Low‐rank approximation of tensors via sparse optimization

The goal of this paper is to find a low‐rank approximation for a given nth tensor. Specifically, we give a computable strategy on calculating the rank of a given tensor, based on approximating the solution to an NP‐hard problem. In this paper, we formulate a sparse optimization problem via an l₁‐regularization to find a low‐rank approximation of tensors. To solve this sparse optimization problem, we propose a rescaling algorithm of the proximal alternating minimization and study the theoretical convergence of this algorithm. Furthermore, we discuss the probabilistic consistency of the sparsity result and suggest a way to choose the regularization parameter for practical computation. In the simulation experiments, the performance of our algorithm supports that our method provides an efficient estimate on the number of rank‐one tensor components in a given tensor. Moreover, this algorithm is also applied to surveillance videos for low‐rank approximation.     

An integral formula for the heat kernel of tubular neighborhoods of complete (connected) Riemannian manifolds

Let M be a complete connected Riemannian manifold and let N be a submanifold of M. Let v: E _v»N be the normal bundle of N and exp _v : E _v»M its exponential map.Let (exp _infv ^/sup-1 , M ₀) be the Fermi chart relative to the submanifold N. Then, by using the Fermi coordinates we obtain an integral formula for the Dirichlet heat kernel p _t ^m (-,-). That is, we obtain a probabilistic representation for the integral _N f(y)p _t ^M (x,y) dywhere f is any measurable function of compact support in M ₀. This representation involves a submanifold semi-classical Brownian Riemannian bridge process. Then applying the integral formula via a Riemannian submersion in [5], we obtain heat kernel formulae for the complex projective space cP ⁿ, the quaternionic projective space QP ⁿ and the Caley line CaP ¹. The case of the Caley plane CaP ² eludes us due to the lack of a submersion theorem.This work is part of a Ph.D. Thesis which was undertaken under Professor K. D. Elworthy, Mathematics Institute, Warwick University, Coventry CV47AL, England, Great Britain.     

Lie Superalgebras Associated with Constant Sectional Curvature

This note describes an observation connecting Riemannian manifolds of constant sectional curvature with a particular class of Lie superalgebras. Specifically, it is shown that the structural equations of a space M with constant sectional curvature, of one variety or another, nearly coincide with some identities satisfied by tensors which can be used to construct some specific families of Lie superalgebras. In particular, one obtains either osp(n,2), spl(n,2), or osp(4,2n) if the Riemannian manifold has constant curvature, constant holomorphic curvature or constant quaternion-holomorphic curvature, respectively.Mathematics Subject Classiffications (2000). 17A70, 53C29, 53C99, 57Rxx     

On the existence of harmonic morphisms from symmetric spaces of rank one

Summary In this paper we give a unified framework for constructing harmonic morphisms from the irreducible Riemannian symmetric spaces ℍH ⁿ, ℂH ⁿ, ℝH ² _t+1, ℍP ⁿ, ℂP ⁿ and ℝP ²ⁿ⁺¹ of rank one. Using this we give a positive answer to the global existence problem for the non-compact hyperbolic cases. This work was supported by The Swedish Natural Science Research Council. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

On the complete decomposition of curvature tensors of Riemannian manifolds with symmetric connection

We give a complete decomposition of the space of curvature tensors with the symmetry properties as the curvature tensor associated with a symmetric connection of Riemannian manifold. We solve the problem under the action ofS0(n). The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined. Several applications for Riemannian manifolds with symmetric connection are given. The group of projective transformations of a Riemannian manifold and its subgroups are considered.     

A new smoothing and regularization Newton method for <Emphasis Type="Italic">P</Emphasis><Subscript>0</Subscript>-NCP

The nonlinear complementarity problem (denoted by NCP(F)) can be reformulated as the solution of a nonsmooth system of equations. In this paper, we propose a new smoothing and regularization Newton method for solving nonlinear complementarity problem with P ₀-function (P ₀-NCP). Without requiring strict complementarity assumption at the P ₀-NCP solution, the proposed algorithm is proved to be convergent globally and superlinearly under suitable assumptions. Furthermore, the algorithm has local quadratic convergence under mild conditions. Numerical experiments indicate that the proposed method is quite effective. In addition, in this paper, the regularization parameter ε in our algorithm is viewed as an independent variable, hence, our algorithm seems to be simpler and more easily implemented compared to many previous methods.     

ADDITIVE FAMILIES OF INVARIANTS OF SKEWSYMMETRIC TENSORS

Abstract

Let d be a positive integer and F be a field of characteristic 0. Suppose that for each positive integer n, I _n is a polynomial invariant of the usual action of GL_n (F) on Λ^d(Fⁿ), such that for t ? Λ^d(F ^k) and s ? Λ^d(F ^l), I _{k + l} (t l s) = I _k(t)I _t (s), where t ⊥ s is defined in §1.4. Then we say that {I_n} is an additive family of invariants of the skewsymmetric tensors of degree d, or, briefly, an additive family of invariants. If not all the I_n are constant we say that the family is non-trivial. We show that in each even degree d there is a non-trivial additive family of invariants, but that this is not so for any odd d. These results are analogous to those in our paper [3] for symmetric tensors. Our proofs rely on the symbolic method for representing invariants of skewsymmetric tensors. To keep this paper self-contained we expound some of that theory, but for the proofs we refer to the book [2] of Grosshans, Rota and Stein.     

Riemannian Metrics with the Prescribed Curvature Tensor and all Its Covariant Derivatives at One Point

On an n-dimensional vector space, equipped with a scalar product, we prescribe (0, 4) -, (0, 5)-, … type tensors R⁽⁰⁾, R⁽¹⁾, …, satisfying the well-known conditions for a curvature tensor and its derivatives and furthermore certain inequalities for the absolute values of the components of R^(k). Then there is an analytic Riemannian metric g on an open ball of the Cartesian space Rⁿ[u¹, …, uⁿ] for which u¹, …, uⁿ are normal coordinates and (▽^(k)R)₀ = R^(k) (k = 0, 1, 2, …) hold under an identification of the tangent space T₀Rⁿ at the origin with the vector space; ▽^(k)R denote the curvature tensor and its covariant derivatives with respect to the Levi-Civita connection ▽ of g, respectively.     

Coordinate gap and bidualily

For a varietyX inP ⁿ, we define a numberb ₂, called coordinate gap number,and prove thatb ₂>2 only ifX is not reflexive. Then, for a smooth surface inP ³, we obtain a concrete sufficient and necessary condition forb ₂>2, which enables us to discuss the biduality of surfaces inP ³.     

Examples of λ-invariants

In this paper, we will examine some of the implications of the results in [C1] for the Iwasawa invariants, λ _p , of the cyclotomic fields Q(ζ _n ), wherep + n. In particular, a number of examples, for various primesp, are given.     

Conjugate gradient algorithms for best rank-1 approximation of tensors

Motivated by considerations of pure state entanglement in quantum information, we consider the problem of finding the best rank-1 approximation to an arbitrary r -th order tensor. Reformulating the problem as an optimization problem on the Lie group SU (n₁) ⊗ … ⊗ SU (n_r) of so-called local unitary transformations and exploiting its intrinsic geometry yields a new approach, which finally leads to Riemannian variant of the conjugate gradient algorithm. Numerical simulations support that our method offers an alternative to the higher-order power method for computing the best rank-1 approximation to a tensor. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Discriminants in characteristic two

The Kronecker-Weierstrass theory of pencils is extended to give a necessary and sufficient condition that two 2×m×n tensors are equivalent. The connection between equivalence class representatives and the triple transitivity of PGL(2,F) is discussed. One consequence of the discussion is that the number of inequivalent 2×3×n tensors is finite. An efficient algorithm is given for testing the condition which ultimately depends on a fast pattern matching algorithm.     

The Embedding W n,1 into C 0 on Compact Riemannian Manifolds

We study the best constant in the inequality corresponding to the Sobolev embedding W ^n,1(R ⁿ ) into the space of bounded continuous functions C ₀(R ⁿ ). Then, we adapt this inequality on compact Riemannian manifolds and discuss on its optimality.     

A random 1-011-011-01algorithm for depth first search

In this paper we present a fast parallel algorithm for constructing a depth first search tree for an undirected graph. The algorithm is anRNC algorithm, meaning that it is a probabilistic algorithm that runs in polylog time using a polynomial number of processors on aP-RAM. The run time of the algorithm isO(T _MM(n) log³ n), and the number of processors used isP _MM (n) whereT _MM(n) andP _MM(n) are the time and number of processors needed to find a minimum weight perfect matching on ann vertex graph with maximum edge weightn.This research was done while the first author was visiting the Mathematical Research Institute in Berkeley. Research supported in part by NSF grant 8120790.Supported by Air Force Grant AFOSR-85-0203A.     

A parallel derangement generation algorithm

A parallel algorithm to generate allD _n derangements ofn distinct elements is presented in this paper. The algorithm requiresO([D _n /P]nlogn) time whenP processors are available on a Single Instruction Multiple Data Stream (SIMD) computer.     

Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram

The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian symmetric space from its Satake diagram, in a way that is suited for the use with computer algebra systems; an example implementation for Maple Version 10 can be found on . As an example application, the totally geodesic submanifolds of the Riemannian symmetric space SU(3)/SO(3) are classified.      

On a class of Einstein hypersurfaces immersed in a Riemannian manifold

Let x:M→ be an isometric immersion of a hypersurface M into an (n+1)-dimensional Riemannian manifold and let ρ _i  (i∈{1,...,n}) be the principal curvatures of M. We denote by E and P the distinguished vector field and the curvature vector field of M, respectively, in the sense of [8].?If M is structured by a P-parallel connection [7], then it is Einsteinian. In this case, all the curvature 2-forms are exact and other properties induced by E and P are stated.?The principal curvatures ρ _i are isoparametric functions and the set (ρ₁,...,ρ _n ) defines an isoparametric system [10].?In the last section, we assume that, in addition, M is endowed with an almost symplectic structure. Then, the dual 1-form π=P ^♭ of P is symplectic harmonic. If M is compact, then its 2nd Betti number b ₂≥1. Received: April 7, 1999; in final form: January 7, 2000?Published online: May 10, 2001     

Recursive events in random sequences

Let ω be a Kolmogorov–Chaitin random sequence with ω_{1: n} denoting the first n digits of ω. Let P be a recursive predicate defined on all finite binary strings such that the Lebesgue measure of the set {ω|∃nP(ω_{1: n} )} is a computable real α. Roughly, P holds with computable probability for a random infinite sequence. Then there is an algorithm which on input indices for any such P and α finds an n such that P holds within the first n digits of ω or not in ω at all. We apply the result to the halting probability Ω and show that various generalizations of the result fail. Received: 1 December 1998 / Published online: 3 October 2001     

The number of triangles covering the center of an n-set

Let the points P ₁, P ₂, ..., P _nbe given in the plane such that there are no three on a line. Then there exists a point of the plane which is contained in at least n ³/27 (open) P _iP_jP_ktriangles. This bound is the best possible.     

Notes on Gromov’s systolic estimate

These notes cover some of the main results of Gromov’s paper Filling Riemannian manifolds. The goal of these notes is to make the results and proofs accessible to more people. The main result is that if (M,g) is a Riemannian manifold of dimension n, then there is a non-contractible curve in (M,g) of length at most C _n Vol(M,g)^1/n .