tag:blogger.com,1999:blog-11620213981371019582024-02-20T15:00:22.100+01:00Mathematical ThinkingAntonio Falcohttp://www.blogger.com/profile/18177869047336526550noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-1162021398137101958.post-19396299727507322062019-03-31T12:42:00.001+02:002019-03-31T12:42:39.643+02:00Principal bundle structure of matrix manifolds by Marie Billaud-Friess, Antonio Falco, Anthony Nouy<div style="text-align: justify;">
<b>Abstract </b><br />
<br />
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 $\mathbb{G}_r(\mathbb{R}^k)$ of linear subspaces of dimension $r<k$ in $\mathbb{R}^k$ which avoids the use of equivalence classes. The set $\mathbb{G}_r(\mathbb{R}^k)$ is equipped with an atlas which provides it with the structure of an analytic manifold modelled on $\mathbb{R}^{(k-r)\times r}$.
Then we define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{k \times r})$ of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^k)$ and typical fibre $\mathrm{GL}_r$, the general linear group of invertible matrices in $\mathbb{R}^{k\times k}$. Finally, we define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ of non-full rank matrices and prove that the resulting
manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^n) \times \mathbb{G}_r(\mathbb{R}^m)$ and typical fibre $\mathrm{GL}_r$.
The atlas of $\mathcal{M}_r(\mathbb{R}^{n \times m})$ 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 $\mathcal{M}_r(\mathbb{R}^{n \times m})$ equipped with the topology induced by the atlas is
proven to be an embedded submanifold of the matrix space $\mathbb{R}^{n \times m}$
equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space $\mathbb{R}^{n \times m}$, seen as the union of manifolds $\mathcal{M}_r(\mathbb{R}^{n \times m})$, as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.
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You can download the paper from <a href="https://arxiv.org/pdf/1705.04093">https://arxiv.org/pdf/1705.04093</a></div>Antonio Falcohttp://www.blogger.com/profile/18177869047336526550noreply@blogger.com0tag:blogger.com,1999:blog-1162021398137101958.post-13268082792154307982015-05-25T21:07:00.001+02:002015-05-25T21:10:14.297+02:00Geometric Structures in Tensor Representations (Final Release) by A. Falcó, W. Hackbusch and A. Nouy<div style="text-align: justify;">
<b>Abstract </b><br />
<br />
The main goal of this paper is to study the geometric structures associated with the representation of tensors in subspace based formats. To do this we use a property of the so-called minimal subspaces which allows us to describe the tensor representation by means of a rooted tree. By using the tree structure and the dimensions of the associated minimal subspaces, we introduce, in the underlying algebraic tensor space, the set of tensors in a tree-based format with either bounded or fixed tree-based rank. This class contains the Tucker format and the Hierarchical Tucker format (including the Tensor Train format). In particular, we show that the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank of an algebraic tensor product of normed vector spaces is an analytic Banach manifold. Indeed, the manifold geometry for the set of tensors with fixed tree-based rank is induced by a fibre bundle structure and the manifold geometry for the set of tensors with bounded tree-based rank is given by a finite union of connected components where each of them is a manifold of tensors in the tree-based format with a fixed tree-based rank. The local chart representation of these manifolds is often crucial for an algorithmictreatment of high-dimensional PDEs and minimization problems. In order to describe the relationship between these manifolds and the natural ambient space, we introduce the definition of topological tensor spaces in the tree-based format. We prove under natural conditions that any tensor of the topological tensor space under consideration admits best approximations in the manifold of tensors in the tree-based format with<br />
bounded tree-based rank. In this framework, we also show that the tangent (Banach) space at a given tensor is a complemented subspace in the natural ambient tensor Banach space and hence the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank is an immersed submanifold. This fact allows us to extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces.<br />
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You can download the paper from <a href="http://arxiv.org/pdf/1505.03027">http://arxiv.org/pdf/1505.03027</a></div>
Antonio Falcohttp://www.blogger.com/profile/18177869047336526550noreply@blogger.com0