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1. Encoding robust representation for graph generation

   https://doi.org/10.1109/IJCNN.2019.8851705  

   Zou, Dongmian; Lerman, Gilad (July 2019, International Joint Conference on Neural Networks (IJCNN))

   Generative networks have made it possible to generate meaningful signals such as images and texts from simple noise. Recently, generative methods based on GAN and VAE were developed for graphs and graph signals. However, the mathematical properties of these methods are unclear, and training good generative models is difficult. This work proposes a graph generation model that uses a recent adaptation of Mallat's scattering transform to graphs. The proposed model is naturally composed of an encoder and a decoder. The encoder is a Gaussianized graph scattering transform, which is robust to signal and graph manipulation. The decoder is a simple fully connected network that is adapted to specific tasks, such as link prediction, signal generation on graphs and full graph and signal generation. The training of our proposed system is efficient since it is only applied to the decoder and the hardware requirements are moderate. Numerical results demonstrate state-of-the-art performance of the proposed system for both link prediction and graph and signal generation. 

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2. Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds

   Perlmutter, Michael; Gao, Feng; Wolf, Guy; Hirn, Matthew (January 2020, Proceedings of Machine Learning Research)

   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. 

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3. Geometric scattering on measure spaces

   https://doi.org/10.1016/j.acha.2024.101635  

   Chew, Joyce; Hirn, Matthew; Krishnaswamy, Smita; Needell, Deanna; Perlmutter, Michael; Steach, Holly; Viswanath, Siddharth; Wu, Hau-Tieng (May 2024, Applied and Computational Harmonic Analysis)

   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. 

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4. Norm bounds for a scattering transform on graphs

   Bernhard G. Bodmann and Iris Emilsdottir (January 2021, Oberwolfach reports)

   Ingrid Daubechies, Gitta Kutyniok (Ed.)

   This proceedings article describes the construction of a scattering transform for graphs and elementary properties of it. In particular, the norm of the signal obtained in the layers of the transform decreases exponentially with the depth of the layer. 

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5. ADAPTIVE TIME–FREQUENCY SCATTERING FOR PERIODIC MODULATION RECOGNITION IN MUSIC SIGNALS

   Wang, C; Benetos, E; Lostanlen, V; Chew, E (November 2019, International Society for Music Information Retrieval Conference)

   Vibratos, tremolos, trills, and flutter-tongue are techniques frequently found in vocal and instrumental music. A com- mon feature of these techniques is the periodic modulation in the time–frequency domain. We propose a representa- tion based on time–frequency scattering to model the inter- class variability for fine discrimination of these periodic modulations. Time–frequency scattering is an instance of the scattering transform, an approach for building invari- ant, stable, and informative signal representations. The proposed representation is calculated around the wavelet subband of maximal acoustic energy, rather than over all the wavelet bands. To demonstrate the feasibility of this approach, we build a system that computes the represen- tation as input to a machine learning classifier. Whereas previously published datasets for playing technique analy- sis focus primarily on techniques recorded in isolation, for ecological validity, we create a new dataset to evaluate the system. The dataset, named CBF-periDB, contains full- length expert performances on the Chinese bamboo flute that have been thoroughly annotated by the players them- selves. We report F-measures of 99% for flutter-tongue, 82% for trill, 69% for vibrato, and 51% for tremolo detec- tion, and provide explanatory visualisations of scattering coefficients for each of these techniques. 

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