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.

提出了一种拓扑主成分分析 (tPCA) 方法，通过持久 Laplacian 技术和 L$_{2,1}$ 范数正则化的组合来解决多尺度和多类别异质性问题，进一步引入了 k - 最近邻持久 Laplacian 技术来提高鲁棒性。在 11 个多样的 scRNA-seq 数据集上验证了 tPCA 和 kNN-tPCA 方法的有效性，并展示了这些方法在无监督 PCA 增强、UMAP、tSNE 和 NMF 等方面的显著优势。

K 最近邻诱导的拓扑 PCA 用于 scRNA 序列数据分析

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

The success of many machine learning (ML) methods depends crucially on having
large amounts of labeled data. However, obtaining enough labeled data can be
expensive, time-consuming, and subject to ethical constraints for many
applications. One approach that has shown tremendous value in addressing this
challenge is semi-supervised learning (SSL); this technique utilizes both
labeled and unlabeled data during training, often with much less labeled data
than unlabeled data, which is often relatively easy and inexpensive to obtain.
In fact, SSL methods are particularly useful in applications where the cost of
labeling data is especially expensive, such as medical analysis, natural
language processing (NLP), or speech recognition. A subset of SSL methods that
have achieved great success in various domains involves algorithms that
integrate graph-based techniques. These procedures are popular due to the vast
amount of information provided by the graphical framework and the versatility
of their applications. In this work, we propose an algebraic topology-based
semi-supervised method called persistent Laplacian-enhanced graph MBO (PL-MBO)
by integrating persistent spectral graph theory with the classical
Merriman-Bence- Osher (MBO) scheme. Specifically, we use a filtration procedure
to generate a sequence of chain complexes and associated families of simplicial
complexes, from which we construct a family of persistent Laplacians. Overall,
it is a very efficient procedure that requires much less labeled data to
perform well compared to many ML techniques, and it can be adapted for both
small and large datasets. We evaluate the performance of the proposed method on
data classification, and the results indicate that the proposed technique
outperforms other existing semi-supervised algorithms.

本文提出一种称为 PL-MBO 的代数拓扑半监督学习方法，它将持久谱图理论与经典的 MBO 方法相结合，通过一系列链复杂体和关联的简单复杂体构建一族持久拉普拉斯变换，从而实现在标记数据量较少的情况下实现良好的性能。