Manifold Filter-Combine Networks

TL;DR

This work introduces Manifold Filter-Combine Networks (MFCNs), a flexible, spectral-filter based framework for learning on data that lie on or near unknown manifolds. By approximating the manifold with a sparse graph on point clouds, MFCNs implement a filter–combine pipeline that parallels GNNs while operating in the manifold spectral domain. The authors establish convergence guarantees for spectral filters and full MFCNs to continuum limits as the sample size grows, with rates that depend on the ambient manifold dimension and data density, and show that depth can scale linearly under proper weight normalization. They also propose Infogain, a data-driven method for selecting diffusion scales per channel, and validate the approach through convergence tests and tasks including ellipsoid regression and high-dimensional melanoma data classification, where wavelet-based MFCNs often outperform traditional baselines. Overall, the paper provides a principled, scalable framework for manifold deep learning with theoretical guarantees and practical algorithms for point-cloud data.

Abstract

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.

Figures (10)

• Figure 1: Illustration of Manifold Filter-Combine Networks steps. Starting from a $C$-dimensional row vector-valued function evaluated at $N$ points, the MFCN layer in turn filters, combines features, combines filters, applies a point-wise nonlinearity, and reshapes. (For conciseness, we do not visualize the activation step).
• Figure 2: Discretization error for spectral filter $w(\lambda)=e^{-\lambda}$ applied to the sum of two spherical harmonics, for a $k$-NN graph construction. The median error of 10 runs is shown in blue, against a gray band of the 25th- to 75th-percentile error range.
• Figure 3: Convergence of first eight, non-zero eigenvalues on the sphere, for a $k$-NN graph construction, all 10 runs combined. The blue lines are the first three eigenvalues that converge to the same limit (since the first non-zero eigenvalue of the spherical Laplacian has multiplicity three). Similarly, the green lines are the next five eigenvalues.
• Figure 4: High-level illustration of MFCNs as applied to point-cloud data. An MFCN may have multiple cycles of steps (i) to (v) (boxed). Note that step (i) is typically combinatorially expansive (i.e. transforms the raw feature matrix $\mathbf{X} \in \mathbb{R}^{N \times C} \rightarrow \mathbb{R}^{N \times C \times J}$, where $N$ is the total number of nodes across all batched graphs, $C$ is the number of channels/features, and $J$ is the number of filters). However, steps (ii) and (iii) may be expansive or reductional (as drawn), depending on the whether the number of input features is less than or greater than the number of output features specified as hyperparameters. After filter-combine cycles, final pooling layers can be applied in node-wise and/or feature-wise fashion (e.g. as linear, scattering moments, max, mean, or 'top-k' pooling layers, etc.). Finally, the resulting features may be fed into a regressor or classifier head, depending on the learning task.
• Figure 5: Illustration of a random bandlimited function constructed from the first 20 nontrivial eigenvectors of an ellipsoid (as estimated from the Laplacian of a $k$-NN graph) (left), and evaluated pointwise in a uniform sample of 1024 points (right).

Theorems & Definitions (16)

• Definition 1: $\epsilon$-graphs
• Definition 2: $k$-NN graphs
• Definition 3: Bandlimited functions and bandlimited spectral filters
• Definition 4: Lipschitz Constant
• Theorem 1: Theorems 2.4, 2.5, 2.7, and 2.9 of Calder2019
• Lemma 1
• Theorem 2
• Remark 1
• Theorem 3
• proof