Abstract

The Bregman distance Bξx(y,x)B_{\xi_x}(y,x), ξx∈∂J(y),\xi_x \in \partial J(y), associated to a convex sub-differentiable functional JJ is known to be in general non-symmetric in its arguments xx, yy. In this note we address the question when Bregman distances can be bounded against each other when the arguments are switched, i.e., if some constant C>0C>0 exists such that for all x,yx,y on a convex set MM it holds that 1CBξx(y,x)≤Bξy(x,y)≤CBξx(y,x).\frac{1}{C} B_{\xi_x}(y,x) \leq B_{\xi_y}(x,y) \leq C B_{\xi_x}(y,x). We state sufficient conditions for such an inequality and prove in particular that it holds for the pp-powers of the ℓp\ell_p and LpL^p-norms when 1<p<∞1 < p <\infty.

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Bregman distance
  • symmetry
  • strong monotonicity
  • convexity

Mathematics Subject Classification

52A41, 47H05, 26B25

References

  1. I. Ekeland, R. Témam: Convex Analysis and Variational Problems, SIAM, Philadelphia, (1999).https://doi.org/10.1137/1.9781611971088Google Scholar
  2. J. Flemming: A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations, J. Inverse Ill-Posed Problems 26(5) (2018) 639–646.https://doi.org/10.1515/jiip-2017-0116Google Scholar
  3. B. Hofmann, B. Kaltenbacher, C. Pöschl, O. Scherzer: A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems 23(3) (2007) 987–1010.https://doi.org/10.1088/0266-5611/23/3/009Google Scholar
  4. S. Kindermann: Convex Tikhonov regularization in Banach spaces: new results on convergence rates, J. Inverse Ill-Posed Problems 24(3) (2016) 341–350.https://doi.org/10.1515/jiip-2015-0038Google Scholar
  5. E. Resmerita, O. Scherzer: Error estimates for non-quadratic regularization and the relation to enhancement, Inverse Problems 22(3) (2006) 801–814.https://doi.org/10.1088/0266-5611/22/3/004Google Scholar
  6. Z. B. Xu, G. F. Roach: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces, J. Math. Anal. Appl. 157(1) (1991) 189–210.https://doi.org/10.1016/0022-247X(91)90144-OGoogle Scholar

Suggested citation

S. Kindermann. “A Note on the Approximate Symmetry of Bregman Distances.” Journal of Convex Analysis 26 (2019), No. 3, 991–999. https://doi.org/10.68381/jca26053

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.