A Least Squares Formulation for a Class of Generalized Eigenvalue Problems in Machine Learning

Abstract

Many machine learning algorithms can be formulated as a generalized eigenvalue problem. One major limitation of such formulation is that the generalized eigenvalue problem is computationally expensive to solve especially for large-scale problems. In this paper, we show that under a mild condition, a class of generalized eigenvalue problems in machine learning can be formulated as a least squares problem. This class of problems include classical techniques such as Canonical Correlation Analysis (CCA), Partial Least Squares (PLS), Linear Discriminant Analysis (LDA), as well as Hypergraph Spectral Learning (HSL). As a result, various regularization techniques such as the 1-norm and 2-norm regularization can be readily incorporated into the formulation to improve model sparsity and generalization ability. In addition, the least squares formulation leads to efficient and scalable implementations based on the iterative conjugate gradient type algorithms. We report experimental results that confirm the established equivalence relationship. We also demonstrate the efficiency and effectiveness of the equivalent least squares formulations on large-scale problems. The presented analysis provides significant new insights into the relationship between the generalized eigenvalue and least squares problems in machine learning.