提出几何自适应的非负流形因子分解方法，用于解析具有非线性结构的数据。
Curvature Corrected Nonnegative Manifold Data Factorization
- 基于流形曲率修正的非负因子分解，融合黎曼几何特性
- 在扩散张量成像数据上验证，有效提取可解释成分
- 适合处理带非线性结构的科学数据，如医学影像
具有潜在非线性结构的数据广泛存在于多个应用领域，亟需适配非线性数据结构的新分析方法。黎曼流形为这类工具的构建提供了丰富环境，因为流形值数据在多种科学场景中出现，且黎曼几何为几何数据分析提供了坚实的理论基础。低秩近似（如非负矩阵分解，NMF）是许多欧氏空间数据分析方法的基础，因此将此类分解推广至流形值数据，对进一步发展流形数据分析至关重要。本文提出曲率修正的非负流形数据因子分解（CC-NMDF），一种面向流形值数据、可提取可解释因子的几何感知方法，类似于非负矩阵分解。我们设计了一种高效的迭代算法以计算CC-NMDF，并在真实世界的扩散张量磁共振成像（DT-MRI）数据上进行了验证。
原文摘要 · Abstract (English)
Data with underlying nonlinear structure are collected across numerous application domains, necessitating new data processing and analysis methods adapted to nonlinear domain structure. Riemannanian manifolds present a rich environment in which to develop such tools, as manifold-valued data arise in a variety of scientific settings, and Riemannian geometry provides a solid theoretical grounding for geometric data analysis. Low-rank approximations, such as nonnegative matrix factorization (NMF), are the foundation of many Euclidean data analysis methods, so adaptations of these factorizations for manifold-valued data are important building blocks for further development of manifold data analysis. In this work, we propose curvature corrected nonnegative manifold data factorization (CC-NMDF) as a geometry-aware method for extracting interpretable factors from manifold-valued data, analogous to nonnegative matrix factorization. We develop an efficient iterative algorithm for computing CC-NMDF and demonstrate our method on real-world diffusion tensor magnetic resonance imaging data.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。