Principal Bundle Structure of Matrix Manifolds

Marie Billaud-Friess,Anthony Nouy,Antonio Falcó

doi:10.3390/math9141669

Marie Billaud-Friess, Anthony Nouy + Show 1 more

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https://doi.org/10.3390/math9141669

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Abstract

In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold Gr(Rk) of linear subspaces of dimension r<k in Rk, which avoids the use of equivalence classes. The set Gr(Rk) is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on R(k−r)×r. Then, we define an atlas for the set Mr(Rk×r) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rk) and typical fibre GLr, the general linear group of invertible matrices in Rk×k. Finally, we define an atlas for the set Mr(Rn×m) of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rn)×Gr(Rm) and typical fibre GLr. The atlas of Mr(Rn×m) is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set Mr(Rn×m) equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space Rn×m equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space Rn×m, seen as the union of manifolds Mr(Rn×m), as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

Highlights

Low-rank matrices appear in many applications involving high-dimensional data.Low-rank models are commonly used in statistics, machine learning or data analysis
The proposed geometric description results in a description of the matrix space rank: Mr (Rn×m), seen as the union of manifolds Mr (Rn×m ), as an analytic manifold equipped with a topology for which the matrix rank is a continuous map
The set Mr (Rn×m ) is here endowed with the structure of analytic principal bundle with an explicit description of local charts. This results in a description of the matrix space Rn×m as an analytic manifold with a topology induced by local charts that is different from τRn×m and for which the rank is a continuous map

Summary

Introduction

Low-rank matrices appear in many applications involving high-dimensional data. Low-rank models are commonly used in statistics, machine learning or data analysis The purpose of this paper is to propose a new geometric description of the sets of matrices with fixed rank, which is amenable for numerical use, and relies on the natural parametrization of matrices in Mr (Rn×m ) given by. The set Mr (Rn×m ) is here endowed with the structure of analytic principal bundle with an explicit description of local charts This results in a description of the matrix space Rn×m as an analytic manifold with a topology induced by local charts that is different from τRn×m and for which the rank is a continuous map. A splitting algorithm relying on the geometric description of the set of fixed rank matrices proposed in this paper has been introduced for dynamical low-rank approximation [20]. Before introducing the main results and outline of the paper, we recall some elements of geometry

Elements of Geometry

Main Results and Outline

Lie Group Structure of Neighbourhoods UZ

Lie Group Structure of Neighbourhoods V Z

Lie Group Structure of Neighbourhoods U Z

Methods