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Single-cell omics enable the profiles of cells, which contain large numbersof biological features, to be quantified. Cluster analysis, a dimensionalityreduction process, is used to reduce the dimensions of the data to make itcomputationally tractable. In these analyses, cells are represented as vectorsin $n$-Dimensional space, where each dimension corresponds to a certain cellfeature. The distance between cells is used as a surrogate measure ofsimilarity, providing insight into the cell's state, function, and geneticmechanisms. However, as cell profiles are clustered in 3D or higher-dimensionalspace, it remains unknown which distance metric provides the most accuratespatiotemporal representation of similarity, limiting the interpretability ofthe data. I propose and prove a generalized proposition and set of corollariesthat serve as a criterion to determine which of the standard distance measuresis most accurate for conveying cell profile heterogeneity. Each distance methodis evaluated via statistical, geometric, and topological proofs, which areformalized into a set of criteria. In this paper, I present the putative,first-ever method to elect the most accurate and precise distance metrics withany profiling modality, which are determined to be the Wasserstein distance andcosine similarity metrics, respectively, in general cases. I also identifyspecial cases in which the criterion may select non-standard metrics. Combiningthe metric properties selected by the criterion, I develop a novel, custom,optimal distance metric that demonstrates superior computational efficiency,peak annotation, motif identification, and footprinting for transcriptionfactor binding sites when compared with leading methods.

On Selecting Distance Metrics in $n$-Dimensional Normed Vector Spaces of  Cells: A Novel Criterion and Similarity Measure Towards Efficient and  Accurate Omics Analysis