Duality for Bregman projections onto translated cones and affine subspaces

Bauschke, Heinz H. (2002) Duality for Bregman projections onto translated cones and affine subspaces. [Preprint]

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      In 2001, Della Pietra, Della Pietra, and Lafferty suggested a dual characterization of he Bregman projection onto linear constraints, which has already been applied by Collins, Schapire, and Singer to boosting algorithms and maximum likelihood logistic regression. The proof provided by Della Pietra \emph{et al.} is fairly complicated, and their statement features a curious nonconvex component. In this note, the Della Pietra \emph{et al.} characterization is proved differently, using the powerful framework of convex analysis. Assuming a standard constraint qualification, the proof presented here is not only much shorter and cleaner, but it also reveals the strange nonconvex component as a reformulation of a \emph{convex} (dual) optimization problem. Furthermore, the setting is extended from an affine subspace to a translated cone, and the convex function inducing the Bregman distance is only required to be Legendre. Various remarks are made on limitations and possible extensions, and an illustrative example in the Euclidean plane is provided.

      Item Type: Preprint
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: affine subspace, Bregman distance, Bregman projection, convex cone, convex duality, Legendre function, orthogonal complement
      Subjects: 90-xx Economics, operations research, programming, games > 90Cxx Mathematical programming
      41-xx Approximations and expansions > 41Axx Approximations and expansions
      52-xx Convex and discrete geometry > 52Axx General convexity
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 28 Oct 2003
      Last Modified: 21 Apr 2010 11:13

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