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1. Lefschetz fibrations with arbitrary signature

   https://doi.org/10.4171/JEMS/1326  

   Baykur, R İnanç; Hamada, Noriyuki (June 2024, Journal of the European Mathematical Society)

   We develop techniques to construct explicit symplectic Lefschetz fibrations over the2-sphere with any prescribed signature\sigmaand any spin type when\sigmais divisible by16. This solves a long-standing conjecture on the existence of such fibrations with positive signature. As applications, we produce symplectic4-manifolds that are homeomorphic but not diffeomorphic to connected sums ofS^2 \times S^2, with the smallest topology known to date, as well as larger examples as symplectic Lefschetz fibrations. 

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2. Manifold Learning with Arbitrary Norms

   https://doi.org/10.1007/s00041-021-09879-2  

   Kileel, Joe; Moscovich, Amit; Zelesko, Nathan; Singer, Amit (October 2021, Journal of Fourier Analysis and Applications)

   null (Ed.)
3. Posterior Matching for Arbitrary Conditioning

   Strauss, Ryan and (January 2022, Advances in neural information processing systems)
4. Arbitrary Conditional Distributions with Energy

   Strauss, Ryan R.; Oliva, Junier B. (December 2021, Advances in neural information processing systems)

   Modeling distributions of covariates, or density estimation, is a core challenge in unsupervised learning. However, the majority of work only considers the joint distribution, which has limited utility in practical situations. A more general and useful problem is arbitrary conditional density estimation, which aims to model any possible conditional distribution over a set of covariates, reflecting the more realistic setting of inference based on prior knowledge. We propose a novel method, Arbitrary Conditioning with Energy (ACE), that can simultaneously estimate the distribution p(x_u | x_o) for all possible subsets of unobserved features x_u and observed features x_o. ACE is designed to avoid unnecessary bias and complexity — we specify densities with a highly expressive energy function and reduce the problem to only learning one-dimensional conditionals (from which more complex distributions can be recovered during inference). This results in an approach that is both simpler and higher-performing than prior methods. We show that ACE achieves state-of-the-art for arbitrary conditional likelihood estimation and data imputation on standard benchmarks. 

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5. Bounds for the smallest eigenvalue of the NTK for arbitrary spherical data of arbitrary dimension

   Karhadkar, Kedar; Murray, Michael; Montufar, Guido (September 2024, 38th Conference on Neural Information Processing Systems (NeurIPS 2024))

   Bounds on the smallest eigenvalue of the neural tangent kernel (NTK) are a key ingredient in the analysis of neural network optimization and memorization. How- ever, existing results require distributional assumptions on the data and are limited to a high-dimensional setting, where the input dimension d0 scales at least log- arithmically in the number of samples n. In this work we remove both of these requirements and instead provide bounds in terms of a measure of distance between data points: notably these bounds hold with high probability even when d0 is held constant versus n. We prove our results through a novel application of the hemisphere transform. 

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