Abstract : Nonlinear eigenvalue problems fall into two major categories
in terms of eigenvalue-dependency or eigenvector-dependency,
short-named NEP and NEPv, respectively. NEPv is the next natural
topic after NEP. The most well-known origin of NEPv is
from Kohn-Sham density functional theory in electronic structure
calculations, but most recent sources are various machine learning
models in the form of optimization on matrix manifolds,
core-periphery detection in networks, rate-splitting multiple access in
wireless communication, among others. In this minisymposium, speakers
will present recent advancements in algorithms, numerical analysis
and applications of NEPv and discuss emerging challenges.
[03905] A Self-Consistent-Field Iteration for OCCA
Format : Talk at Waseda University
Author(s) :
Leihong Zhang (Soochow University)
Li Wang (University of Texas at Arlington)
Zhaojun Bai (University of California, Davis)
Rencang Li (University of Texas at Arlington)
Abstract : In this talk, we propose an efficient algorithm for solving the orthogonal canonical correlation analysis (OCCA). Within an alternating optimization scheme, a customized self-consistent-field (SCF) iteration for a core trace-fractional sub-maximization over orthogonality constraint is devised and analyzed. The SCF iteration is further extended to deal with the multi-view OCCA. Experiments on real-world applications of multi-view learning will be reported.
[04633] Mathematical Analysis and Numerical Approximations of Density Functional Theory Models for Metallic Systems
Format : Talk at Waseda University
Author(s) :
Xiaoying Dai (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Abstract : In this talk, we will introduce our study on the energy minimization model arising in the ensemble
Kohn-Sham density functional theory for metallic systems, in which a pseudo-eigenvalue matrix and
a general smearing approach are involved. We investigate the invariance of the energy functional and
the existence of the minimizer of the ensemble Kohn-Sham model. We propose an adaptive two-
parameter step size strategy and the corresponding preconditioned conjugate gradient methods to
solve the energy minimization model. Under some mild but reasonable assumptions, we prove the
global convergence for the gradients of the energy functional produced by our algorithms. Numerical
experiments show that our algorithms are efficient, especially for large scale metallic systems. In
particular, our algorithms produce convergent numerical approximations for some metallic systems,
for which the traditional self-consistent field iterations fail to converge. This is a joint work with Dr. Bin Yang and Prof. Aihui Zhou.
[05119] Convergence of SCF Iteration for Eigenvector-dependent Nonlinear Eigenvalue Problems
Format : Talk at Waseda University
Author(s) :
Ding Lu (University of Kentucky)
Abstract : The self-consistent field (SCF) iteration is a widely used method for solving eigenvector-dependent nonlinear eigenvalue problems (NEPv). Despite its simplicity, SCF is prone to slow convergence and can even fail to converge entirely. Extensive research has been devoted to analyzing the algorithm's convergence, and in this talk, we will share new insights on the local convergence analysis of SCF. First, for NEPv with a unitarily-invariant coefficient matrix, we will establish a precise estimation of the local convergence factor of SCF. This estimation enables us to prove the convergence of a level-shift scheme, which can help address the convergence issues of SCF. Second, for a class of NEPv without the unitary invariance property, we will show how to transform the problem into a unitarily invariant NEPv locally at the eigenbasis of interest. We will then establish the convergence rate of SCF based on this transformed problem. This work is a joint effort with Zhaojun Bai and Ren-Cang Li.
[04225] Geometric inexact Newton method for generalized singular values
Format : Online Talk on Zoom
Author(s) :
Weiwei Xu (Nanjing University of Information Science and Technology)
Michael K. Ng (The University of Hong Kong)
Zhengjian Bai (Xiamen University)
Abstract : We give new model formulations for computing arbitrary generalized singular value of a Grassmann matrix pair or a real matrix pair, where we need to solve matrix optimization problems with unitary constraints or orthogonal constraints. We propose a geometric inexact Newton-CG method for solving these optimization problems. Under some mild assumptions, we establish the global and quadratic convergence of the proposed method for the complex case. We illustrate its effectiveness by some numerical examples.
[05381] Nonlinear spectral graph theory: an overview of graph properties
Format : Online Talk on Zoom
Author(s) :
Francesco Tudisco (Gran Sasso Science Institute)
Dong Zhang (Peking University)
Piero Deidda (Gran Sasso Science Institute)
Abstract : It is well-known that a variety of combinatorial graph quantities are approximated by the spectrum of relevant graph matrices, such as the Laplacian or the adjacency matrix. While very useful and widely used, these approximations are far from being tight. A generalization of the graph spectrum can be defined by considering nonlinear eigenvalue problems defined in terms of pairs of homogeneous convex functions. For quadratic functions, this definition boils down to the standard linear (matrix) eigenvalue problem. In this talk, we review several basic definitions and properties of nonlinear eigenproblems defined in terms of generic homogeneous pairs, and we then show how the corresponding eigenvalues can be used to provide tight approximations to fundamental graph quantities, including the multi-way Cheeger constant, the graph packing radius, the graph's max and min cuts, and graph's modularity.
[05445] Solving Non-Convex Problems without Relaxation: Unexpected Usefulness of NEPv on Optimization Theory
Format : Online Talk on Zoom
Author(s) :
Jeonghun Park (Yonsei University)
Abstract : Non-convex optimization problems arise in many applications of wireless communications and signal processing. To solve this, one typical principle is relaxing the original problem, obtaining a "convexified" solution, and repeat the process until convergence. In this talk, breaking away from this conventional approach, we develop an unorthodox method called spectral method. We also show that in some applications, this new approach offers significantly better performances with less complexity.
[05390] Trace Minimization Principles on Matrix Manifolds
Format : Talk at Waseda University
Author(s) :
Xin Liang (Tsinghua University)
Ren-Cang Li (University of Texas at Arlington)
Li Wang (University of Texas at Arlington)
Leihong Zhang (Soochow University)
Abstract : This talk is concerned with establishing a trace minimization principle for two Hermitian matrix pairs: when is $\inf_X{\rm trace}(\widehat AX^{\rm H}AX)$ subject to $\widehat BX^{\rm H}BX=I$ finite? Sufficient and necessary conditions are obtained and, when the infimum is finite, an explicit formula for it is presented in terms of the finite eigenvalues of the matrix pairs.
[05272] Bound states in the continuum for a class of infinite matrices
Format : Talk at Waseda University
Author(s) :
Ya Yan Lu (City University of Hong Kong)
Abstract : For certain infinite matrix $A$, the equation $Ax=\lambda x$ may represent a boundary value problem (BVP) for $\lambda$ in a real interval $C$. Such an equation models the scattering of incident waves by some objects. Without the incident waves, $Ax=\lambda x$ is still an eigenvalue problem. A bound state in the continuum (BIC) is a special eigenpair with the eigenvalue $\lambda \in C$. In that case, the BVP for the same $\lambda$ has no uniqueness. A nonlinear version is $(A + D(x)) x = \lambda x$, where $D(x)$ is a diagonal matrix with the $(i,i)$ entry being $d_i |x_i|^2$. In this talk, we discuss linear and nonlinear BICs for a class of infinite matrices.
