Manifolds Of Rank Research Articles

The Shifted Wave Equation on Non-flat Harmonic Manifolds

We solve the shifted wave equation ∂2∂t2φ(x,t)=(Δx+ρ2)φ(x,t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\\partial ^2}{\\partial t^2}\\varphi (x,t)=(\\Delta _x+\\rho ^2)\\varphi (x,t) \\end{aligned}$$\\end{document}on a non-compact simply connected harmonic manifold with mean curvature of the horospheres 2ρ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\rho >0$$\\end{document}. We give an explicit representation of the solution as the inverse dual Abel transform of the spherical means of their initial conditions using the local injectivity of the Abel transform and symmetry properties of the spherical mean value operator. Furthermore, we investigate the shifted wave equation using the Fourier transform on harmonic manifolds of rank one. Additionally, we obtain a result analogous to the classical Paley–Wiener theorem and use it to show an asymptotic Huygens principle as well as asymptotic equidistribution of the energy of a solution of the shifted wave equation under assumptions on the Harish–Chandra type c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{c}$$\\end{document}-function.

Moment maps and isoparametric hypersurfaces in spheres — Grassmannian cases

We expect that every Cartan–Münzner polynomial of degree four can be described as a squared-norm of a moment map for a Hamiltonian action. Our expectation is known to be true for Hermitian cases, that is, those obtained from the isotropy representations of compact irreducible Hermitian symmetric spaces of rank two. In this paper, we prove that our expectation is true for the Cartan–Münzner polynomials obtained from the isotropy representations of Grassmannian manifolds of rank two over R, C or H. The quaternion cases are the first non-Hermitian examples that our expectation is verified.

Del Pezzo foliations with log canonical singularities

In this paper, we classify del Pezzo foliations on projective manifolds of rank at least 3 and with log canonical singularities in the sense of McQuillan.

Non-minimality of spirals in sub-Riemannian manifolds

We show that in analytic sub-Riemannian manifolds of rank 2 satisfying a commutativity condition spiral-like curves are not length minimizing near the center of the spiral. The proof relies upon the delicate construction of a competing curve.

Higher genera Catalan numbers and Hirota equations for extended nonlinear Schr\xf6dinger hierarchy

We consider the Dubrovin–Frobenius manifold of rank 2 whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces, ribbon graphs, Grothendieck’s dessins d’enfants, strictly monotone Hurwitz numbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhou conjectured that the full partition function of this Dubrovin–Frobenius manifold is a tau-function of the extended nonlinear Schrödinger hierarchy, an extension of a particular rational reduction of the Kadomtsev–Petviashvili hierarchy. We prove a version of their conjecture specializing the Givental–Milanov method that allows to construct the Hirota quadratic equations for the partition function, and then deriving from them the Lax representation.

Learning deep linear neural networks: Riemannian gradient flows and convergence to global minimizers

Abstract We study the convergence of gradient flows related to learning deep linear neural networks (where the activation function is the identity map) from data. In this case, the composition of the network layers amounts to simply multiplying the weight matrices of all layers together, resulting in an overparameterized problem. The gradient flow with respect to these factors can be re-interpreted as a Riemannian gradient flow on the manifold of rank-$r$ matrices endowed with a suitable Riemannian metric. We show that the flow always converges to a critical point of the underlying functional. Moreover, we establish that, for almost all initializations, the flow converges to a global minimum on the manifold of rank $k$ matrices for some $k\leq r$.

Rank $2r$ Iterative Least Squares: Efficient Recovery of Ill-Conditioned Low Rank Matrices from Few Entries

We present a new, simple, and computationally efficient iterative method for low rank matrix completion. Our method is inspired by the class of factorization-type iterative algorithms, but substantially differs from them in the way the problem is cast. Precisely, given a target rank $r$, instead of optimizing on the manifold of rank $r$ matrices, we allow our interim estimated matrix to have a specific overparametrized rank $2r$ structure. Our algorithm, denoted R2RILS, for rank $2r$ iterative least squares, thus has low memory requirements, and at each iteration it solves a computationally cheap sparse least squares problem. We motivate our algorithm by its theoretical analysis for the simplified case of a rank 1 matrix. Empirically, R2RILS is able to recover ill-conditioned low rank matrices from very few observations---near the information limit---and it is stable to additive noise.

Quotient Geometry with Simple Geodesics for the Manifold of Fixed-Rank Positive-Semidefinite Matrices

This paper explores the well-known identification of the manifold of rank $p$ positive-semidefinite matrices of size $n$ with the quotient of the set of full-rank $n$-by-$p$ matrices by the orthogo...

Information Constrained Control Analysis of Eye Gaze Distribution Under Workload

We describe a novel model of human eye gaze behavior under workload, derived from the basic principle of information constrained control. The model assumes two distributions over the visual field: A saliency distribution, which is nongoal oriented, and a reward task-related distribution. The eye gaze behavior is determined by the tradeoff between these two distributions, where the goal is to preserve the task-related constraints, while remaining as close as possible to the saliency distribution representing a comfort zone. Based on minimum Kullback–Liebler divergence principles, the model gives rise to a family of gaze distributions controlled by a single tradeoff parameter. The model was evaluated experimentally in a driving simulator that consisted of an immersive environment with clear tasks and accurate monitoring capabilities. The findings confirm the theoretical predictions with respect to the low rank manifold and order relations in the data. We show that the model can be used to visualize the unknown reward function associated with a task, and predict human workload based on gaze pattern.

Bounds for $$L_p$$-Discrepancies of Point Distributions in Compact Metric Measure Spaces

Upper bounds for the $$L_p$$-discrepancies of point distributions in compact metric measure spaces are proved for all exponents $$0<p<\infty $$ and $$p=\infty $$ under very simple conditions on the volume of metric balls as a function of radii. Particularly, these conditions hold for all compact Riemannian manifolds. The obtained upper bounds are of the sharp order, at least, for $$2\leqslant p<\infty $$ and Riemannian symmetric manifolds of rank one.

A Geometric Approach to Dynamical Model Order Reduction

Any model order reduced dynamical system that evolves a modal decomposition to approximate the discretized solution of a stochastic PDE can be related to a vector field tangent to the manifold of fixed rank matrices. The dynamically orthogonal (DO) approximation is the canonical reduced-order model for which the corresponding vector field is the orthogonal projection of the original system dynamics onto the tangent spaces of this manifold. The embedded geometry of the fixed rank matrix manifold is thoroughly analyzed. The curvature of the manifold is characterized and related to the smallest singular value through the study of the Weingarten map. Differentiability results for the orthogonal projection onto embedded manifolds are reviewed and used to derive an explicit dynamical system for tracking the truncated singular value decomposition (SVD) of a time-dependent matrix. It is demonstrated that the error made by the DO approximation remains controlled under the minimal condition that the original solution stays close to the low rank manifold, which translates into an explicit dependence of this error on the gap between singular values. The DO approximation is also justified as the dynamical system that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution. Riemannian matrix optimization is investigated in this extrinsic framework to provide algorithms that adaptively update the best low rank approximation of a smoothly varying matrix. The related gradient flow provides a dynamical system that converges to the truncated SVD of an input matrix for almost every initial datum.

Fast Hybrid Algorithm for Big Matrix Recovery

Large-scale Nuclear Norm penalized Least Square problem (NNLS) is frequently encountered in estimation of low rank structures. In this paper we accelerate the solution procedure by combining non-smooth convex optimization with smooth Riemannian method. Our methods comprise of two phases. In the first phase, we use Alternating Direction Method of Multipliers (ADMM) both to identify the fix rank manifold where an optimum resides and to provide an initializer for the subsequent refinement. In the second phase, two superlinearly convergent Riemannian methods: Riemannian NewTon (NT) and Riemannian Conjugate Gradient descent (CG) are adopted to improve the approximation over a fix rank manifold. We prove that our Hybrid method of ADMM and NT (HADMNT) converges to an optimum of NNLS at least quadratically. The experiments on large-scale collaborative filtering datasets demonstrate very competitive performance of these fast hybrid methods compared to the state-of-the-arts.

A Quadratically Convergent Algorithm for Structured Low-Rank Approximation

Structured Low-Rank Approximation is a problem arising in a wide range of applications in Numerical Analysis and Engineering Sciences. Given an input matrix $$M$$ , the goal is to compute a matrix $$M'$$ of given rank $$r$$ in a linear or affine subspace $$E$$ of matrices (usually encoding a specific structure) such that the Frobenius distance $$\left\| M-M' \right\| $$ is small. We propose a Newton-like iteration for solving this problem, whose main feature is that it converges locally quadratically to such a matrix under mild transversality assumptions between the manifold of matrices of rank $$r$$ and the linear/affine subspace $$E$$ . We also show that the distance between the limit of the iteration and the optimal solution of the problem is quadratic in the distance between the input matrix and the manifold of rank $$r$$ matrices in $$E$$ . To illustrate the applicability of this algorithm, we propose a Maple implementation and give experimental results for several applicative problems that can be modeled by Structured Low-Rank Approximation: univariate approximate GCDs (Sylvester matrices), low-rank matrix completion (coordinate spaces) and denoising procedures (Hankel matrices).

On rational Frobenius manifolds of rank three with symmetries

We study Frobenius manifolds of rank three and dimension one that are related to submanifolds of certain Frobenius manifolds arising in mirror symmetry of elliptic orbifolds. We classify such Frobenius manifolds that are defined over an arbitrary field $\mathbb{K} \subset \mathbb{C}$ via the theory of modular forms. By an arithmetic property of an elliptic curve $\mathbb{E}_\tau$ defined over $\mathbb K$ associated to such a Frobenius manifold, it is proved that there are only two such Frobenius manifolds defined over $\mathbb C$ satisfying a certain symmetry assumption and thirteen Frobenius manifolds defined over $\mathbb Q$ satisfying a weak symmetry assumption on the potential.

Analytic continuation of holomorphic maps respecting varieties of minimal rational tangents and applications to rational homogeneous manifolds

In a series of works, one of the authors has developed with J.- M. Hwang a geometric theory of uniruled projective manifolds, especially those of Picard number 1, basing on the study of varieties of minimal rational tangents. A fundamental result in this theory is a principle of analytic continuation under very mild assumptions, called Cartan-Fubini extension, of biholomorphisms between connected open subsets of two Fano manifolds of Picard number 1 which preserve varieties of minimal rational tangents. In this article we develop a generalization of Cartan-Fubini extension for non-equidimensional holomorphic immersions from a connected open subset of a Fano manifold of Picard number 1 into a uniruled projective manifold, under the assumptions that the map sends varieties of minimal rational tangents onto linear sections of varieties of minimal rational tangents and that it satisfies a mild geometric condition formulated in terms of second fundamental forms on varieties of minimal rational tangents. Formerly such a result was known only in the very special case of irreducible Hermitian symmetric manifolds of rank at least two, and the proof relied on the existence of flattening coordinates, viz., Harish-Chandra coordinates, with respect to which the varieties of minimal rational tangents form a constant family. The proof of the main result, which is based on the deformation theory of rational curves, is differential-geometric in nature and is applicable to the general situation of uniruled projective manifolds without any assumption on the existence of special coordinate systems. As an application, we give a characterization of standard embeddings for certain pairs of rational homogeneous manifolds in terms of embeddings of varieties of minimal rational tangents.

Some Einstein–Randers metrics on homogeneous spaces

In this paper we study invariant Einstein–Randers metrics on some homogeneous Riemannian manifolds of rank ≥ 2 . By finding out all the invariant Killing vector fields on the corresponding homogeneous Einstein Riemannian manifolds, we obtain a complete description of all the invariant Einstein–Randers metrics on those homogeneous manifolds. Since the underlying Riemannian Einstein metrics are generically not diagonal, this presents a large number of non-Riemannian homogeneous Einstein–Randers spaces of nonconstant flag curvature.

THE GEOMETRY OF 3-QUASI-SASAKIAN MANIFOLDS

3-quasi-Sasakian manifolds were studied systematically by the authors in a recent paper as a suitable setting unifying 3-Sasakian and 3-cosymplectic geometries. This paper throws new light on their geometric structure which appears to be generally richer compared to the 3-Sasakian subclass. In fact, it turns out that they are multiply foliated by four distinct fundamental foliations. The study of the transversal geometries with respect to these foliations allows us to link the 3-quasi-Sasakian manifolds to the more famous hyper-Kähler and quaternionic-Kähler geometries. Furthermore, we strongly improve the splitting results previously obtained; we prove that any 3-quasi-Sasakian manifold of rank 4l + 1 is 3-cosymplectic and any 3-quasi-Sasakian manifold of maximal rank is 3-α-Sasakian.

Optimal cubature formulas on compact homogeneous manifolds

We find lower bounds for the rate of convergence of optimal cubature formulas on sets of differentiable functions on compact homogeneous manifolds of rank I or two-point homogeneous spaces. It is shown that these lower bounds are sharp in the power scale in the case of S 2 , the unit sphere in R 3 .

A natural Lie super-algebra bundle on rank 3 WSD manifolds

We prove the existence of a natural ∗ -Lie super-algebra bundle on any orientable WSD manifold of rank 3. We describe in detail the associated Lie super-algebra L 3 , C of global sections. We show that L 3 , C is a product of sl ( 4 , C ) with the full special linear super-algebras of some graded vector spaces isotypical with respect to a natural action of so ( 3 , R ) . We give an explicit description of a geometrically natural real form of L 3 , C . This real form is made up of so ( 3 , R ) -invariant operators which preserve the Poincaré pairing on the bundle of forms.

Holomorphic invariant forms of a bounded domain

Given a complete ortho-normal system ϕ = (ϕ0,ϕ1,ϕ2,…) of L 2 H(\( \mathcal{D} \)), the space of holomorphic and absolutely square integrable functions in the bounded domain \( \mathcal{D} \) of ℂ n , we construct a holomorphic imbedding \( \iota _\phi :\mathcal{D} \to \mathfrak{F}(n,\infty ) \), the complex infinite dimensional Grassmann manifold of rank n. It is known that in \( \mathfrak{F}(n,\infty ) \) there are n closed (μ, μ)-forms τμ (μ = 1,…,n) which are invariant under the holomorphic isometric automorphism of \( \mathfrak{F}(n,\infty ) \) and generate algebraically all the harmonic differential forms of \( \mathfrak{F}(n,\infty ) \). So we obtain in \( \mathcal{D} \) a set of (μ, μ)-forms ι ϕ * τμ (μ = 1,…, n), which are independent of the system ϕ chosen and are invariant under the bi-holomorphic transformations of \( \mathcal{D} \). Especially the differential metric ds 2 1 associated to the Kahler form ι ϕ * τ1 is a Kahler metric which differs from the Bergman metric ds 2 of \( \mathcal{D} \) in general, but in case that the Bergman metric is an Einstein metric, ds 1 2 differs from ds 2 only by a positive constant factor.