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1. Three travel time inverse problems on simple Riemannian manifolds

   https://doi.org/10.1090/proc/16453  

   Ilmavirta, Joonas; Liu, Boya; Saksala, Teemu (June 2023, Proceedings of the American Mathematical Society)

   Jiaping Wang (Ed.)

   We provide new proofs based on the Myers–Steenrod theorem to confirm that travel time data, travel time difference data and the broken scattering relations determine a simple Riemannian metric on a disc up to the natural gauge of a boundary fixing diffeomorphism. Our method of the proof leads to a Lipschitz-type stability estimate for the first two data sets in the class of simple metrics. 

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2. Lower semi-continuity of Lagrangian volume

   Çineli, Erman; Ginzburg, Viktor L; Gürel, Başak Z (December 2024, Israel journal of mathematics)

   We study lower semi-continuity properties of the volume, i.e., the surface area, of a closed Lagrangian manifold with respect to the Hofer- and γ-distance on a class of monotone Lagrangian submanifolds Hamiltonian isotopic to each other. We prove that volume is γ-lower semi-continuous in two cases. In the first one the volume form comes from a Kähler metric with a large group of Hamiltonian isometries, but there are no additional constraints on the Lagrangian submanifold. The second one is when the volume is taken with respect to any compatible metric, but the Lagrangian submanifold must be a torus. As a consequence, in both cases, the volume is Hofer lower semi-continuous. 

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3. First‐order asymptotic perturbation theory for extensions of symmetric operators

   https://doi.org/10.1112/jlms.13005  

   Latushkin, Yuri; Sukhtaiev, Selim (November 2024, Journal of the London Mathematical Society)

   Abstract This work offers a new prospective on asymptotic perturbation theory for varying self‐adjoint extensions of symmetric operators. Employing symplectic formulation of self‐adjointness, we use a version of resolvent difference identity for two arbitrary self‐adjoint extensions that facilitates asymptotic analysis of resolvent operators via first‐order expansion for the family of Lagrangian planes associated with perturbed operators. Specifically, we derive a Riccati‐type differential equation and the first‐order asymptotic expansion for resolvents of self‐adjoint extensions determined by smooth one‐parameter families of Lagrangian planes. This asymptotic perturbation theory yields a symplectic version of the abstract Kato selection theorem and Hadamard–Rellich‐type variational formula for slopes of multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. The latter, in turn, gives a general infinitesimal version of the celebrated formula equating the spectral flow of a path of self‐adjoint extensions and the Maslov index of the corresponding path of Lagrangian planes. Applications are given to quantum graphs, periodic Kronig–Penney model, elliptic second‐order partial differential operators with Robin boundary conditions, and physically relevant heat equations with thermal conductivity. 

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4. Mean Field Approximations via Log-Concavity

   https://doi.org/10.1093/imrn/rnad302  

   Lacker, Daniel; Mukherjee, Sumit; Yeung, Lane Chun (December 2023, International Mathematics Research Notices)

   Abstract We propose a new approach to deriving quantitative mean field approximations for any probability measure $$P$$ on $$\mathbb {R}^{n}$$ with density proportional to $$e^{f(x)}$$, for $$f$$ strongly concave. We bound the mean field approximation for the log partition function $$\log \int e^{f(x)}dx$$ in terms of $$\sum _{i \neq j}\mathbb {E}_{Q^{*}}|\partial _{ij}f|^{2}$$, for a semi-explicit probability measure $$Q^{*}$$ characterized as the unique mean field optimizer, or equivalently as the minimizer of the relative entropy $$H(\cdot \,|\,P)$$ over product measures. This notably does not involve metric-entropy or gradient-complexity concepts which are common in prior work on nonlinear large deviations. Three implications are discussed, in the contexts of continuous Gibbs measures on large graphs, high-dimensional Bayesian linear regression, and the construction of decentralized near-optimizers in high-dimensional stochastic control problems. Our arguments are based primarily on functional inequalities and the notion of displacement convexity from optimal transport. 

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5. Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space

   Jiang, Yiheng; Chewi, Sinho; Pooladian, Aram-Alexandre (June 2024, Conference on Learning Theory (COLT) 2024)

   We develop a theory of finite-dimensional polyhedral subsets over the Wasserstein space and optimization of functionals over them via first-order methods. Our main application is to the problem of mean-field variational inference, which seeks to approximate a distribution π over ℝd by a product measure π⋆. When π is strongly log-concave and log-smooth, we provide (1) approximation rates certifying that π⋆ is close to the minimizer π⋆⋄ of the KL divergence over a \emph{polyhedral} set ⋄, and (2) an algorithm for minimizing KL(⋅‖π) over ⋄ with accelerated complexity O(κ√log(κd/ε2)), where κ is the condition number of π. 

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