skip to main content

This content will become publicly available on May 1, 2025

Title: Geometric scattering on measure spaces
The scattering transform is a multilayered, wavelet-based transform initially introduced as a mathematical model of convolutional neural networks (CNNs) that has played a foundational role in our understanding of these networks’ stability and invariance properties. In subsequent years, there has been widespread interest in extending the success of CNNs to data sets with non- Euclidean structure, such as graphs and manifolds, leading to the emerging field of geometric deep learning. In order to improve our understanding of the architectures used in this new field, several papers have proposed generalizations of the scattering transform for non-Euclidean data structures such as undirected graphs and compact Riemannian manifolds without boundary. Analogous to the original scattering transform, these works prove that these variants of the scattering transform have desirable stability and invariance properties and aim to improve our understanding of the neural networks used in geometric deep learning. In this paper, we introduce a general, unified model for geometric scattering on measure spaces. Our proposed framework includes previous work on compact Riemannian manifolds without boundary and undirected graphs as special cases but also applies to more general settings such as directed graphs, signed graphs, and manifolds with boundary. We propose a new criterion that identifies to which groups a useful representation should be invariant and show that this criterion is sufficient to guarantee that the scattering transform has desirable stability and invariance properties. Additionally, we consider finite measure spaces that are obtained from randomly sampling an unknown manifold. We propose two methods for constructing a data-driven graph on which the associated graph scattering transform approximates the scattering transform on the underlying manifold. Moreover, we use a diffusion-maps based approach to prove quantitative estimates on the rate of convergence of one of these approximations as the number of sample points tends to infinity. Lastly, we showcase the utility of our method on spherical images, a directed graph stochastic block model, and on high-dimensional single-cell data.  more » « less
Award ID(s):
Author(s) / Creator(s):
; ; ; ; ; ; ;
Publisher / Repository:
Date Published:
Journal Name:
Applied and Computational Harmonic Analysis
Page Range / eLocation ID:
Subject(s) / Keyword(s):
Geometric scattering graph signal processing manifold scattering single cell data pointcloud classification
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Convolutional neural networks (CNNs) are revolutionizing imaging science for two- and three-dimensional images over Euclidean domains. However, many data sets are intrinsically non-Euclidean and are better modeled through other mathematical structures, such as graphs or manifolds. This state of affairs has led to the development of geometric deep learning, which refers to a body of research that aims to translate the principles of CNNs to these non-Euclidean structures. In the process, various challenges have arisen, including how to define such geometric networks, how to compute and train them efficiently, and what are their mathematical properties. In this letter we describe the geometric wavelet scattering transform, which is a type of geometric CNN for graphs and manifolds consisting of alternating multiscale geometric wavelet transforms and nonlinear activation functions. As the name suggests, the geometric wavelet scattering transform is an adaptation of the Euclidean wavelet scattering transform, first introduced by S. Mallat, to graph and manifold data. Like its Euclidean counterpart, the geometric wavelet scattering transform has several desirable properties. In the manifold setting these properties include isometric invariance up to a user specified scale and stability to small diffeomorphisms. Numerical results on manifold and graph data sets, including graph and manifold classification tasks as well as others, illustrate the practical utility of the approach. 
    more » « less
  2. Lu, Jianfeng ; Ward, Rachel (Ed.)
    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. 
    more » « less
  3. The scattering transform is a multilayered wavelet-based architecture that acts as a model of convolutional neural networks. Recently, several works have generalized the scattering transform to graph-structured data. Our work builds on these constructions by introducing windowed and nonwindowed geometric scattering transforms for graphs based on two very general classes wavelets, which are in most cases based on asymmetric matrices. We show that these transforms have many of the same theoretical guarantees as their symmetric counterparts. As a result, the proposed construction unifies and extends known theoretical results for many of the existing graph scattering architectures. Therefore, it helps bridge the gap between geometric scattering and other graph neural networks by introducing a large family of networks with provable stability and invariance guarantees. These results lay the groundwork for future deep learning architectures for graph-structured data that have learned filters and also provably have desirable theoretical properties. 
    more » « less
  4. The manifold scattering transform is a deep feature extractor for data defined on a Riemannian manifold. It is one of the first examples of extending convolutional neural network-like operators to general manifolds. The initial work on this model focused primarily on its theoretical stability and invariance properties but did not provide methods for its numerical implementation except in the case of two-dimensional surfaces with predefined meshes. In this work, we present practical schemes, based on the theory of diffusion maps, for implementing the manifold scattering transform to datasets arising in naturalistic systems, such as single cell genetics, where the data is a high-dimensional point cloud modeled as lying on a low-dimensional manifold. We show that our methods are effective for signal classification and manifold classification tasks. 
    more » « less
  5. Abstract

    A variational formulation for accelerated optimization on normed vector spaces was recently introduced in Wibisono et al. (PNAS 113:E7351–E7358, 2016), and later generalized to the Riemannian manifold setting in Duruisseaux and Leok (SJMDS, 2022a). This variational framework was exploited on normed vector spaces in Duruisseaux et al. (SJSC 43:A2949–A2980, 2021) using time-adaptive geometric integrators to design efficient explicit algorithms for symplectic accelerated optimization, and it was observed that geometric discretizations which respect the time-rescaling invariance and symplecticity of the Lagrangian and Hamiltonian flows were substantially less prone to stability issues, and were therefore more robust, reliable, and computationally efficient. As such, it is natural to develop time-adaptive Hamiltonian variational integrators for accelerated optimization on Riemannian manifolds. In this paper, we consider the case of Riemannian manifolds embedded in a Euclidean space that can be characterized as the level set of a submersion. We will explore how holonomic constraints can be incorporated in discrete variational integrators to constrain the numerical discretization of the Riemannian Hamiltonian system to the Riemannian manifold, and we will test the performance of the resulting algorithms by solving eigenvalue and Procrustes problems formulated as optimization problems on the unit sphere and Stiefel manifold.

    more » « less