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.
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Pyramid transform of manifold data via subdivision operators
Abstract Multiscale transforms have become a key ingredient in many data processing tasks. With technological development we observe a growing demand for methods to cope with nonlinear data structures such as manifold values. In this paper we propose a multiscale approach for analyzing manifold-valued data using a pyramid transform. The transform uses a unique class of downsampling operators that enable a noninterpolating subdivision schemes as upsampling operators. We describe this construction in detail and present its analytical properties, including stability and coefficient decay. Next, we numerically demonstrate the results and show the application of our method to denoising and anomaly detection.
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- Award ID(s):
- 2009753
- NSF-PAR ID:
- 10417487
- Date Published:
- Journal Name:
- IMA Journal of Numerical Analysis
- Volume:
- 43
- Issue:
- 1
- ISSN:
- 0272-4979
- Page Range / eLocation ID:
- 387 to 413
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imanum/drab095
