(Probably) Concave Graph Matching
In this paper, we address the graph matching problem. Following the recent works of Zaslavskiy et al. (2009);Vestner et al. (2017) we analyze and generalize the idea of concave relaxations. We introduce the concepts of conditionally concave and probably conditionally concave energies on polytopes and show that they encapsulate many instances of the graph matching problem, including matching Euclidean graphs and graphs on surfaces. We further prove that local minima of probably conditionally concave energies on general matching polytopes (e.g., doubly stochastic) are with high probability extreme points of the matching polytope (e.g., permutations).
Update: I have considered the author response. Some of my previous questions have been answered. I have updated my review and "overall score".  ==========================================================  This paper focuses on graph matching problems with quadratic objective functions. These objectives are usually equivalent to a quadratic assignment problem (QAP) [1] on the permutation matrix space. The authors focused on the Frank-Wolfe algorithm (Algorithm 1) on the objectives with the relaxation from the permutation matrix space to the doubly stochastic matrix space DS. To analyze the performances of this algorithm, the authors defined conditionally concave (or negative definite) and used these definitions to do some theoretical analysis on the local optima convergence of this algorithm. Then the authors extended this definition to a probabilistic setting and did some analysis for a more general form of graph matching problems. At the end, the authors provided another algorithm (Algorithm 2) for the one sided permutation problems where the probably conditionally concavity does not hold.  Graph matching problems are very popular now in the machine learning community. In my opinion, the authors proposed some very interesting and novel definitions, and these definitions may be very useful for doing relaxations for graph matching problems. However, it will be better if more analysis on the optimality of the convergence local optima can be shown instead of just showing some sufficient conditions for the algorithm to guarantee (or with high probability) to converge to extreme point local optima.   Strengths of this paper:  1. Proposed the definition for (probably) conditionally concave and (probably) conditionally negative definite. These definitions can clearly capture the possibility for the Frank-Wolfe algorithm to locally optimize the objective function.  2. The theoretical analysis on these new concavity definitions provide us important insight on the convergence performances of Algorithm 1, under certain circumstances.  Weakness:  1. As I just mentioned, the paper only analyzed, under which cases will the Algorithm 1 converges to permutations as local minima. However, it will be better if the quality of this kind of local minima could be analyzed (e.g. the approximation ratio of these local minima, under certain assumptions).  2. This paper is not very easy to follow. First, many definitions are used before they are defined. For example, on line 59 in Theorem 1, the authors used the definition of  "conditionally positive definite function of order 1", which is defined on line 126 in Definition 2. Also, the author used the definition "\epsilon-negative definite" on line 162, which is defined on line 177 in Definition 3. It will be better if the authors could define those important concepts before using them. Second, the introduction is a little bit too long (more than 2.5 pages) and many parts of that are repeated in Section 2 and 3. It might be better to restructure the first 3 sections for a little bit. Third, it will be good if more captions could be added to the figures in the experiment section so that the readers could understand the results more easily.  3. For the description of Theorem 3, from the proof it seems that we need to use Equation 12 as a condition. It will be better if this information is included in the theorem description to avoid confusions (though I know this is mentioned right before the theorem, it is still better to have it in the theorem description). Also, for the constant c_1, it is better to give it an explicit formula in the theorem.   Reference:  [1] Burkard, R. E., Cela, E., Pardalos, P. M., and Pitsoulis, L. S. (1998). The quadratic assignment problem. In Handbook of combinatorial optimization, pages 1713–1809. Springer.