Kernel Functional Maps

• Published in: Computer graphics forum Vol. 37; no. 5; pp. 27 - 36
• Main Authors: Wang, L., Gehre, A., Bronstein, M. M., Solomon, J.
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.08.2018
• Subjects: 1.3.5 [Computer Graphics]: Computer graphics—Computational Geometry and Object Modeling; Algorithms; Categories and Subject Descriptors (according to ACM CCS); Kernel functions; Mathematical analysis; Operators (mathematics); Optimization; Regularization; Vectors (mathematics)
• ISSN: 0167-7055; 1467-8659
• DOI: 10.1111/cgf.13488

Functional maps provide a means of extracting correspondences between surfaces using linear‐algebraic machinery. While the functional framework suggests efficient algorithms for map computation, the basic technique does not incorporate the intuition that pointwise modifications of a descriptor function (e.g. composition of a descriptor and a nonlinearity) should be preserved under the mapping; the end result is that the basic functional maps problem can be underdetermined without regularization or additional assumptions on the map. In this paper, we show how this problem can be addressed through kernelization, in which descriptors are lifted to higher‐dimensional vectors or even infinite‐length sequences of values. The key observation is that optimization problems for functional maps only depend on inner products between descriptors rather than descriptor values themselves. These inner products can be evaluated efficiently through use of kernel functions. In addition to deriving a kernelized version of functional maps including a recent extension in terms of pointwise multiplication operators, we provide an efficient conjugate gradient algorithm for optimizing our generalized problem as well as a strategy for low‐rank estimation of kernel matrices through the Nyström approximation.