Maximum entropy and feasibility methods for convex and nonconvex inverse problems

Abstract

We discuss informally two approaches to solving convex and nonconvex feasibility problems – via entropy optimization and via algebraic iterative methods. We shall highlight the advantages and disadvantages of each and give various related applications and limiting examples. While some of the results are very classical, they are not as well-known to practitioners as they should be. A key role is played by the Fenchel conjugate.

Keywords:

• entropy optimization
• Fenchel duality theorem
• inverse problems
• feasibility problems
• alternating projection and reflection methods

AMS Subject Classifications::

• Primary: 45Q05
• Secondary: 49N45, 90C25, 49N15, 90C90

Acknowledgements

Thanks are due to many but especially to Adrian Lewis, Heinz Bauschke, Russell Luke and Brailey Sims who have been close collaborators on matters relating to this study over many years. Thanks are also due to Francisco Aragon for a careful reading of this manuscript. This research was partially supported by ARC Grant #DP1093769.

Notes

Notes

1. A companion lecture is at http://www.carma.newcastle.edu.au/~jb616/inverse.pdf.

2. More is an unrealistic task; all details may be found in the references!

3. In MacHale 50.

4. This possibly apocryphal anecdote is taken from The American Heritage Book of English Usage, p. 158.

5. Given noise, modelling, measurement and numerical errors, the problem may well not be feasible in practice.

6. This is ensured by the condition that φ″(t) > 0.

7. Letter from Saul Gass. Dantzig's reminiscence is quoted from 31.

8. The solution is actually the absolutely continuous part of a measure in C(Ω)* 13.

9. Fenchel like other early researchers in nonlinear duality theory missed the need for a (CQ) in his 1951 Princeton Notes.

10. This is trivial if d = ∞.

11. Here n = 2 for images, 3 for holographic imaging, etc.

12. Observation of the modulus of the diffracted image in crystallography. Similarly, for optical aberration correction.

13. My former PDF, he was a NASA Graduate student.