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   Celaya, A; Riviere, B; Fuentes, D (November 2023, arXiv)
2. FMG-Net and W-Net: Multigrid Inspired Deep Learning Architectures for Medical Imaging Segmentation

   Celaya, Adrian; Riviere, Beatrice; Fuentes, David (December 2023, https://research.latinxinai.org/workshops/neurips/neurips-2023.html)

   Accurate medical imaging segmentation is critical for precise and effective medi- cal interventions. However, despite the success of convolutional neural networks (CNNs) in medical image segmentation, they still face challenges in handling fine-scale features and variations in image scales. These challenges are particularly evident in complex and challenging segmentation tasks, such as the BraTS multi- label brain tumor segmentation challenge. In this task, accurately segmenting the various tumor sub-components, which vary significantly in size and shape, remains a significant challenge, with even state-of-the-art methods producing substantial errors. Therefore, we propose two architectures, FMG-Net and W-Net, that incor- porate the principles of geometric multigrid methods for solving linear systems of equations into CNNs to address these challenges. Our experiments on the BraTS 2020 dataset demonstrate that both FMG-Net and W-Net outperform the widely used U-Net architecture regarding tumor subcomponent segmentation accuracy and training efficiency. These findings highlight the potential of incorporating the principles of multigrid methods into CNNs to improve the accuracy and efficiency of medical imaging segmentation. 

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3. Enriched string-net models and their excitations

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   Boundaries of Walker-Wang models have been used to construct commuting projector models which realize chiral unitary modular tensor categories (UMTCs) as boundary excitations. Given a UMTC $\mathcal{A}$representing the Witt class of an anomaly, the article \cite{MR4640433} gave a commuting projector model associated to an $\mathcal{A}$-enriched unitary fusion category $\mathcal{X}$on a 2D boundary of the 3D Walker-Wang model associated to $\mathcal{A}$. That article claimed that the boundary excitations were given by the enriched center/Müger centralizer $Z^{\mathcal{A}}\left(\mathcal{X}\right)$of $\mathcal{A}$in $Z\left(\mathcal{X}\right)$.In this article, we give a rigorous treatment of this 2D boundary model, and we verify this assertion using topological quantum field theory (TQFT) techniques, including skein modules and a certain semisimple algebra whose representation category describes boundary excitations. We also use TQFT techniques to show the 3D bulk point excitations of the Walker-Wang bulk are given by the Müger center $Z_2\left(\mathcal{A}\right)$, and we construct bulk-to-boundary hopping operators $Z_2\left(\mathcal{A}\right)\to Z^{\mathcal{A}}\left(\mathcal{X}\right)$reflecting how the UMTC of boundary excitations $Z^{\mathcal{A}}\left(\mathcal{X}\right)$is symmetric-braided enriched in $Z_2\left(\mathcal{A}\right)$.This article also includes a self-contained comprehensive review of the Levin-Wen string net model from a unitary tensor category viewpoint, as opposed to the skeletal $6j$symbol viewpoint. 

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