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The Extrinsic Geometry of Continuous Time Matrix Algorithms

Feppon, F. and P.F.J. Lermusiaux, 2018. The extrinsic geometry of dynamical systems tracking nonlinear matrix projections. SIAM Journal on Matrix Analysis and Applications, sub-judice.

A geometric framework is introduced for the systematic study of a class of maps called oblique projections. These include orthogonal projections, i.e. maps that project points of a finite dimensional ambient vector space to the closest point of an embedded manifold, but also their generalizations to non euclidean ambient spaces. Typical everyday examples involve the truncated SVD, the polar decomposition, linear subspace filters, and projectors over the dominant eigenspaces of symmetric and non symmetric matrices. A methodology is proposed for the systematic derivation of the differential of these maps, and of convergent continuous time matrix algorithms that allow to dynamically track their values on time dependent matrices. It is shown that these maps are characterized by a bundle of normal spaces that provide the image manifold with a differentiable structure. Generalizations of classical properties of embedded Riemannian manifolds, such as the Gauss equation and the Weingarten identity, are found by replacing the ambient scalar product with the duality bracket. Previous differentiability results obtained for orthogonal projections onto embedded Riemannian manifolds are extended to oblique projections in the non euclidean setting. The framework is applied to the study of the maps above and their image manifold in an embedded setting, that include the Stiefel and the Orthogonal group, the Isospectral and the Grassman manifold, and the bi-Grassman manifold or the set of fixed rank linear projectors.