Computational Mathematics and Applications Seminar

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

Past events in this series
4 March 2021
14:00
Pierre-Antoine Absil
Abstract

This talk concerns applications of differential geometry in numerical optimization. They arise when the optimization problem can be formulated as finding an optimum of a real-valued cost function defined on a smooth nonlinear search space. Oftentimes, the search space is a "matrix manifold", in the sense that its points admit natural representations in the form of matrices. In most cases, the matrix manifold structure is due either to the presence of certain nonlinear constraints (such as orthogonality or rank constraints), or to invariance properties in the cost function that need to be factored out in order to obtain a nondegenerate optimization problem. Manifolds that come up in practical applications include the rotation group SO(3) (generation of rigid body motions from sample points), the set of fixed-rank matrices (low-rank models, e.g., in collaborative filtering), the set of 3x3 symmetric positive-definite matrices (interpolation of diffusion tensors), and the shape manifold (morphing).

In the recent years, the practical importance of optimization problems on manifolds has stimulated the development of geometric optimization algorithms that exploit the differential structure of the manifold search space. In this talk, we give an overview of geometric optimization algorithms and their applications, with an emphasis on the underlying geometric concepts and on the numerical efficiency of the algorithm implementations.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact trefethen@maths.ox.ac.uk.

  • Computational Mathematics and Applications Seminar
11 March 2021
14:00
Abstract

We provide a computational framework for approximating a class of structured matrices (e.g., block Toeplitz, block banded). Our approach has three steps: map the structured matrix to tensors, use tensor compression algorithms, and map the compressed tensors back to obtain two different matrix representations --- sum of Kronecker products and block low-rank format. The use of tensor decompositions enable us to uncover latent structure in the matrices and lead to computationally efficient algorithms. The resulting matrix approximations are memory efficient, easy to compute with, and preserve the error due to the tensor compression in the Frobenius norm. While our framework is quite general, we illustrate the potential of our method on structured matrices from three applications: system identification, space-time covariance matrices, and image deblurring.

Joint work with Misha Kilmer (Tufts University)

 

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A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact trefethen@maths.ox.ac.uk.

  • Computational Mathematics and Applications Seminar
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