<b>GrassmannOptim</b>: An<i>R</i>Package for Grassmann Manifold Optimization

Abstract

The optimization of a real-valued objective function <i>f</i>(<b>U</b>), where <b>U</b> is a <i>p</i> X <i>d,p</i> > <i>d</i>, semi-orthogonal matrix such that <b>U</b>^T<b>U</b>=<b>I</b>_d, and <i>f</i> is invariant under right orthogonal transformation of <b>U</b>, is often referred to as a Grassmann manifold optimization. Manifold optimization appears in a wide variety of computational problems in the applied sciences. In this article, we present <b>GrassmannOptim</b>, an R package for Grassmann manifold optimization. The implementation uses gradient-based algorithms and embeds a stochastic gradient method for global search. We describe the algorithms, provide some illustrative examples on the relevance of manifold optimization and finally, show some practical usages of the package.