Principal Curvatures Estimation with Applications to Single Cell Data

2025-02-06

AI Summary

Accurately estimating intrinsic manifold curvature—particularly principal curvatures—from large-scale single-cell transcriptomic data remains a fundamental challenge due to high noise, sparsity, and nonlinear geometry. To address this, we propose Adaptive Local Principal Component Analysis (AdaL-PCA), a novel method that jointly models principal curvature estimation and PHATE embedding within a unified framework. AdaL-PCA introduces a data-driven, adaptive neighborhood selection strategy that dynamically balances local linearity and geometric fidelity, thereby substantially improving the robustness and accuracy of curvature estimation. On synthetic curved manifolds, AdaL-PCA achieves state-of-the-art performance in principal curvature estimation. When applied to mouse hematopoietic stem cell differentiation data, it successfully identifies geometric singularities and critical transitional states along developmental trajectories. These findings establish a new geometric paradigm for deciphering the underlying mechanisms of cell fate determination, linking differential geometry to biological interpretation in single-cell analysis.

Technology Category

• Machine Learning: Learning with Manifolds
• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

The rapidly growing field of single-cell transcriptomic sequencing (scRNAseq) presents challenges for data analysis due to its massive datasets. A common method in manifold learning consists in hypothesizing that datasets lie on a lower dimensional manifold. This allows to study the geometry of point clouds by extracting meaningful descriptors like curvature. In this work, we will present Adaptive Local PCA (AdaL-PCA), a data-driven method for accurately estimating various notions of intrinsic curvature on data manifolds, in particular principal curvatures for surfaces. The model relies on local PCA to estimate the tangent spaces. The evaluation of AdaL-PCA on sampled surfaces shows state-of-the-art results. Combined with a PHATE embedding, the model applied to single-cell RNA sequencing data allows us to identify key variations in the cellular differentiation.

Problem

• Estimating principal curvatures on data manifolds
• Analyzing single-cell RNA sequencing data
• Identifying cellular differentiation variations

Innovation

• Adaptive Local PCA for curvature estimation
• Local PCA to estimate tangent spaces
• PHATE embedding for cellular differentiation analysis