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1. HALF-SPACE MACDONALD PROCESSES

   https://doi.org/10.1017/fmp.2020.3  

   BARRAQUAND, GUILLAUME; BORODIN, ALEXEI; CORWIN, IVAN (January 2020, Forum of Mathematics, Pi)

   null (Ed.)

   Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the integrability of many probabilistic systems, including the Kardar–Parisi–Zhang (KPZ) equation and a number of other models in its universality class. In this paper, we develop the structural theory behind half-space variants of these models and the corresponding half-space Macdonald processes. These processes are built using a Littlewood summation identity instead of the Cauchy identity, and their analysis is considerably harder than their full-space counterparts. We compute moments and Laplace transforms of observables for general half-space Macdonald measures. Introducing new dynamics preserving this class of measures, we relate them to various stochastic processes, in particular the log-gamma polymer in a half-quadrant (they are also related to the stochastic six-vertex model in a half-quadrant and the half-space ASEP). For the polymer model, we provide explicit integral formulas for the Laplace transform of the partition function. Nonrigorous saddle-point asymptotics yield convergence of the directed polymer free energy to either the Tracy–Widom (associated to the Gaussian orthogonal or symplectic ensemble) or the Gaussian distribution depending on the average size of weights on the boundary. 

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2. Sequential Estimation of Gaussian Process-Based Deep State-Space Models

   https://doi.org/10.1109/TSP.2023.3303648  

   Liu, Yuhao; Ajirak, Marzieh; Djurić, Petar M (January 2023, IEEE Transactions on Signal Processing)

   We consider the problem of sequential estimation of the unknowns of state-space and deep state-space models that include estimation of functions and latent processes of the models. The proposed approach relies on Gaussian and deep Gaussian processes that are implemented via random feature-based Gaussian processes. In these models, we have two sets of unknowns, highly nonlinear unknowns (the values of the latent processes) and conditionally linear unknowns (the constant parameters of the random feature-based Gaussian processes). We present a method based on particle filtering where the parameters of the random feature-based Gaussian processes are integrated out in obtaining the predictive density of the states and do not need particles. We also propose an ensemble version of the method, with each member of the ensemble having its own set of features. With several experiments, we show that the method can track the latent processes up to a scale and rotation. 

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3. Parametric description of intermittent probability distribution functions in solar wind and magnetohydrodynamic turbulence

   https://doi.org/10.1093/mnras/stae1065  

   Palacios, Juan C.; Perez, Jean C.; Bourouaine, Sofiane (April 2024, Monthly Notices of the Royal Astronomical Society)

   ABSTRACT In this work, we find empirical evidence that the scale-dependent statistical properties of solar wind and magnetohydrodynamic (MHD) turbulence can be described in terms of a family of parametric probability distribution functions (PDFs) known as Normal Inverse Gaussian (NIG). Understanding these PDFs is one of the most important goals in turbulence theory, as they are inherently connected to the intermittent properties of solar wind turbulence. We investigate the properties of PDFs of Elsasser increments based on a large statistical sample from solar wind observations and high-resolution numerical simulations of MHD turbulence. In order to measure the PDFs and their corresponding properties, three experiments are presented: fast and slow solar wind for experimental data and a simulation of reduced MHD (RMHD) turbulence. Conditional statistics on a 23-yr-long sample of WIND data near 1 au and high-resolution pseudo-spectral simulation of steadily driven RMHD turbulence on a $2048^3$ mesh are used to construct scale-dependent PDFs. The empirical PDFs are fitted to NIG distributions, which depend on four free parameters. Our analysis shows that NIG distributions accurately capture the evolution of the PDFs, with scale-dependent parameters, from large scales characterized by a Gaussian distribution, turning to exponential tails within the inertial range and stretched exponentials at dissipative scales. We also show that empirically-measured NIG parameters exhibit well-defined scaling properties that are similar across the three empirical data sets, which may be indicative of universal behaviour. 

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4. Extended Variational Inference for Propagating Uncertainty in Convolutional Neural Networks

   https://doi.org/10.1109/MLSP.2019.8918747  

   Dera, Dimah; Rasool, Ghulam; Bouaynaya, Nidhal (December 2019, IEEE 29th International Workshop on Machine Learning for Signal Processing)

   Model confidence or uncertainty is critical in autonomous systems as they directly tie to the safety and trustworthiness of the system. The quantification of uncertainty in the output decisions of deep neural networks (DNNs) is a challenging problem. The Bayesian framework enables the estimation of the predictive uncertainty by introducing probability distributions over the (unknown) network weights; however, the propagation of these high-dimensional distributions through multiple layers and non-linear transformations is mathematically intractable. In this work, we propose an extended variational inference (eVI) framework for convolutional neural network (CNN) based on tensor Normal distributions (TNDs) defined over convolutional kernels. Our proposed eVI framework propagates the first two moments (mean and covariance) of these TNDs through all layers of the CNN. We employ first-order Taylor series linearization to approximate the mean and covariances passing through the non-linear activations. The uncertainty in the output decision is given by the propagated covariance of the predictive distribution. Furthermore, we show, through extensive simulations on the MNIST and CIFAR-10 datasets, that the CNN becomes more robust to Gaussian noise and adversarial attacks. 

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5. Dense gas in a giant molecular filament

   https://doi.org/10.1051/0004-6361/202037928  

   Wang, Y.; Beuther, H.; Schneider, N.; Meidt, S. E.; Linz, H.; Ragan, S.; Zucker, C.; Battersby, C.; Soler, J. D.; Schinnerer, E.; et al (September 2020, Astronomy & Astrophysics)

   null (Ed.)

   Context. Recent surveys of the Galactic plane in the dust continuum and CO emission lines reveal that large (≳50 pc) and massive (≳10 5 M ⊙ ) filaments, know as giant molecular filaments (GMFs), may be linked to Galactic dynamics and trace the mid-plane of the gravitational potential in the Milky Way. Yet our physical understanding of GMFs is still poor. Aims. We investigate the dense gas properties of one GMF, with the ultimate goal of connecting these dense gas tracers with star formation processes in the GMF. Methods. We imaged one entire GMF located at l ~ 52–54° longitude, GMF54 (~68 pc long), in the empirical dense gas tracers using the HCN(1–0), HNC(1–0), and HCO + (1–0) lines, and their 13 C isotopologue transitions, as well as the N 2 H + (1–0) line. We studied the dense gas distribution, the column density probability density functions (N-PDFs), and the line ratios within the GMF. Results. The dense gas molecular transitions follow the extended structure of the filament with area filling factors between 0.06 and 0.28 with respect to 13 CO(1–0). We constructed the N-PDFs of H 2 for each of the dense gas tracers based on their column densities and assumed uniform abundance. The N-PDFs of the dense gas tracers appear curved in log–log representation, and the HCO + N-PDF has the flattest power-law slope index. Studying the N-PDFs for sub-regions of GMF54, we found an evolutionary trend in the N-PDFs that high-mass star-forming and photon-dominated regions have flatter power-law indices. The integrated intensity ratios of the molecular lines in GMF54 are comparable to those in nearby galaxies. In particular, the N 2 H + / 13 CO ratio, which traces the dense gas fraction, has similar values in GMF54 and all nearby galaxies except Ultraluminous Infrared Galaxies. Conclusions. As the largest coherent cold gaseous structure in our Milky Way, GMFs, are outstanding candidates for connecting studies of star formation on Galactic and extragalactic scales. By analyzing a complete map of the dense gas in a GMF we have found that: (1) the dense gas N-PDFs appear flatter in more evolved regions and steeper in younger regions, and (2) its integrated dense gas intensity ratios are similar to those of nearby galaxies. 

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