Linear Constrained Rayleigh Quotient Optimization: Theory and Algorithms

  title={Linear Constrained Rayleigh Quotient Optimization: Theory and Algorithms},
  author={Yunshen Zhou and Zhaojun Bai and Ren-Cang Li},
We consider the following constrained Rayleigh quotient optimization problem (CRQopt) $$ \min_{x\in \mathbb{R}^n} x^{T}Ax\,\,\mbox{subject to}\,\, x^{T}x=1\,\mbox{and}\,C^{T}x=b, $$ where $A$ is an $n\times n$ real symmetric matrix and $C$ is an $n\times m$ real matrix. Usually, $m\ll n$. The problem is also known as the constrained eigenvalue problem in the literature because it becomes an eigenvalue problem if the linear constraint $C^{T}x=b$ is removed. We start by equivalently transforming… 

Figures and Tables from this paper

A Linearly Constrained Power Iteration for Spectral Semi-Supervised Classification on Signed Graphs

This work considers the problem of semi-supervised node classification and extends the method of Xu et al. to a multiclass setting and deviates from the orthogonal iteration by replacing the QR-decomposition for orthonormalization by a projection operation.

CMS: a novel surrogate model with hierarchical structure based on correlation mapping

A novel surrogate model, termed correlation mapping surrogate (CMS), is proposed based on the Rayleigh quotient and the multi-fidelity surrogate framework, demonstrating its satisfactory feasibility, practicality, and stability.



A Constrained Eigenvalue Problem

In this paper we consider the following mathematical and computational problem. Given the quantities A: (n + m)-by-(n + m) matrix, symmetric, n > 0 N: (n + m)-by-m matrix with full rank

LSMR: An Iterative Algorithm for Sparse Least-Squares Problems

Robust and Efficient Computation of Eigenvectors in a Generalized Spectral Method for Constrained Clustering

The proposed theoretical and computational solutions can be applied to eigenproblems of positive semi-definite pencils arising in other machine learning algorithms, such as generalized linear discriminant analysis in dimension reduction and multisurface classification via eigenvectors.

A semidefinite framework for trust region subproblems with applications to large scale minimization

A dual simplex type method is studied that solves (TRS) as a parametric eigenvalue problem and the essential cost of the algorithm is the matrix-vector multiplication and, thus, sparsity can be exploited.

On constrained spectral clustering and its applications

This paper presents a more natural and principled formulation of constrained spectral clustering, which explicitly encodes the constraints as part of a constrained optimization problem, and demonstrates an innovative use of encoding large number of constraints: transfer learning via constraints.

Vandermonde matrices with Chebyshev nodes

Matrix Perturbation Theory

X is the vector space which acts in the n-dimensional (complex) vector space R.1.1 and is related to Varepsilon by the following inequality.

On Meinardus' examples for the conjugate gradient method

A closed formula for the CG residuals for all 1 ≤ k < N- 1 on Meinardus' example is obtained, and in particular it implies that the bound is always within a factor of -√2 of the actual residuals.

On the Generalized Lanczos Trust-Region Method

A priori upper bounds for the convergence to both the optimal objective value as well as the optimal solution are developed and it is argued that these bounds can be efficiently estimated numerically and serve as stopping criteria for better numerical performance.

Segmentation given partial grouping constraints

It is demonstrated not only that it is possible to integrate both image structures and priors in a single grouping process, but also that objects can be segregated from the background without specific object knowledge.