K-Nearest-Neighbors Induced Topological PCA for scRNA Sequence Data
  Analysis
Sean Cottrell Yuta Hozumi Guo-Wei Wei
Single-cell RNA sequencing (scRNA-seq) is widely used to reveal heterogeneity
in cells, which has given us insights into cell-cell communication, cell
differentiation, and differential gene expression. However, analyzing scRNA-seq
data is a challenge due to sparsity and the large number of genes involved.
Therefore, dimensionality reduction and feature selection are important for
removing spurious signals and enhancing downstream analysis. Traditional PCA, a
main workhorse in dimensionality reduction, lacks the ability to capture
geometrical structure information embedded in the data, and previous graph
Laplacian regularizations are limited by the analysis of only a single scale.
We propose a topological Principal Components Analysis (tPCA) method by the
combination of persistent Laplacian (PL) technique and L$_{2,1}$ norm
regularization to address multiscale and multiclass heterogeneity issues in
data. We further introduce a k-Nearest-Neighbor (kNN) persistent Laplacian
technique to improve the robustness of our persistent Laplacian method. The
proposed kNN-PL is a new algebraic topology technique which addresses the many
limitations of the traditional persistent homology. Rather than inducing
filtration via the varying of a distance threshold, we introduced kNN-tPCA,
where filtrations are achieved by varying the number of neighbors in a kNN
network at each step, and find that this framework has significant implications
for hyper-parameter tuning. We validate the efficacy of our proposed tPCA and
kNN-tPCA methods on 11 diverse benchmark scRNA-seq datasets, and showcase that
our methods outperform other unsupervised PCA enhancements from the literature,
as well as popular Uniform Manifold Approximation (UMAP), t-Distributed
Stochastic Neighbor Embedding (tSNE), and Projection Non-Negative Matrix
Factorization (NMF) by significant margins.