Joint M-Best-Diverse Labelings as a Parametric Submodular Minimization
We consider the problem of jointly inferring the M -best diverse labelings for a binary (high-order) submodular energy of a graphical model. Recently, it was shown that this problem can be solved to a global optimum, for many practically interesting diversity measures. It was noted that the labelings are, so-called, nested. This nestedness property also holds for labelings of a class of parametric submodular minimization problems, where different values of the global parameter γ give rise to different solutions. The popular example of the parametric submodular minimization is the monotonic parametric max-flow problem, which is also widely used for computing multiple labelings. As the main contribution of this work we establish a close relationship between diversity with submodular energies and the parametric submodular minimization. In particular, the joint M -best diverse labelings can be obtained by running a non-parametric submodular minimization (in the special case -max-flow) solver for M different values of γ in parallel, for certain diversity measures. Importantly, the values for γ can be computed in a closed form in advance, prior to any optimization. These theoretical results suggest two simple yet efficient algorithms for the joint M -best diverse problem, which outperform competitors in terms of runtime and quality of results. In particular, as we show in the paper, the new methods compute the exact M -best diverse labelings faster than a popular method of Batra et al., which in some sense only obtains approximate solutions.
The authors describe an improved sequential and a parallel algorithm for the case joint diverse best M solutions problem for binary submodular energy functions and nodewise diversity functions that are permutation invariant and concave.
This work builds primarily on the work of Kirillov but feels a bit incremental in some ways as the main theoretical result is very reminiscent of previous work.  The fact that it leads to a better algorthimic approach is important, I think that the comparison with previous work could be stated more clearly in the draft.  I would have liked to see an actual complexity analysis of the algorithms.  That is, the authors claim that it leads to an improvement that scales with M, but that doesn't seem to be presented theoretically.  What is the computational complexity of the algorithms involved?  Does this improve the theoretical asymptotic complexity, or is it just a constant time improvement?  In the worst case I think that they may be the samThis kind of analysis should be possible to state for the simple pairwise binary models discussed in this work.  The same goes for the parallel algorithm.    The draft is difficult to read because of a number of typos.  This can certainly be corrected with some effort.  Also, it appears that the main theoretical contribution is imprecisely stated.  This should be corrected as it makes it difficult to read/check the proofs.  Miscellaneous typos/comments:  -  line 61, "and proposes a solver to it" -> "and proposes a solver for it" -  line 66, "e.g. Hamming distance" -> "e.g., Hamming distance" -  line 104, "i.e. each variable may take only" -> "i.e., each variable may take only", this typo occurs in many places throughout the draft. -  line 135-136, "In this paragraph we briefly summarize existing efficient minimization approaches for (2)" ->  this is the end of the paragraph so it doesn't make any sense. -  line 142, "this algorithm requires to sequentially solve M" ->  "this algorithm requires sequentially solving M" -  line 161, "which plays a crucial role for obtaining the main results of this work" -> "which plays a crucial role in obtaining the main results of this work" -  line 176, I'm not sure that the definition of \Gamma here makes sense given that \Gamma should be a subset of R^{|V|}.  In fact, I think that there is a problem with the definition of the parametric problem in line 165 as "i" does not appear anywhere in the definition. -  line 204, "monotonous" -> "monotone"