More Like this

1. Approximate isoperimetry for convex polytopes

   https://doi.org/10.1112/mtk.70070  

   Ball, Keith; Böröczky, Károly J; Naor, Assaf (January 2026, Mathematika)

   Abstract For all with , the smallest possible isoperimetric quotient of an ‐dimensional convex polytope that has facets is shown to be bounded from above and from below by positive universal constant multiples of . For all and , it is shown that every ‐dimensional origin‐symmetric convex polytope that has vertices admits an affine image whose isoperimetric quotient is at most a universal constant multiple of , which is sharp. The weak isomorphic reverse isoperimetry conjecture is proved for ‐dimensional convex polytopes that have facets by demonstrating that any such polytope has an image under a volume‐preserving matrix and a convex body such that the isoperimetric quotient of is at most a universal constant multiple of , and also is at least a positive universal constant. 

   more » « less
2. Shortest closed curve to contain a sphere in its convex hull

   https://doi.org/10.1112/blms.13066  

   Ghomi, Mohammad; Wenk, James (July 2024, Bulletin of the London Mathematical Society)

   Abstract We show that in Euclidean 3‐space any closed curve which contains the unit sphere within its convex hull has length , and characterize the case of equality. This result generalizes the authors' recent solution to a conjecture of Zalgaller. Furthermore, for the analogous problem in dimensions, we include the estimate by Nazarov, which is sharp up to the constant . 

   more » « less
3. Local types of (Γ,G)‐bundles and parahoric group schemes

   https://doi.org/10.1112/plms.12544  

   Damiolini, Chiara; Hong, Jiuzu (August 2023, Proceedings of the London Mathematical Society)

   Abstract Let be a simple algebraic group over an algebraically closed field . Let be a finite group acting on . We classify and compute the local types of ‐bundles on a smooth projective ‐curve in terms of the first nonabelian group cohomology of the stabilizer groups at the tamely ramified points with coefficients in . When , we prove that any generically simply connected parahoric Bruhat–Tits group scheme can arise from a ‐bundle. We also prove a local version of this theorem, that is, parahoric group schemes over the formal disc arise from constant group schemes via tamely ramified coverings. 

   more » « less
4. Gradient descent with adaptive stepsize converges (nearly) linearly under fourth-order growth

   https://doi.org/10.1007/s10107-025-02290-5  

   Davis, Damek; Drusvyatskiy, Dmitriy; Jiang, Liwei (October 2025, Mathematical Programming)

   Abstract A prevalent belief among optimization specialists is that linear convergence of gradient descent is contingent on the function growing quadratically away from its minimizers. In this work, we argue that this belief is inaccurate. We show that gradient descent with an adaptive stepsize converges at a local (nearly) linear rate on any smooth function that merely exhibits fourth-order growth away from its minimizer. The adaptive stepsize we propose arises from an intriguing decomposition theorem: any such function admits a smooth manifold around the optimal solution—which we call the ravine—so that the function grows at least quadratically away from the ravine and has constant order growth along it. The ravine allows one to interlace many short gradient steps with a single long Polyak gradient step, which together ensure rapid convergence to the minimizer. We illustrate the theory and algorithm on the problems of matrix sensing and factorization and learning a single neuron in the overparameterized regime. 

   more » « less
5. Robust and provably monotonic networks

   https://doi.org/10.1088/2632-2153/aced80  

   Kitouni, Ouail; Nolte, Niklas; Williams, Mike (August 2023, Machine Learning: Science and Technology)

   Abstract The Lipschitz constant of the map between the input and output space represented by a neural network is a natural metric for assessing the robustness of the model. We present a new method to constrain the Lipschitz constant of dense deep learning models that can also be generalized to other architectures. The method relies on a simple weight normalization scheme during training that ensures the Lipschitz constant of every layer is below an upper limit specified by the analyst. A simple monotonic residual connection can then be used to make the model monotonic in any subset of its inputs, which is useful in scenarios where domain knowledge dictates such dependence. Examples can be found in algorithmic fairness requirements or, as presented here, in the classification of the decays of subatomic particles produced at the CERN Large Hadron Collider. Our normalization is minimally constraining and allows the underlying architecture to maintain higher expressiveness compared to other techniques which aim to either control the Lipschitz constant of the model or ensure its monotonicity. We show how the algorithm was used to train a powerful, robust, and interpretable discriminator for heavy-flavor-quark decays, which has been adopted for use as the primary data-selection algorithm in the LHCb real-time data-processing system in the current LHC data-taking period known as Run 3. In addition, our algorithm has also achieved state-of-the-art performance on benchmarks in medicine, finance, and other applications. 

   more » « less