In this work we propose to combine the advantages of learning-based and
combinatorial formalisms for 3D shape matching. While learning-based shape
matching solutions lead to state-of-the-art matching performance, they do not
ensure geometric consistency, so that obtained matchings are locally unsmooth.
On the contrary, axiomatic methods allow to take geometric consistency into
account by explicitly constraining the space of valid matchings. However,
existing axiomatic formalisms are impractical since they do not scale to
practically relevant problem sizes, or they require user input for the
initialisation of non-convex optimisation problems. In this work we aim to
close this gap by proposing a novel combinatorial solver that combines a unique
set of favourable properties: our approach is (i) initialisation free, (ii)
massively parallelisable powered by a quasi-Newton method, (iii) provides
optimality gaps, and (iv) delivers decreased runtime and globally optimal
results for many instances.

结合学习和组合形式主义，提出一种新的组合求解器，应用于三维形状匹配，解决了学习方法和公理化方法之间的局部不平滑性和问题规模以及非凸优化问题的计算成本的问题。

快速离散优化用于几何一致的三维形状匹配

Fast Discrete Optimisation for Geometrically Consistent 3D Shape  Matching

We consider the MAP-inference problem for graphical models, which is a valued
constraint satisfaction problem defined on real numbers with a natural
summation operation. We propose a family of relaxations (different from the
famous Sherali-Adams hierarchy), which naturally define lower bounds for its
optimum. This family always contains a tight relaxation and we give an
algorithm able to find it and therefore, solve the initial non-relaxed NP-hard
problem.
The relaxations we consider decompose the original problem into two
non-overlapping parts: an easy LP-tight part and a difficult one. For the
latter part a combinatorial solver must be used. As we show in our experiments,
in a number of applications the second, difficult part constitutes only a small
fraction of the whole problem. This property allows to significantly reduce the
computational time of the combinatorial solver and therefore solve problems
which were out of reach before.

针对基于图模型的最大后验概率推断问题，本研究提出了一系列不同于 Sherali-Adams 层次结构的松弛方法，将问题分解为一个简单的 LP-tight 部分和一个困难的组合求解器部分，实验证明对小部分问题可以显著降低计算时间。