对称正定流形上函数型数据的几何特征学习
Geometric Feature Learning for Functional Data Valued on the Symmetric Positive Definite Manifold
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中文总结 AI 辅助
本文提出函数型神经网络MatFAE，用于学习SPD流形上的轨迹，通过内在层和函数层编码动态信息，并在fMRI数据上验证了其高效性与可解释性。
中文摘要 AI 辅助
我们在此开发了一种函数型神经网络，称为MatFAE，用于学习对称正定（SPD）矩阵黎曼流形上的轨迹。MatFAE具有内在层，将流形值函数映射为欧几里得向量值函数，随后是一个函数层，将它们投影到有限维欧几里得空间。与大多数用于离散时间序列的神经网络不同，MatFAE将每个序列视为连续函数，因此可以在其潜在表示中编码轨迹动态（例如，一阶导数）。此外，函数层中函数权重的形态通过揭示输入函数型数据中对潜在表示贡献最大的区域，提供了可解释性。我们论证了每个内在层的设计原理和性质，并详细说明了在反向传播过程中如何处理矩阵分解。我们将MatFAE应用于一系列fMRI数据集，展示了其从高维SPD轨迹中高效学习信息丰富表示的能力，以及其在真实世界神经影像分析中的实用价值。
英文摘要
We here develop a functional neural network, termed MatFAE, for learning trajectories on the Riemannian manifold of symmetric positive definite (SPD) matrices. MatFAE features intrinsic layers that map manifold-valued functions to Euclidean vector-valued functions, followed by a functional layer that projects them into a finite-dimensional Euclidean space. Unlike most neural networks for discrete-time sequences, MatFAE treats each sequence as a continuous function and can therefore encode trajectory dynamics (e.g., first-order derivatives) in its latent representations. Additionally, the morphology of the functional weights in the functional layer offers interpretability by revealing the regions of the input functional data that contribute most to the latent representations. We justify the design principles and properties of each intrinsic layer and detail how matrix factorization is handled during backpropagation. We apply MatFAE to a range of fMRI datasets, demonstrating its ability to efficiently learn informative representations from high-dimensional SPD trajectories and its practical value for real-world neuroimaging analysis.
发表机构
- Trinity College Dublin（都柏林圣三一学院）
机构由 AI 辅助整理，请以论文原文为准。
