Hodge Theory on Metric Spaces
This work extends Hodge theory to a broader class of spaces, potentially impacting the mathematical foundations of vision and pattern recognition, though the results are theoretical and no concrete applications or numbers are provided.
The authors develop a version of Hodge theory for metric spaces with a probability measure, motivated by applications in computer vision and pattern recognition. The appendix provides an example of a separable, compact metric space with infinite-dimensional α-scale homology.
Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We believe that this constitutes a step towards understanding the geometry of vision. The appendix by Anthony Baker provides a separable, compact metric space with infinite dimensional α-scale homology.