用地图式结构在流形上直接做机器学习，提升可解释性和稳定性。
Atlas-based Manifold Representations for Interpretable Riemannian Machine Learning
- 构建可微分的地图系统，实现流形上的优化计算
- 在克莱因瓶分类和造血数据中准确率更高、更稳定
- 适合需要理解数据几何结构的研究者
尽管流形假设广受关注，现有流形学习方法通常将数据降维至ℝ^D，当嵌入维数D趋近于流形真实维数d时会丢失关键几何特征。相比之下，直接学习潜在流形作为可微分地图的方法尚未充分探索。本文提出一种通用数据结构，维护可微分地图，支持在流形上进行黎曼优化。同时设计无监督启发式算法，从点云数据中学习可微分地图。实验表明该方法在特定场景下具有更高的效率与精度。在克莱因瓶上的监督分类任务以及造血数据的RNA速度分析中，展示了更好的可解释性与鲁棒性。
原文摘要 · Abstract (English)
Despite the popularity of the manifold hypothesis, current manifold-learning methods do not support machine learning directly on the latent $d$-dimensional data manifold, as they primarily aim to perform dimensionality reduction into $\mathbb{R}^D$, losing key manifold features when the embedding dimension $D$ approaches $d$. On the other hand, methods that directly learn the latent manifold as a differentiable atlas have been relatively underexplored. In this paper, we aim to give a proof of concept of the effectiveness and potential of atlas-based methods. To this end, we implement a generic data structure to maintain a differentiable atlas that enables Riemannian optimization over the manifold. We complement this with an unsupervised heuristic that learns a differentiable atlas from point cloud data. We experimentally demonstrate that this approach has advantages in terms of efficiency and accuracy in selected settings. Moreover, in a supervised classification task over the Klein bottle and in RNA velocity analysis of hematopoietic data, we showcase the improved interpretability and robustness of our approach.
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