arXiv subjects

Ke Ye

Publications and source records attributed to Ke Ye.

At least 73 records · Page 4Linked to original sources

arXiv2019

Optimization on flag manifolds

A flag is a sequence of nested subspaces. Flags are ubiquitous in numerical analysis, arising in finite elements, multigrid, spectral, and pseudospectral methods for numerical PDE; they arise in the form of Krylov subspaces in matrix computations, and as multiresolution analysis in wavelets constructions. They are common in statistics too --- principal component, canonical correlation, and correspondence analyses may all be viewed as methods for extracting flags from a data set. The main goal of this article is to develop the tools needed for optimizing over a set of flags, which is a smooth manifold called the flag manifold, and it contains the Grassmannian as the simplest special case. We will derive closed-form analytic expressions for various differential geometric objects required for Riemannian optimization algorithms on the flag manifold; introducing various systems of extrinsic coordinates that allow us to parameterize points, metrics, tangent spaces, geodesics, distance, parallel transport, gradients, Hessians in terms of matrices and matrix operations; and thereby permitting us to formulate steepest descent, conjugate gradient, and Newton algorithms on the flag manifold using only standard numerical linear algebra.

math.OC↗

arXiv2019

Tensor network ranks

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 \emph{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 rank --- the latter two being the most straightforward generalizations of the former. To remedy this, we show that one may vastly expand these classical notions of ranks: Given any undirected graph $G$, there is a notion of $G$-rank associated with $G$, which provides us with as many different kinds of ranks as there are undirected graphs. In particular, the popular tensor network states in physics (e.g., \textsc{mps}, \textsc{ttns}, \textsc{peps}) may be regarded as functions of a specific $G$-rank for various choices of $G$. Among other things, we will see that a function, matrix, or tensor may have very high matrix, tensor, or multilinear rank and yet very low $G$-rank for some $G$. In fact the difference is in the orders of magnitudes and the gaps between $G$-ranks and these classical ranks are arbitrarily large for some important objects in computer science, mathematics, and physics. Furthermore, we show that there is a $G$ such that almost every tensor has $G$-rank exponentially lower than its rank or the dimension of its ambient space.

math.NA↗

arXiv2019

Hankel tensor decompositions and ranks

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 $n$-dimensional 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 true for generic Hankel tensors when the order is even or three. Some open questions are also posed.

math.AG↗

arXiv2018

Semi-Riemannian Manifold Optimization

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-degenerate semi-Riemannian manifolds as well, and can be carefully leveraged to adapt Riemannian optimization algorithms to the semi-Riemannian setting. In particular, we discuss the metric independence of manifold optimization algorithms, and illustrate that the weaker but more general semi-Riemannian geometry often suffices for the purpose of optimizing smooth functions on smooth manifolds in practice.

math.OC↗

arXiv2018

3d TQFTs from Argyres-Douglas theories

We construct a new class of three-dimensional topological quantum field theories (3d TQFTs) by considering generalized Argyres-Douglas theories on $S^1 \times M_3$ with a non-trivial holonomy of a discrete global symmetry along the $S^1$. For the minimal choice of the holonomy, the resulting 3d TQFTs are non-unitary and semisimple, thus distinguishing themselves from theories of Chern-Simons and Rozansky-Witten types respectively. Changing the holonomy performs a Galois transformation on the TQFT, which can sometimes give rise to more familiar unitary theories such as the $(G_2)_1$ and $(F_4)_1$ Chern-Simons theories. Our construction is based on an intriguing relation between topologically twisted partition functions, wild Hitchin characters, and chiral algebras which, when combined together, relate Coulomb branch and Higgs branch data of the same 4d $\mathcal{N}=2$ theory. We test our proposal by applying localization techniques to the conjectural $\mathcal{N}=1$ UV Lagrangian descriptions of the $(A_1,A_2)$, $(A_1,A_3)$ and $(A_1,D_3)$ theories.

hep-th↗

arXiv2018

The Grassmannian of affine subspaces

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; furthermore; it affords an analogue of Schubert calculus and its (co)homology and homotopy groups may be readily determined. On the other hand, like the Euclidean space, the affine Grassmannian serves as a concrete computational platform on which various distances, metrics, probability densities may be explicitly defined and computed via numerical linear algebra. Moreover, many standard problems in machine learning and statistics --- linear regression, errors-in-variables regression, principal components analysis, support vector machines, or more generally any problem that seeks linear relations among variables that either best represent them or separate them into components --- may be naturally formulated as problems on the affine Grassmannian.

math.DG↗

arXiv2018

Numerical algorithms on the affine Grassmannian

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, these algorithms are in the style of Edelman--Arias--Smith --- they rely only on standard numerical linear algebra and are readily computable.

stat.ME↗

arXiv2018

Geometric distance between positive definite matrices of different dimensions

We show how the Riemannian distance on $\mathbb{S}^n_{++}$, the cone of $n\times n$ real 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 $n$-dimensional ellipsoids, and inner products on $\mathbb{R}^n$, $n \times n$ covariance matrices of nondegenerate probability distributions, this gives us a natural way to define a geometric distance between a pair of such objects of different dimensions.

math.NA↗

arXiv2018

Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters

We use Coulomb branch indices of Argyres-Douglas theories on $S^1 \times L(k,1)$ to quantize moduli spaces ${\cal M}_H$ of wild/irregular Hitchin systems. In particular, we obtain formulae for the "wild Hitchin characters" -- the graded dimensions of the Hilbert spaces from quantization -- for four infinite families of ${\cal M}_H$, giving access to many interesting geometric and topological data of these moduli spaces. We observe that the wild Hitchin characters can always be written as a sum over fixed points in ${\cal M}_H$ under the $U(1)$ Hitchin action, and a limit of them can be identified with matrix elements of the modular transform $ST^kS$ in certain two-dimensional chiral algebras. Although naturally fitting into the geometric Langlands program, the appearance of chiral algebras, which was known previously to be associated with Schur operators but not Coulomb branch operators, is somewhat surprising.

hep-th↗

arXiv2018

Topology of tensor ranks

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-$r$ tensors and the set of symmetric rank-$r$ symmetric tensors are both path-connected if $r$ is 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-$r$ tensors is path-connected if it has the expected dimension but the corresponding result for symmetric rank-$r$ symmetric $d$-tensors depends on the order $d$: connected when $d$ is odd but not when $d$ is even. Border rank and symmetric border rank over $\mathbb{R}$ have essentially the same path-connectedness properties as rank and symmetric rank over $\mathbb{R}$. When $r$ is greater than the complex generic rank, we are unable to discern any general pattern: For example, we show that border-rank-three tensors in $\mathbb{R}^2 \otimes \mathbb{R}^2 \otimes \mathbb{R}^2$ fall into four connected components. For multilinear rank, the manifold of $d$-tensors of multilinear rank $(r_1,\dots,r_d)$ in $\mathbb{C}^{n_1} \otimes \cdots \otimes \mathbb{C}^{n_d}$ is always path-connected, and the same is true in $\mathbb{R}^{n_1} \otimes \cdots \otimes \mathbb{R}^{n_d}$ unless $n_i = r_i = \prod_{j \ne i} r_j$ for some $i\in\{1, \dots, d\}$. Beyond path-connectedness, we determine, over both $\mathbb{R}$ and $\mathbb{C}$, the fundamental and higher homotopy groups of the set of tensors of a fixed small rank, and, taking advantage of Bott periodicity, those of the manifold of tensors of a fixed multilinear rank. We also obtain analogues of these results for symmetric tensors of a fixed symmetric rank or a fixed symmetric multilinear rank.

math.AG↗

arXiv2018

Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality

In this paper, we show the equivalence between two seemingly distinct 2d TQFTs: one comes from the "Coulomb branch index" of the class S theory $T[Σ,G]$ on $L(k,1) \times S^1$, the other is the $^LG$ "equivariant Verlinde formula", or equivalently partition function of $^LG_{\mathbb{C}}$ complex Chern-Simons theory on $Σ\times S^1$. We first derive this equivalence using the M-theory geometry and show that the gauge groups appearing on the two sides are naturally $G$ and its Langlands dual $^LG$. When $G$ is not simply-connected, we provide a recipe of computing the index of $T[Σ,G]$ as summation over indices of $T[Σ,\tilde{G}]$ with non-trivial background 't Hooft fluxes, where $\tilde{G}$ is the simply-connected group with the same Lie algebra. Then we check explicitly this relation between the Coulomb index and the equivariant Verlinde formula for $G=SU(2)$ or $SO(3)$. In the end, as an application of this newly found relation, we consider the more general case where $G$ is $SU(N)$ or $PSU(N)$ and show that equivariant Verlinde algebra can be derived using field theory via (generalized) Argyres-Seiberg duality. We also attach a Mathematica notebook that can be used to compute the $SU(3)$ equivariant Verlinde coefficients.

hep-th↗

arXiv2017

Argyres-Douglas matter and S-duality: Part II

We study S-duality of Argyres-Douglas theories obtained by compactification of 6d (2,0) theories of ADE type on a sphere with irregular punctures. The weakly coupled descriptions are given by the degeneration limit of auxiliary Riemann sphere with marked points, among which three punctured sphere represents isolated superconformal theories. We also discuss twisted irregular punctures and their S-duality.

hep-th↗

arXiv2017

On the Chiral Ring and Vacua of ${\cal N}=1$ Adjoint SQCD

We analyze classical and quantum chiral ring relations of four dimensional ${\cal N} = 1$ adjoint SQCD with superpotential turned on for the adjoint field. In particular, for the mass deformed theory we obtain the complete on shell vacuum expectation value for various gauge invariant chiral operators and find non-trivial gaugino condensations. When approaching to massless limit nontrivial flat directions in the moduli space of vacua appear, where the Coulomb branch can be naturally classified and the Higgs branch receives quantum corrections. We argue that the solution of the chiral ring is in one-to-one correspondence with supersymmetric vacua, provided that an additional Konishi anomaly equation is included.

hep-th↗

arXiv2017

Cohomology of Cryo-Electron Microscopy

The goal of cryo-electron microscopy (EM) is to reconstruct the 3-dimensional structure of a molecule from a collection of its 2-dimensional projected images. In this article, we show that the basic premise of cryo-EM --- patching together 2-dimensional projections to reconstruct a 3-dimensional object --- is naturally one of Cech cohomology with SO(2)-coefficients. We deduce that every cryo-EM reconstruction problem corresponds to an oriented circle bundle on a simplicial complex, allowing us to classify cryo-EM problems via principal bundles. In practice, the 2-dimensional images are noisy and a main task in cryo-EM is to denoise them. We will see how the aforementioned insights can be used towards this end.

cs.CV↗

arXiv2016

New classes of matrix decompositions

The idea of decomposing a matrix into a product of structured matrices such as triangular, orthogonal, diagonal matrices is a milestone of numerical computations. In this paper, we describe six new classes of matrix decompositions, extending our work in arXiv:1307.5132. We prove that every $n\times n$ matrix is a product of finitely many bidiagonal, skew symmetric (when n is even), generic, companion matrices and generalized Vandermonde matrices, respectively. We also prove that a generic $n\times n$ centrosymmetric matrix is a product of finitely many symmetric Toeplitz (resp. persymmetric Hankel) matrices. We determine an upper bound of the number of structured matrices needed to decompose a matrix for each case.

math.AG↗

arXiv2016

Schubert varieties and distances between subspaces of different dimensions

We resolve a basic problem on subspace distances that often arises in applications: How can the usual Grassmann distance between equidimensional subspaces be extended to subspaces of different dimensions? We show that a natural solution is given by the distance of a point to a Schubert variety within the Grassmannian. This distance reduces to the Grassmann distance when the subspaces are equidimensional and does not depend on any embedding into a larger ambient space. Furthermore, it has a concrete expression involving principal angles, and is efficiently computable in numerically stable ways. Our results are largely independent of the Grassmann distance --- if desired, it may be substituted by any other common distances between subspaces. Our approach depends on a concrete algebraic geometric view of the Grassmannian that parallels the differential geometric perspective that is well-established in applied and computational mathematics.

math.NA↗

arXiv2016

Fast structured matrix computations: tensor rank and Cohn--Umans method

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 of the generalized method, we apply it to find the fastest algorithms for forming structured matrix-vector product, the basic operation underlying iterative algorithms for structured matrices. The structures we study include Toeplitz, Hankel, circulant, symmetric, skew-symmetric, f-circulant, block-Toeplitz-Toeplitz-block, triangular Toeplitz matrices, Toeplitz-plus-Hankel, sparse/banded/triangular. Except for the case of skew-symmetric matrices, for which we have only upper bounds, the algorithms derived using the generalized Cohn-Umans method in all other instances are the fastest possible in the sense of having minimum bilinear complexity. We also apply this framework to a few other bilinear operations including matrix-matrix, commutator, simultaneous matrix products, and briefly discuss the relation between tensor nuclear norm and numerical stability.

math.NA↗

arXiv2016

A 3d-3d appetizer

We test the 3d-3d correspondence for theories that are labelled by Lens spaces. We find a full agreement between the index of the 3d ${\cal N}=2$ "Lens space theory" $T[L(p,1)]$ and the partition function of complex Chern-Simons theory on $L(p,1)$. In particular, for $p=1$, we show how the familiar $S^3$ partition function of Chern-Simons theory arises from the index of a free theory. For large $p$, we find that the index of $T[L(p,1)]$ becomes a constant independent of $p$. In addition, we study $T[L(p,1)]$ on the squashed three-sphere $S^3_b$. This enables us to see clearly, at the level of partition function, to what extent $G_\mathbb{C}$ complex Chern-Simons theory can be thought of as two copies of Chern-Simons theory with compact gauge group $G$.

hep-th↗