将神经网络拓展到相关矩阵流形，提升对称正定矩阵的表示能力。
Riemannian Networks over Full-Rank Correlation Matrices
- 基于五种新相关几何构造流形神经网络
- 实现两类相关几何下的精确反向传播
- 在多种任务中优于现有对称正定与格拉斯曼网络
对称正定（SPD）流形上的表示在多个应用中受到广泛关注。相比之下，全秩相关矩阵这一对称正定矩阵的归一化替代流形却尚未得到充分研究。本文提出在相关矩阵流形上构建黎曼神经网络，利用最近提出的五种相关几何结构。系统性地将基础网络层（如多项式逻辑回归、全连接层、卷积层）扩展至这些几何空间，并为其中两种相关几何提出了精确的反向传播方法。实验表明，该方法在与现有SPD和格拉斯曼网络的对比中表现更优。
原文摘要 · Abstract (English)
Representations on the Symmetric Positive Definite (SPD) manifold have garnered significant attention across different applications. In contrast, the manifold of full-rank correlation matrices, a normalized alternative to SPD matrices, remains largely underexplored. This paper introduces Riemannian networks over the correlation manifold, leveraging five recently developed correlation geometries. We systematically extend basic layers, including Multinomial Logistic Regression (MLR), Fully Connected (FC), and convolutional layers, to these geometries. Besides, we present methods for accurate backpropagation for two correlation geometries. Experiments comparing our approach against existing SPD and Grassmannian networks demonstrate its effectiveness.
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。