Abstract
Principal angles between linear subspaces have been studied for their application to statistics, numerical linear algebra, and other areas. In 2005, Iusem and Seeger defined critical angles within a single convex cone as an extension of antipodality in a compact set. Then, in 2016, Seeger and Sossa extended that notion to two cones. This was motivated in part by an application to regression analysis, but also allows their cone theory to encompass linear subspaces which are themselves convex cones. One obstacle to computing the maximal critical angle between cones is that, in general, the maximum will not occur at a pair of generators of the cones. We show that in the special case where the maximal angle between the cones is nonobtuse, it does suffice to check only the generators. This special case can be checked at essentially no extra cost, and we incorporate that information into an improved algorithm to find the maximal angle.
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The author thanks Alberto Seeger for taking the time to answer questions, and for his comments on a draft of this work.
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Communicated by Jinyun Yuan.
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Orlitzky, M. When a maximal angle among cones is nonobtuse. Comp. Appl. Math. 39, 83 (2020). https://doi.org/10.1007/s40314-020-1115-y
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DOI: https://doi.org/10.1007/s40314-020-1115-y
Keywords
- Critical angle
- Maximal angle
- Nash angle
- Convex cone
- Polyhedral cone
- Principal angle
- Nonconvex optimization