# K-Nearest-Neighbors Induced Topological PCA for scRNA Sequence Data Analysis

arXiv: https://arxiv.org/abs/2310.14521 · HTML: https://mlsanity.com/paper/2310.14521/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 47 references that are in the map.
- First posted on arXiv 23 Oct 2023.

## Builds on (most cited first, 4 of 4)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Persistent Laplacians: properties, algorithms and implications](https://mlsanity.com/paper/2012.02808/) (2020)
- [CCP: Correlated Clustering and Projection for Dimensionality Reduction](https://mlsanity.com/paper/2206.04189/) (2022)
- [Non-Greedy L21-Norm Maximization for Principal Component Analysis](https://mlsanity.com/paper/1603.08293/) (2016)

## Built on by (most cited first, 2 of 2)

- [Multiscale Grassmann Manifolds for Single-Cell Data Analysis](https://mlsanity.com/paper/2511.11717/) (2025)
- [A Hierarchical Sheaf Spectral Embedding Framework for Single-Cell RNA-seq Analysis](https://mlsanity.com/paper/2603.26858/) (2026)

## Abstract (arXiv)

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 L2,1 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.