33 research outputs found
Low-rank Approximation of Linear Maps
This work provides closed-form solutions and minimal achievable errors for a
large class of low-rank approximation problems in Hilbert spaces. The proposed
theorem generalizes to the case of linear bounded operators and p-th Schatten
norms previous results obtained in the finite dimensional case for the
Frobenius norm. The theorem is illustrated in various settings, including
low-rank approximation problems with respect to the trace norm, the 2-induced
norm or the Hilbert-Schmidt norm. The theorem provides also the basics for the
design of tractable algorithms for kernel-based or continuous DM
Principal bundle structure of matrix manifolds
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 of linear subspaces of
dimension in which avoids the use of equivalence classes.
The set is equipped with an atlas which provides
it with the structure of an analytic manifold modelled on
. Then we define an atlas for the set
of full rank matrices and prove that
the resulting manifold is an analytic principal bundle with base
and typical fibre , the general
linear group of invertible matrices in . Finally, we
define an atlas for the set of
non-full rank matrices and prove that the resulting manifold is an analytic
principal bundle with base and typical fibre . The atlas of
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 equipped with the topology
induced by the atlas is proven to be an embedded submanifold of the matrix
space equipped with the subspace topology. The
proposed geometric description then results in a description of the matrix
space , seen as the union of manifolds
, as an analytic manifold equipped with
a topology for which the matrix rank is a continuous map
Convergence results for projected line-search methods on varieties of low-rank matrices via \L{}ojasiewicz inequality
The aim of this paper is to derive convergence results for projected
line-search methods on the real-algebraic variety of real
matrices of rank at most . Such methods extend Riemannian
optimization methods, which are successfully used on the smooth manifold
of rank- matrices, to its closure by taking steps along
gradient-related directions in the tangent cone, and afterwards projecting back
to . Considering such a method circumvents the
difficulties which arise from the nonclosedness and the unbounded curvature of
. The pointwise convergence is obtained for real-analytic
functions on the basis of a \L{}ojasiewicz inequality for the projection of the
antigradient to the tangent cone. If the derived limit point lies on the smooth
part of , i.e. in , this boils down to more
or less known results, but with the benefit that asymptotic convergence rate
estimates (for specific step-sizes) can be obtained without an a priori
curvature bound, simply from the fact that the limit lies on a smooth manifold.
At the same time, one can give a convincing justification for assuming critical
points to lie in : if is a critical point of on
, then either has rank , or