The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
On Smooth Decompositions of Matrices
SIAM Journal on Matrix Analysis and Applications
Adaptive Eigenvalue Computations Using Newton's Method on the Grassmann Manifold
SIAM Journal on Matrix Analysis and Applications
Dynamic inversion of nonlinear maps with applications to nonlinear control and robotics
Dynamic inversion of nonlinear maps with applications to nonlinear control and robotics
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We propose a general framework for tracking the zeros of a time-varying gradient vector field on Riemannian manifolds. Thus, a differential equation, called the time-varying Newton flow, is introduced, whose solutions asymptotically converge to a time-varying family of critical points of the corresponding cost function. A discretization of the differential equation leads to a recursive update scheme for the time-varying critical point. As an application of such techniques we develop new algorithms for computing the principal and minor subspace of a time-varying family of symmetric matrices. Using a convenient local parameterization of the Grassmann manifold, we derive simple expressions for the subspace tracking schemes. Key benefits of the algorithms are (a) the reduced complexity aspects due to efficient parameterizations of the Grassmannian and (b) their guaranteed accuracy during all iterates. Numerical simulations illustrate the feasibility of the approach.