On the Quality of a Semidefinite Programming Bound for Sparse Principal Component Analysis
نویسنده
چکیده
We examine the problem of approximating a positive, semidefinite matrix Σ by a dyad xxT , with a penalty on the cardinality of the vector x. This problem arises in sparse principal component analysis, where a decomposition of Σ involving sparse factors is sought. We express this hard, combinatorial problem as a maximum eigenvalue problem, in which we seek to maximize, over a box, the largest eigenvalue of a symmetric matrix that is linear in the variables. This representation allows to use the techniques of robust optimization, to derive a bound based on semidefinite programming. The quality of the bound is investigated using a technique inspired by Nemirovski and Ben-Tal (2002).
منابع مشابه
Sparse Structured Principal Component Analysis and Model Learning for Classification and Quality Detection of Rice Grains
In scientific and commercial fields associated with modern agriculture, the categorization of different rice types and determination of its quality is very important. Various image processing algorithms are applied in recent years to detect different agricultural products. The problem of rice classification and quality detection in this paper is presented based on model learning concepts includ...
متن کاملEigenvalue Maximization in Sparse PCA
We examine the problem of approximating a positive, semidefinite matrix Σ by a dyad xxT , with a penalty on the cardinality of the vector x. This problem arises in the sparse principal component analysis problem, where a decomposition of Σ involving sparse factors is sought. We express this hard, combinatorial problem as a maximum eigenvalue problem, in which we seek to maximize, over a box, th...
متن کاملApproximating Semidefinite Packing Programs
In this paper we define semidefinite packing programs and describe an algorithm to approximately solve these problems. Semidefinite packing programs arise in many applications such as semidefinite programming relaxations for combinatorial optimization problems, sparse principal component analysis, and sparse variance unfolding techniques for dimension reduction. Our algorithm exploits the struc...
متن کاملA path following interior-point algorithm for semidefinite optimization problem based on new kernel function
In this paper, we deal to obtain some new complexity results for solving semidefinite optimization (SDO) problem by interior-point methods (IPMs). We define a new proximity function for the SDO by a new kernel function. Furthermore we formulate an algorithm for a primal dual interior-point method (IPM) for the SDO by using the proximity function and give its complexity analysis, and then we sho...
متن کاملApproximation bounds for sparse principal component analysis
We produce approximation bounds on a semidefinite programming relaxation for sparse principal component analysis. These bounds control approximation ratios for tractable statistics in hypothesis testing problems where data points are sampled from Gaussian models with a single sparse leading component. We study approximation bounds for a semidefinite relaxation of the sparse eigenvalue problem, ...
متن کامل