Ke Ye
24Publications
6H-index
141Citations
Publications 24
#1Ke YeH-Index: 6
#2Ken Sze-Wai WongH-Index: 1
Last.Lek-Heng LimH-Index: 19
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A flag is a sequence of nested subspace. Flags are ubiquitous in numerical analysis, arising in finite elements, multigrid, spectral, and pseudospectral methods for numerical PDE; they arise as Krylov subspaces in matrix computations, and as multiresolution analysis in wavelets constructions. They are common in statistics too --- prinicipal component, canonical correlation, and correspondence analyses may all be viewed as methods for extracting flags from a data set that in turn shed light on th...
#1Yang Qi (U of C: University of Chicago)
#2Pierre ComonH-Index: 42
Last.Ke Ye (CAS: Chinese Academy of Sciences)H-Index: 6
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Computations of low-rank approximations of tensors often involve path-following optimization algorithms. In such cases, a correct solution may only be found if there exists a continuous path connecting the initial point to a desired solution. We will investigate the existence of such a path in sets of low-rank tensors for various notions of ranks, including tensor rank, border rank, multilinear rank, and their counterparts for symmetric tensors.
#1Lek-Heng LimH-Index: 19
#2Ken Sze-Wai WongH-Index: 1
Last.Ke YeH-Index: 6
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The affine Grassmannian is a noncompact smooth manifold that parameterizes all affine subspaces of a fixed dimension. It is a natural generalization of Euclidean space, points being zero-dimensional affine subspaces. We will realize the affine Grassmannian as a matrix manifold and extend Riemannian optimization algorithms including steepest descent, Newton method, and conjugate gradient, to real-valued functions on the affine Grassmannian. Like their counterparts for the Grassmannian, they rely ...
#1Lek-Heng LimH-Index: 19
#2Ken Sze-Wai WongH-Index: 1
Last.Ke YeH-Index: 6
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The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being zero-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show that, like the Grassmannian, the affine Grassmannian has rich geometrical and topological properties: It has the structure of a homogeneous space, a differential manifold, an algebraic variety, a vector bundle, a classifying space, among many more structures; furt...
#1Lek-Heng LimH-Index: 19
#2Rodolphe SepulchreH-Index: 43
Last.Ke YeH-Index: 6
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We show how the Riemannian distance on \mathbb{S}^n_{++} the cone of n\times nreal symmetric or complex Hermitian positive definite matrices, may be used to naturally define a distance between two such matrices of different dimensions. Given that \mathbb{S}^n_{++}also parameterizes ndimensional ellipsoids, and inner products on \mathbb{R}^n n \times ncovariance matrices of nondegenerate probability distributions, this gives us a natural way to define a geometric distance betwee...
#1Pierre ComonH-Index: 42
#2Lek-Heng LimH-Index: 19
Last.Ke YeH-Index: 6
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We study path-connectedness and homotopy groups of sets of tensors defined by tensor rank, border rank, multilinear rank, as well as their symmetric counterparts for symmetric tensors. We show that over \mathbb{C} the set of rank-rtensors and the set of symmetric rank-rsymmetric tensors are both path-connected if ris not more than the complex generic rank; these results also extend to border rank and symmetric border rank over \mathbb{C} Over \mathbb{R} the set of rank-rtens...
#1Ke Ye (U of C: University of Chicago)H-Index: 6
#2Lek-Heng Lim (U of C: University of Chicago)H-Index: 19
We discuss a generalization of the Cohn–Umans method, a potent technique developed for studying the bilinear complexity of matrix multiplication by embedding matrices into an appropriate group algebra. We investigate how the Cohn–Umans method may be used for bilinear operations other than matrix multiplication, with algebras other than group algebras, and we relate it to Strassen’s tensor rank approach, the traditional framework for investigating bilinear complexity. To demonstrate the utility o...
#1Ke YeH-Index: 6
#2Lek-Heng LimH-Index: 19
In problems involving approximation, completion, denoising, dimension reduction, estimation, interpolation, modeling, order reduction, regression, etc, we argue that the near-universal practice of assuming that a function, matrix, or tensor (which we will see are all the same object in this context) has low rank may be ill-justified. There are many natural instances where the object in question has high rank with respect to the classical notions of rank: matrix rank, tensor rank, multilinear ran...
#1Tingran GaoH-Index: 6
#2Lek-Heng LimH-Index: 19
Last.Ke YeH-Index: 6
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We introduce in this paper a manifold optimization framework that utilizes semi-Riemannian structures on the underlying smooth manifolds. Unlike in Riemannian geometry, where each tangent space is equipped with a positive definite inner product, a semi-Riemannian manifold allows the metric tensor to be indefinite on each tangent space, i.e., possessing both positive and negative definite subspaces; differential geometric objects such as geodesics and parallel-transport can be defined on non-dege...
#1Jiawang NieH-Index: 24
#2Ke YeH-Index: 6
Hankel tensors are generalizations of Hankel matrices. This article studies the relations among various ranks of Hankel tensors. We give an algorithm that can compute the Vandermonde ranks and decompositions for all Hankel tensors. For a generic ndimensional Hankel tensor of even order or order three, we prove that the the cp rank, symmetric rank, border rank, symmetric border rank, and Vandermonde rank all coincide with each other. In particular, this implies that the Comon's conjecture is t...
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