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1. Gradient descent algorithms for Bures-Wasserstein barycenters

   Chewi, Sinh; Maunu, Tyler; Rigollet, Philipp; Stromme, Austin J. (January 2020, Proceedings of Machine Learning Research)

   Abernethy, Jacob; Agarwal, Shivani (Ed.)

   We study first order methods to compute the barycenter of a probability distribution $$P$$ over the space of probability measures with finite second moment. We develop a framework to derive global rates of convergence for both gradient descent and stochastic gradient descent despite the fact that the barycenter functional is not geodesically convex. Our analysis overcomes this technical hurdle by employing a Polyak-Ł{}ojasiewicz (PL) inequality and relies on tools from optimal transport and metric geometry. In turn, we establish a PL inequality when $$P$$ is supported on the Bures-Wasserstein manifold of Gaussian probability measures. It leads to the first global rates of convergence for first order methods in this context. 

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2. Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifolds

   Sun, Xingzhi; Liao, Danqi; MacDonald, Kincaid; Zhang, Yanlei; Huguet, Guillaume; Wolf, Guy; Adelstein, Ian; Rudner, Tim G; Krishnaswamy, Smita (January 2025, PMLR)

   apid growth of high-dimensional datasets in fields such as single-cell RNA sequencing and spatial genomics has led to unprecedented opportunities for scientific discovery, but it also presents unique computational and statistical challenges. Traditional methods struggle with geometry-aware data generation, interpolation along meaningful trajectories, and transporting populations via feasible paths. To address these issues, we introduce Geometry-Aware Generative Autoencoder (GAGA), a novel framework that combines extensible manifold learning with generative modeling. GAGA constructs a neural network embedding space that respects the intrinsic geometries discovered by manifold learning and learns a novel warped Riemannian metric on the data space. This warped metric is derived from both the points on the data manifold and negative samples off the manifold, allowing it to characterize a meaningful geometry across the entire latent space. Using this metric, GAGA can uniformly sample points on the manifold, generate points along geodesics, and interpolate between populations across the learned manifold. GAGA shows competitive performance in simulated and real-world datasets, including a 30% improvement over SOTA in single-cell population-level trajectory inference. 

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3. On a generalized conjecture of Hopf with symmetry

   https://doi.org/10.1112/S0010437X16008150  

   Amann, Manuel; Kennard, Lee (February 2017, Compositio Mathematica)

   A famous conjecture of Hopf states that $$\mathbb{S}^{2}\times \mathbb{S}^{2}$$ does not admit a Riemannian metric with positive sectional curvature. In this article, we prove that no manifold product $$N\times N$$ can carry a metric of positive sectional curvature admitting a certain degree of torus symmetry. 

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4. Four-manifolds of pinched sectional curvature

   https://doi.org/10.1016/j.aim.2018.07.003  

   Cao, Xiaodong; Tran, Hung (January 2022, Pacific journal of mathematics)

   In this paper, we study closed four-dimensional manifolds. In particular, we show that under various pinching curvature conditions (for example, the sectional curvature is no more than 5 6 of the smallest Ricci eigenvalue), the manifold is definite. If restricting to a metric with harmonic Weyl tensor, then it must be self-dual or anti-self-dual under the same conditions. Similarly, if restricting to an Einstein metric, then it must be either the complex projective space with its Fubini-Study metric, the round sphere, or their quotients. Furthermore, we also classify Einstein manifolds with positive intersection form and an upper bound on the sectional curvature. 

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5. Bergman metrics with constant holomorphic sectional curvatures

   https://doi.org/10.1515/crelle-2025-0013  

   Huang, Xiaojun; Li, Song-Ying; Treuer, John N (March 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract The paper studies complex manifolds whose Bergman metrics are incomplete but have constant holomorphic sectional curvature.We will construct a real analytic unbounded domain in ${\mathbb{C}}^2$\mathbb{C}^{2}whose Bergman metric is well-defined and has a positive constant holomorphic sectional curvature, which appears to be the first example of this kind.We will answer a long standing folklore conjecture that a Stein manifold has a negative constant holomorphic sectional curvature if and only if it is biholomorphic to a ball with a pluripolar set removed.Together with the uniqueness of a moment problem in the appendix of the paper provided by John Treuer, we will show that, under natural assumptions, there is no complex manifold whose Bergman metric is flat. 

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