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Discrete Optimization in Computer Vision: Theory and Practice

Final Report Summary - DOICV (Discrete Optimization in Computer Vision: Theory and Practice)

Below I highlight outcomes of the DOiCV project which I view as the most significant.

1. THEORETICAL DEVELOPMENTS.
A substantial progress has been made on the theory of discrete optimization:

- [Kolmogorov,Krokhin,Rolinek FOCS'15; SICOMP’17] settled the complexity of general-valued CSPs modulo the complexity of ordinary CSPs. When combined with other results (in particular by Bulatov and by Zhuk at FOCS’17), it yields a complete complexity classification of general-valued CSPs, and closes one line of research. Note that settling the complexity of ordinary CSPs (completed by Bulatov and by Zhuk) has been seen a major development, as evidenced by best paper awards at FOCS’17 and by the Presburger award to D. Zhuk.

- In [Kolmogorov,Thapper,Zivny SICOMP'15] we provided a complete characterization of languages that can be solved exactly by a natural Linear Programming (LP) relaxation of the problem.

- In [Kazda,Kolmogorov,Rolinek SODA'17; TALG’18] we developed a polynomial time algorithm for "edge CSPs" with a certain class of Delta-matroid constraints. It extends Edmonds's blossom-shrinking algorithm for computing perfect matchings in a graph. We can handle all previously known tractable classes of Delta-matroids, as well as new classes (such as even Delta-matroids). One implication of our result is resolving the complexity classification of Boolean CSPs restricted to planar instances.

2. PRACTICAL DEVELOPMENTS.
Another goal of the project was to design and implement efficient optimization algorithms for real-world applications. Selected achievements on this objective are as follows:

- [Kolmogorov PAMI'15] developed an efficient message passing algorithm for arbitrary graphical models called "SRMP". For some classes of problems it showed a state-of-the-art performance.

- In a series of papers we proposed a new very practical technique for solving relaxations of discrete optimization problems, which I expect to become a standard approach:
(a) in [Swoboda,Kolmogorov CVPR’20] we developed a method that uses the Frank-Wolfe technique inside an (inexact) proximal point algorithm;
(b) in [Kolmogorov,Pock arXiv’21] we studied its accelerated version and established converge rates;
(c) in [Kolmogorov arXiv’20] I showed how the Frank-Wolfe subproblem can be solved more efficiently by a recursive technique.

- In [Mohapatra et al. CVPR’18, best paper honorable mention award] we developed algorithms (together with matching lower bounds in a certain model) for optimizing loss functions that are commonly used in information retrieval systems.

- [Kolmogorov, Pock, Rolinek SIIMS’16] studied the problem of minimizing Total Variation on tree-structured graphs. In particular, it improved the worst-case complexity for an important special case.

- [Shah,Kolmogorov,Lampert CVPR'15] improved the speed of training Structural Support Vector Machines (SSVMs) on some real applications by several orders of magnitude over previous state-of-the-art.

3. OTHER DEVELOPMENTS
Three side projects were not directly related to DOiCV:

- In [Kolmogorov FOCS'16; SICOMP’18], [Achlioptas et al. SICOMP’19], [Harris, Iliopoulos, Kolmogorov arXiv’20] we provided a deeper understanding of the algorithmic version of the Lovasz Local Lemma (LLL). LLL can be loosely connected to the topic of DOiCV project, since it can be applied to certain CSPs. However, the involved techniques (the probabilistic method) are entirely different from those envisaged in the proposal.

- In [Kolmogorov COLT’18], [Harris, Kolmogorov arXiv’20] we studied the problem of estimating parameters of the Gibbs distribution given an access to a sampling oracle. This does not directly fit the topic of the project, as there is no optimization component. However, there is an indirect connection: the proposal considers MAP estimation in graphical models, while estimating the partition function of a graphical model is the probabilistic counterpart of the MAP problem.