流形滤波-组合网络
Manifold Filter-Combine Networks
AI总结：
本文提出流形滤波-组合网络（MFCN），以稀疏图逼近底层流形，在高维点云上实现并证明其连续极限一致性，且经真实与合成数据验证有效。
摘要：
为了更好地理解流形神经网络（MNNs），我们引入了流形滤波-组合网络（MFCNs）。我们的滤波-组合框架与图神经网络（GNNs）中流行的聚合-组合范式相对应，并自然地提出了许多有趣的 MNN 族，它们可被解释为各种流行 GNN 的流形对应物。我们提出了一种在高维点云上实现 MFCN 的方法，该方法依赖于用稀疏图逼近底层流形。随后，我们证明了该方法具有一致性：随着数据点数量趋于无穷大，它会收敛到连续统极限；并且我们在真实和合成数据集上通过数值实验展示了其有效性。
英文摘要：
In order to better understand manifold neural networks (MNNs), we introduce Manifold Filter-Combine Networks (MFCNs). Our filter-combine framework parallels the popular aggregate-combine paradigm for graph neural networks (GNNs) and naturally suggests many interesting families of MNNs which can be interpreted as manifold analogues of various popular GNNs. We propose a method for implementing MFCNs on high-dimensional point clouds that relies on approximating an underlying manifold by a sparse graph. We then prove that our method is consistent in the sense that it converges to a continuum limit as the number of data points tends to infinity, and we numerically demonstrate its effectiveness on real-world and synthetic data sets.
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