Metric Learning Using Iwasawa Decomposition

Finding a good metric over the input space plays a fundamental role in machine learning. Most existing techniques use the Mahalanobis metric without incorporating the geometry of positive matrices and experience difficulties in the optimization procedure. In this paper we introduce the use of Iwasawa decomposition, a unique and effective parametrization of symmetric positive definite (SPD) matrices, for performing metric learning tasks. Unlike other previously employed factorizations, the use of the Iwasawa decomposition is able to reformulate the semidefinite programming (SDP) problems as smooth convex nonlinear programming (NLP) problems with much simpler constraints. We also introduce a modified Iwasawa coordinates for rank-deficient positive semidefinite (PSD) matrices which enables the unifying of the metric learning and linear dimensionality reduction. We show that the Iwasawa decomposition can be easily used in most recent proposed metric learning algorithms and have applied it to the Neighbourhood Components Analysis (NCA). The experimental results on several public domain datasets are also presented.