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1. The variational method of moments

   https://doi.org/10.1093/jrsssb/qkad025  

   Bennett, Andrew; Kallus, Nathan (April 2023, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Abstract The conditional moment problem is a powerful formulation for describing structural causal parameters in terms of observables, a prominent example being instrumental variable regression. We introduce a very general class of estimators called the variational method of moments (VMM), motivated by a variational minimax reformulation of optimally weighted generalized method of moments for finite sets of moments. VMM controls infinitely for many moments characterized by flexible function classes such as neural nets and kernel methods, while provably maintaining statistical efficiency unlike existing related minimax estimators. We also develop inference algorithms and demonstrate the empirical strengths of VMM estimation and inference in experiments. 

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2. Moment-based metrics for molecules computable from cryogenic electron microscopy images

   https://doi.org/10.1017/S2633903X24000023  

   Zhang, Andy; Mickelin, Oscar; Kileel, Joe; Verbeke, Eric J; Marshall, Nicholas F; Gilles, Marc Aurèle; Singer, Amit (January 2024, Biological Imaging)

   Abstract Single-particle cryogenic electron microscopy (cryo-EM) is an imaging technique capable of recovering the high-resolution three-dimensional (3D) structure of biological macromolecules from many noisy and randomly oriented projection images. One notable approach to 3D reconstruction, known as Kam’s method, relies on the moments of the two-dimensional (2D) images. Inspired by Kam’s method, we introduce a rotationally invariant metric between two molecular structures, which does not require 3D alignment. Further, we introduce a metric between a stack of projection images and a molecular structure, which is invariant to rotations and reflections and does not require performing 3D reconstruction. Additionally, the latter metric does not assume a uniform distribution of viewing angles. We demonstrate the uses of the new metrics on synthetic and experimental datasets, highlighting their ability to measure structural similarity. 

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3. Locally Robust Semiparametric Estimation

   https://doi.org/10.3982/ECTA16294  

   Chernozhukov, V.; Escanciano, J.; Ichimura, N.; Newey, W.; Robins, J. (July 2022, Econometrica)

   Imbens, G. (Ed.)

   Many economic and causal parameters depend on nonparametric or high dimensional first steps. We give a general construction of locally robust/orthogonal moment functions for GMM, where first steps have no effect, locally, on average moment functions. Using these orthogonal moments reduces model selection and regularization bias, as is important in many applications, especially for machine learning first steps. Also, associated standard errors are robust to misspecification when there is the same number of moment functions as parameters of interest. We use these orthogonal moments and cross-fitting to construct debiased machine learning estimators of functions of high dimensional conditional quantiles and of dynamic discrete choice parameters with high dimensional state variables. We show that additional first steps needed for the orthogonal moment functions have no effect, globally, on average orthogonal moment functions. We give a general approach to estimating those additional first steps.We characterize double robustness and give a variety of new doubly robust moment functions.We give general and simple regularity conditions for asymptotic theory. 

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4. On a hybrid continuum-kinetic model for complex fluids

   https://doi.org/10.1007/s42985-022-00198-9  

   Chertock, A.; Degond, P.; Dimarco, G.; Lukáčová-Medvid’ová, M.; Ruhi, A. (September 2022, Partial Differential Equations and Applications)

   Abstract In the present work, we first introduce a general framework for modelling complex multiscale fluids and then focus on the derivation and analysis of a new hybrid continuum-kinetic model. In particular, we combine conservation of mass and momentum for an isentropic macroscopic model with a kinetic representation of the microscopic behavior. After introducing a small scale of interest, we compute the complex stress tensor by means of the Irving-Kirkwood formula. The latter requires an expansion of the kinetic distribution around an equilibrium state and a successive homogenization over the fast in time and small in space scale dynamics. For a new hybrid continuum-kinetic model the results of linear stability analysis indicate a conditional stability in the relevant low speed regimes and linear instability for high speed regimes for higher modes. Extensive numerical experiments confirm that the proposed multiscale model can reflect new phenomena of complex fluids not being present in standard Newtonian fluids. Consequently, the proposed general technique can be successfully used to derive new interesting systems combining the macro and micro structure of a given physical problem. 

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5. Negative discrete moments of the derivative of the Riemann zeta‐function

   https://doi.org/10.1112/blms.13092  

   Bui, Hung_M; Florea, Alexandra; Milinovich, Micah_B (May 2024, Bulletin of the London Mathematical Society)

   Abstract We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta‐function averaged over a subfamily of zeros of the zeta function that is expected to be arbitrarily close to full density inside the set of all zeros. For , our bounds for the ‐th moments are expected to be almost optimal. Assuming a conjecture about the maximum size of the argument of the zeta function on the critical line, we obtain upper bounds for these negative moments of the same strength while summing over a larger subfamily of zeta zeros. 

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