Persistent homology is arguably the most successful technique in Topological Data Analysis. It combines homology, a topological feature of a data set, with persistence, which tracks the evolution of homology over different scales. The persistent Laplacian is a recent theoretical development that combines persistence with the combinatorial Laplacian, the higher-order extension of the well-known graph Laplacian. Crucially, the Laplacian encode both the homology of a data set, and some additional geometric information not captured by the homology. Here, we provide the first investigation into the efficacy of the persistence Laplacian as an embedding of data for downstream classification and regression tasks. We extend the persistent Laplacian to cubical complexes so it can be used on images, then evaluate its performance as an embedding method on the MNIST and MoleculeNet datasets, demonstrating that it consistently outperforms persistent homology across tasks.

BACKGROUND: Single-cell sequencing (sc-Seq) experiments are producing increasingly large data sets. However, large data sets do not necessarily contain large amounts of information. RESULTS: Here, we formally quantify the information obtained from a sc-Seq experiment and show that it corresponds to an intuitive notion of gene expression heterogeneity. We demonstrate a natural relation between our notion of heterogeneity and that of cell type, decomposing heterogeneity into that component attributable to differential expression between cell types (inter-cluster heterogeneity) and that remaining (intra-cluster heterogeneity). We test our definition of heterogeneity as the objective function of a clustering algorithm, and show that it is a useful descriptor for gene expression patterns associated with different cell types. CONCLUSIONS: Thus, our definition of gene heterogeneity leads to a biologically meaningful notion of cell type, as groups of cells that are statistically equivalent with respect to their patterns of gene expression. Our measure of heterogeneity, and its decomposition into inter- and intra-cluster, is non-parametric, intrinsic, unbiased, and requires no additional assumptions about expression patterns. Based on this theory, we develop an efficient method for the automatic unsupervised clustering of cells from sc-Seq data, and provide an R package implementation.

Kleinberg introduced three natural clustering properties, or axioms, and showed they cannot be simultaneously satisfied by any clustering algorithm. We present a new clustering property, Monotonic Consistency, which avoids the well-known problematic behaviour of Kleinberg’s Consistency axiom, and the impossibility result. Namely, we describe a clustering algorithm, Morse Clustering, inspired by Morse Theory in Differential Topology, which satisfies Kleinberg’s original axioms with Consistency replaced by Monotonic Consistency. Morse clustering uncovers the underlying flow structure on a set or graph and returns a partition into trees representing basins of attraction of critical vertices. We also generalise Kleinberg’s axiomatic approach to sparse graphs, showing an impossibility result for Consistency, and a possibility result for Monotonic Consistency and Morse clustering.