of approximating a symmetric tensor or a homogeneous polynomial by a linear combination of symmetric rank-1 tensors or powers of linear forms of low symmetric rank. We present two new Riemannian Newton-type methods for low rank approximation of symmetric tensor with complex coefficients.The first method uses the parametrization of the set of tensors of rank at most $r$ by weights and unit vectors.Exploiting the properties of the apolar product on homogeneous polynomials combined with efficient tools from complex optimization, we provide…
Inexact Newton Methods for Solving Generalized Equations on Riemannian Manifolds
本文研究了利用度量正则性求解黎曼流形上广义方程的不精确牛顿方法，在无额外几何假设下证明了其线性和二次局部收敛性及半局部收敛性，并应用于求解向量场奇点与受约束的黎曼质心KKT系统。
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本文提出基于黎曼向量丛的拉格朗日-牛顿方法，用于解决PDE约束下的形状优化问题，并通过数值例子验证了该方法的可行性。
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In the present paper, in order to fnd a singularity of a vector field defined on Riemannian manifolds, we present a new globalization strategy of Newton method and establish its global convergence with superlinear rate. In particular, this globalization generalizes for a general retraction the existing damped Newton's method. The presented global convergence analysis does not require any hypotesesis on singularity of the vector field. We applied the proposed method to solve the truncated singular value problem on the product of two Stiefel manifolds, the dextrous hand grasping problem on the c…
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Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure
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Riemannian optimisation methods for ground states of multicomponent Bose-Einstein condensates
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本文提出了一种结合微分几何和不精确牛顿方法的Riemannian不精确牛顿方法，用于解决结构逆特征值和奇异值问题，并展示了该方法在全局和二次收敛性以及数值鲁棒性和精度方面的表现。
The short-time Fourier transform (STFT), or the discrete Gabor transform (DGT), has been extensively used in signal analysis and processing. Their properties are characterized by a window function. For signal processing, designing a special window called tight window is important because it is known to make DGT-domain processing robust to error. In this paper, we propose a method of designing tight windows that minimize the sidelobe energy. It is formulated as a constrained spectral concentration problem, and a Newton's method on an oblique manifold is derived to efficiently obtain a solution.…
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Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics
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Optimization Techniques on Riemannian Manifolds
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Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence
本文针对含噪线性测量下的低 Tucker 秩张量估计问题，提出黎曼高斯-牛顿方法，首次证明其在含噪环境下的局部二次收敛性，并给出匹配的误差上下界以证明统计最优性，通过张量回归和张量 SVD 应用及仿真验证了理论结果。
Newton's method for nonlinear mappings into vector bundles Part II: Application to variational problems
本文研究流形上变分方程的牛顿法求解，将其转化为无穷维流形到对偶向量丛的求根问题，推导所需微分几何工具并采用仿射协变阻尼策略，通过数值实验验证方法有效性。
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