[Paper Review] Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds

Michael Perlmutter, Feng Gao+2 more|PubMed|May 24, 2019

3D Shape Modeling and Analysis57 references4 citations

TL;DR

This paper introduces a geometric wavelet scattering network for compact Riemannian manifolds, generalizing the Euclidean scattering transform using spectral filtering via the Laplace-Beltrami operator. It achieves isometry invariance and diffeomorphism stability, demonstrating state-of-the-art performance on spherical MNIST (95% accuracy) and FAUST pose/person classification (95% and 76% accuracy), outperforming graph-based scattering and baseline features.

ABSTRACT

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric scattering transform on manifolds. Similar to the Euclidean scattering transform, the geometric scattering transform is based on a cascade of wavelet filters and pointwise nonlinearities. It is invariant to local isometries and stable to certain types of diffeomorphisms. Empirical results demonstrate its utility on several geometric learning tasks. Our results generalize the deformation stability and local translation invariance of Euclidean scattering, and demonstrate the importance of linking the used filter structures to the underlying geometry of the data.

Motivation & Objective

• To extend the Euclidean scattering transform to compact Riemannian manifolds, enabling mathematical analysis of geometric deep learning models.
• To develop a scattering framework invariant to local isometries and stable to diffeomorphisms on non-Euclidean data domains.
• To demonstrate the utility of geometry-aware filter design in manifold-based learning tasks such as classification and feature extraction.
• To unify and generalize existing graph scattering approaches by embedding them in a continuous Riemannian geometry framework.

Proposed method

• Constructs spectral filtering operators using the eigenfunctions of the Laplace-Beltrami operator to generalize convolution on manifolds.
• Defines a cascade of wavelet filters and pointwise nonlinearities (complex modulus) to build a geometric scattering transform.
• Implements a non-expansive transform that is invariant to local isometries and stable to small diffeomorphisms via spectral decomposition.
• Uses the global geometric scattering transform $\overline{S}^L f$ for tasks requiring global invariance, such as manifold classification.
• Applies predefined geometric wavelets derived from the manifold's intrinsic geometry, avoiding end-to-end learning of filters.
• Employs 5-fold cross-validation with RBF kernel SVMs to evaluate classification performance on spherical MNIST and FAUST datasets.

Experimental results

Research questions

• RQ1 Can the scattering transform be generalized from Euclidean space to compact Riemannian manifolds while preserving invariance and stability properties?
• RQ2 How does the use of manifold-specific wavelets affect performance in geometric learning tasks compared to graph-based or Euclidean approaches?
• RQ3 To what extent does the geometric scattering transform achieve invariance to isometries and stability to diffeomorphisms on non-Euclidean data?
• RQ4 How does the geometric scattering framework compare to fully learned geometric deep learning models in terms of accuracy and hyperparameter sensitivity on small datasets?
• RQ5 Can geometric scattering features serve as universal, robust representations for manifold classification tasks?

Key findings

• The geometric wavelet scattering transform achieves 95% classification accuracy on the spherical MNIST dataset, outperforming Haar wavelet scattering (90%) and matching or exceeding learned models like S2CNN.
• On the FAUST dataset, geometric scattering achieves 95% accuracy in pose classification and 76% in person classification, significantly outperforming baseline SHOT features (92% and 61%) and graph scattering (58%).
• The transform is provably invariant to local isometries and Lipschitz stable to small diffeomorphisms, generalizing key theoretical properties of the Euclidean scattering transform.
• The use of geometry-aware wavelets leads to superior performance compared to graph-based scattering, highlighting the importance of embedding manifold structure into filter design.
• Hyperparameter tuning for network depth $L$ is simpler than in learned models, making geometric scattering particularly effective for small datasets like FAUST.
• The framework generalizes to manifolds of any dimension $d$, enabling future application to higher-dimensional geometric data.