Reproduce “Diffeomorphic interpolation for efficient persistence-based topological optimization” on OpenResearch

About this paper

Mathieu Carriere, Marc Theveneau, Théo Lacombe

A method introduces diffeomorphic interpolation to transform sparse topological gradients into dense vector fields, making persistence-based topological optimization efficient and scalable for large datasets. This approach enables the regularization of latent spaces in black-box autoencoders, leading to more interpretable representations and topologically-consistent data generation.

Key insights

• The proposed diffeomorphic updates guarantee a decrease in topological loss functions and their smoothness is quantified by an upper-bounded Lipschitz constant.
• Diffeomorphic interpolation, even when computed on a small subsample, effectively updates the entire point cloud, enabling optimization on datasets previously considered intractable.
• Learning an invertible diffeomorphic flow for black-box autoencoders facilitates post-hoc topological regularization, enhancing latent space interpretability and allowing for generation of new data with desired topological properties.

Summary by alphaXiv.