## Tuesday, May 3, 2011

### Dual decomposition: a recurrent theme?

Some thoughts on the idea of decomposition; any comments or discussions are welcome.

The concept of dual decomposition for inference in graphical models, particularly MRFs, is based on an insight that resounds in almost all aspects of problem solving in computer science. The insight is the ability to decompose a very hard problem into smaller, independent problems that are solvable, and then combine these solutions in a clever way to get a solution to the original problem. The elegance of this approach is that it can be used to solve inference problems with discrete variables that result in a difficult combinatorial optimization problem.

If we think back to one of our favorite sorting algorithms (at least my favorite), mergesort, the idea of "decomposition" pervades this elegant sorting algorithm. Instead of trying to sort a large array of size $N$, break the problem up into smaller sorting problems (the divide approach), solve each of these subproblems by recursively breaking into smaller and smaller subproblems, and then combining cleverly all these solutions (the conquer approach) to the small problems to get a solution to the original sorting problem. Of course, the dual decomposition idea for inference in graphical models has its own details, but essentially one can think of it as a form of "divide and conquer".

There are many great references that I can suggest if you are interested in dual decomposition ideas for inference in graphical models. It is quite amazing that the idea of "decomposition" has not only given us efficient sorting, but has lent itself in developing very strong inference mechanisms to address difficult combinatorial optimization problems.

Epilogue: I am including a few references that give a broad overview of graphical models and the dual decomposition framework. The article by Wainwright et al. titled "Graphical Models, Exponential Families, and Variational Inference" is worth looking at. I would recommend reading the first few chapters of David Sontag's PhD thesis. Sontag, Globerson, and Jaakkola have a book chapter titled "Introduction to Dual Decomposition for Inference" that might also be of interest. Komodakis et al. have a recent PAMI article titled "MRF Energy Minimization and Beyond via Dual Decomposition" which is also a good place to look. Another recent work worth looking is by Yarkony, Ihler and Fowlkes titled "Planar Cycle Covering Graphs". All these papers have a fair bit of mathematics.

1. Hello,

Very good post. I like your way of comparing DD with mergesort. Have you heard of any public available implementation of DD on MRF application. such as segmentation, denoising or inpainting?

2. Also, please recommend some good references. Mostly what I know about DD is from recent pubs at CVPR/ICCV/ECCV.

Thanks

3. Hi siqi,

Thanks for taking the time to read and comment about the post.

I think graph cuts implementation is available that you can use for segmentation (foreground vs background). Sontag and Globerson have an implementation available of their MPLP algorithm, and I think there are some bio applications available in the toolkit as well.

As for references, I have added a few at the end of the post that I think are useful to gain an understanding.

Thanks.