Transport based embeddings with topological guarantees

Abstract

Point clouds arising in image collections, samples from Markov chain Monte Carlo, or states of a random walk, often have a simple underlying geometry which is obscured by noise, high ambient dimension, and the failure of Euclidean distance to reflect similarity. Methods such as UMAP and t-SNE condense such data into usable form, but rely on heuristic choices and provide no guarantee that the output reflects the topology of the input. We introduce a condensation method that comes with such a guarantee. Encoding the data as a positive $m\times n$ stochastic matrix $Q=(q_{ij})$, for instance the transition matrix of a random walk on the point cloud, we define a potential function $\psi(p)=\log \sum_{i} \exp(-KL(p,q_{i\bullet}))$ on the probability simplex $\Delta_n$, where $KL(p,q)$ is the Kullback-Leibler divergence, and prove that $\psi$ is $c$-convex in the sense of Optimal Transport Theory for the cost function $c(p,q)=KL(p,q)$. The associated transport map collapses noisy directions while provably preserving topology: the super-level sets of $\psi$ are homotopy equivalent to those of a $c$-conjugate function, whose image is a condensed, resampleable family of topological spaces which can be interpreted as a continuous analog of an alpha shape. We demonstrate the method by recovering the circle of camera angles from the COIL image dataset, where a standard PCA pipeline produces spurious homology, and the quotient $SO(3)/A_4$ from $45{,}000$ views of a tetrahedron in the SYMSOL pose-estimation benchmark.