arXiv subjects

Ke Ye

Publications and source records attributed to Ke Ye.

80 records · Page 5Linked to original sources

arXiv2016

Inverse tensor eigenvalue problem

A tensor $\mathcal T\in \mathbb T(\mathbb C^n,m+1)$, the space of tensors of order $m+1$ and dimension $n$ with complex entries, has $nm^{n-1}$ eigenvalues (counted with algebraic multiplicities). The inverse eigenvalue problem for tensors is a generalization of that for matrices. Namely, given a multiset $S\in \mathbb C^{nm^{n-1}}/\mathfrak{S}(nm^{n-1})$ of total multiplicity $nm^{n-1}$, is there a tensor in $\mathbb T(\mathbb C^n,m+1)$ such that the multiset of eigenvalues of $\mathcal{T}$ is exact $S$? The solvability of the inverse eigenvalue problem for tensors is studied in this paper. With tools from algebraic geometry, it is proved that the necessary and sufficient condition for this inverse problem to be generically solvable is $m=1,\ \text{or }n=2,\ \text{or }(n,m)=(3,2),\ (4,2),\ (3,3)$.

math.SP↗

arXiv2016

Algorithms for structured matrix-vector product of optimal bilinear complexity

We present explicit algorithms for computing structured matrix-vector products that are optimal in the sense of Strassen, i.e., using a provably minimum number of multiplications. These structures include Toeplitz/Hankel/circulant, symmetric, Toeplitz-plus-Hankel, sparse, and multilevel structures. The last category include \textsc{bttb}, \textsc{bhhb}, \textsc{bccb} but also any arbitrarily complicated nested structures built out of other structures.

math.NA↗

arXiv2014

Multiplicities of eigenvalues of tensors

We study in this article multiplicities of eigenvalues of tensors. There are two natural multiplicities associated to an eigenvalue $λ$ of a tensor: algebraic multiplicity $\operatorname{am}(λ)$ and geometric multiplicity $\operatorname{gm}(λ)$. The former is the multiplicity of the eigenvalue as a root of the characteristic polynomial, and the latter is the dimension of the eigenvariety (i.e., the set of eigenvectors) corresponding to the eigenvalue. We show that the algebraic multiplicity could change along the orbit of tensors by the orthogonal linear group action, while the geometric multiplicity of the zero eigenvalue is invariant under this action, which is the main difficulty to study their relationships. However, we show that for a generic tensor, every eigenvalue has a unique (up to scaling) eigenvector, and both the algebraic multiplicity and geometric multiplicity are one. In general, we suggest for an $m$-th order $n$-dimensional tensor the relationship \[ \operatorname{am}(λ)\geq \operatorname{gm}(λ)(m-1)^{\operatorname{gm}(λ)-1}. \] We show that it is true for serveral cases, especially when the eigenvariety contains a linear subspace of dimension $\operatorname{gm}(λ)$ in coordinate form. As both multiplicities are invariants under the orthogonal linear group action in the matrix counterpart, this generalizes the classical result for a matrix: the algebraic mutliplicity is not smaller than the geometric multiplicity.

math.SP↗

arXiv2014

Every matrix is a product of Toeplitz matrices

We show that every n-by-n matrix is generically a product of [n/2] + 1 Toeplitz matrices and always a product of at most 2n+5 Toeplitz matrices. The same result holds true if the word "Toeplitz" is replaced by "Hankel", and the generic bound [n/2] + 1 is sharp. We will see that these decompositions into Toeplitz or Hankel factors are unusual: We may not in general replace the subspace of Toeplitz or Hankel matrices by an arbitrary (2n-1)-dimensional subspace of n-by-n matrices. Furthermore such decompositions do not exist if we require the factors to be symmetric Toeplitz, persymmetric Hankel, or circulant matrices, even if we allow an infinite number of factors. Lastly, we discuss how the Toeplitz and Hankel decompositions of a generic matrix may be computed by either (i) solving a system of linear and quadratic equations if the number of factors is required to be [n/2] + 1, or (ii) Gaussian elimination in O(n^3) time if the number of factors is allowed to be 2n.

math.AG↗

arXiv2013

The CERN LHC Sensitivity on measuring WZGamma Production and Anomalous WWZGamma Coupling

In this paper we present for the first time a detailed Monte Carlo study of measuring WZGamma production with pure leptonic decays and probing anomalous quartic gauge-boson WWZGamma couplings at the sqrt(s) = 14 TeV LHC, with parton shower and detector simulation effects taken into account. We find that with an integrated luminosity of 100 fb-1 and proper selection cuts, the Standard Model WZGamma signal significance can be improved to as much as 3 sigma. After reviewing previous parametrization on anomalous WWZGamma couplings (see e.g.an/Lambda^2 or k2m/Lambda^2 as shown in Ref. [17]), we propose a more general parametrization scheme with 4 free inputs leading only to genuine WWZGamma aQGC couplings. Finally, our numerical results show that one can reach constraints at 95% confidence level of -5.7 * 10^{-5} GeV^{-2} < k2m/Lambda^2 < 5.5 * 10^{-5} GeV^{-2} and -2.2 x 10^{-5} GeV^{-2} < an/Lambda^2 < 2.4 * 10^{-5} GeV^{-2}, which are more stringent than LEP's results by three orders of magnitude.

hep-ph↗

arXiv2013

Probing W+W-gamma Production and Anomalous Quartic Gauge Boson Couplings at the CERN LHC

Triple gauge boson associated production at the LHC serves as an interesting channel to test the robustness of the Standard Model. Any deviation from its SM prediction may indicate possible existence of relevant new physics, e.g., anomalous quartic gauge boson couplings. In this paper, a Monte-Carlo feasibility study of measuring WWA production with pure leptonic decays and probing anomalous quartic gauge-boson (e.g., WWAA) couplings, is presented in detail for the first time, with parton shower and detector simulation effects taken into account. Our results show that at the sqrt{s} = 14 TeV LHC with an integrated luminosity of 100 (30) fb-1, one can reach a significance of 9 (5) sigma to observe the SM WWA production, and can constrain at the 95% CL the anomalous WWAA coupling parameters, e.g., a_{0,c}^W/Λ^2 (see Ref.[15] for their definitions), at 10^{-5} GeV^{-2}, respectively.

hep-ph↗

arXiv2012

On the geometry of tensor network states

We answer a question of L. Grasedyck that arose in quantum information theory, showing that the limit of tensors in a space of tensor network states need not be a tensor network state. We also give geometric descriptions of spaces of tensor networks states corresponding to trees and loops. Grasedyck's question has a surprising connection to the area of Geometric Complexity Theory, in that the result is equivalent to the statement that the boundary of the Mulmuley-Sohoni type variety associated to matrix multiplication is strictly larger than the projections and re-labelings of matrix multiplication. Tensor Network States are also related to graphical models in algebraic statistics.

math.AG↗

arXiv2011

The Stabilizer Of Immanants

We describe immanants as trivial modules of the symmetric group and show that any homogeneous polynomial of degree n on the space of n by n matrices preserved up to scalar by left and right action by diagonal matrices and conjugation by permutation matrices is a linear combination of immanants. we prove that the identity component of the stabilizer of any immanant (except determinant, permanent) is the expected one. We also prove that for n>5 the stabilizer of the immanant of any non-symmetric partition (except determinant and permanent) is again the expected one.

math.CO↗