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1. Diffusion Scattering Transforms on Graphs

   Gama, Fernando; Ribeiro, Alejandro; Bruna, Joan (May 2019, International Conference on Learning Representations)

   Stability is a key aspect of data analysis. In many applications, the natural notion of stability is geometric, as illustrated for example in computer vision. Scattering transforms construct deep convolutional representations which are certified stable to input deformations. This stability to deformations can be interpreted as stability with respect to changes in the metric structure of the domain. In this work, we show that scattering transforms can be generalized to non-Euclidean domains using diffusion wavelets, while preserving a notion of stability with respect to metric changes in the domain, measured with diffusion maps. The resulting representation is stable to metric perturbations of the domain while being able to capture ''high-frequency'' information, akin to the Euclidean Scattering. 

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2. Stability of Graph Scattering Transforms

   Gama, Fernando; Ribeiro, Alejandro; Bruna, Joan (January 2019, Advances in neural information processing systems)

   Scattering transforms are non-trainable deep convolutional architectures that exploit the multi-scale resolution of a wavelet filter bank to obtain an appropriate representation of data. More importantly, they are proven invariant to translations, and stable to perturbations that are close to translations. This stability property dons the scattering transform with a robustness to small changes in the metric domain of the data. When considering network data, regular convolutions do not hold since the data domain presents an irregular structure given by the network topology. In this work, we extend scattering transforms to network data by using multi-resolution graph wavelets, whose computation can be obtained by means of graph convolutions. Furthermore, we prove that the resulting graph scattering transforms are stable to metric perturbations of the underlying network. This renders graph scattering transforms robust to changes on the network topology, making it particularly useful for cases of transfer learning, topology estimation or time-varying graphs. 

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3. Conformal Transformation of Uniform Domains Under Weights That Depend on Distance to The Boundary

   https://doi.org/10.1515/agms-2022-0141  

   Gibara, Ryan; Shanmugalingam, Nageswari (January 2022, Analysis and Geometry in Metric Spaces)

   Abstract The sphericalization procedure converts a Euclidean space into a compact sphere. In this note we propose a variant of this procedure for locally compact, rectifiably path-connected, non-complete, unbounded metric spaces by using conformal deformations that depend only on the distance to the boundary of the metric space. This deformation is locally bi-Lipschitz to the original domain near its boundary, but transforms the space into a bounded domain. We will show that if the original metric space is a uniform domain with respect to its completion, then the transformed space is also a uniform domain. 

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4. Geometric wavelet scattering on graphs and manifolds

   https://doi.org/10.1117/12.2529615  

   Gao, Feng; Hirn, Matthew; Perlmutter, Michael; Wolf, Guy (September 2019, Proceedings SPIE 11138, Wavelets and Sparsity XVIII)

   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. 

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5. On Stability of Metric Spaces and Kalton’s Property Q

   https://doi.org/10.1093/qmath/haae050  

   Baudier, F; Schlumprecht, Th; ZSÁK, A (December 2024, The Quarterly Journal of Mathematics)

   ABSTRACT The first named author introduced the notion of upper stability for metric spaces in F. Baudier, Barycentric gluing and geometry of stable metrics, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM  116 no. 1, (2022), 48 as a relaxation of stability. The motivation was a search for a new invariant to distinguish the class of reflexive Banach spaces from stable metric spaces in the coarse and uniform category. In this paper we show that property Q does in fact imply upper stability. We also provide a direct proof of the fact that reflexive spaces are upper stable by relating the latter notion to the asymptotic structure of Banach spaces. 

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