Convexity properties of the condition number II
In our previous paper [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1491–1506], we studied the condition metric in the space of maximal rank n × m matrices. Here, we show that this condition metric induces a Lipschitz Riemannian structure on that space. After investigating geodesics in such a nonsmoot...
| Autores: | , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2012 |
| País: | España |
| Institución: | Universidad de Cantabria (UC) |
| Repositorio: | UCrea Repositorio Abierto de la Universidad de Cantabria |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.unican.es:10902/3206 |
| Acceso en línea: | http://hdl.handle.net/10902/3206 |
| Access Level: | acceso abierto |
| Palabra clave: | Condition number Lipschitz Riemannian structure Convexity Self-convexity |
| Sumario: | In our previous paper [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1491–1506], we studied the condition metric in the space of maximal rank n × m matrices. Here, we show that this condition metric induces a Lipschitz Riemannian structure on that space. After investigating geodesics in such a nonsmooth structure, we show that the inverse of the smallest singular value of a matrix is a log-convex function along geodesics. We also show that a similar result holds for the solution variety of linear systems. Some of our intermediate results such as those on the second covariant derivative or Hessian of a function with symmetries on a manifold, and those on piecewise self-convex functions, are of independent interest. Those results were motivated by our investigations on the complexity of path-following algorithms for solving polynomial systems. |
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