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Estimating the sharing of genetic effects across different conditions is important to many statistical analyses of genomic data. The patterns of sharing arising from these data are often highly heterogeneous. To flexibly model these heterogeneous sharing patterns, Urbut et al. (2019) proposed the multivariate adaptive shrinkage (MASH) method to jointly analyze genetic effects across multiple conditions. However, multivariate analyses using MASH (as well as other multivariate analyses) require good estimates of the sharing patterns, and estimating these patterns efficiently and accurately remains challenging. Here we describe new empirical Bayes methods that provide improvements in speed and accuracy over existing methods. The two key ideas are: (1) adaptive regularization to improve accuracy in settings with many conditions; (2) improving the speed of the model fitting algorithms by exploiting analytical results on covariance estimation. In simulations, we show that the new methods provide better model fits, better out-of-sample performance, and improved power and accuracy in detecting the true underlying signals. In an analysis of eQTLs in 49 human tissues, our new analysis pipeline achieves better model fits and better out-of-sample performance than the existing MASH analysis pipeline. We have implemented the new methods, which we call ``Ultimate Deconvolution'', in an R package, udr, available on GitHub.

This paper presents an analysis of properties of two hybrid discretization methods for Gaussian derivatives, based on convolutions with either the normalized sampled Gaussian kernel or the integrated Gaussian kernel followed by central differences. The motivation for studying these discretization methods is that in situations when multiple spatial derivatives of different order are needed at the same scale level, they can be computed significantly more efficiently compared to more direct derivative approximations based on explicit convolutions with either sampled Gaussian kernels or integrated Gaussian kernels. While these computational benefits do also hold for the genuinely discrete approach for computing discrete analogues of Gaussian derivatives, based on convolution with the discrete analogue of the Gaussian kernel followed by central differences, the underlying mathematical primitives for the discrete analogue of the Gaussian kernel, in terms of modified Bessel functions of integer order, may not be available in certain frameworks for image processing, such as when performing deep learning based on scale-parameterized filters in terms of Gaussian derivatives, with learning of the scale levels. In this paper, we present a characterization of the properties of these hybrid discretization methods, in terms of quantitative performance measures concerning the amount of spatial smoothing that they imply, as well as the relative consistency of scale estimates obtained from scale-invariant feature detectors with automatic scale selection, with an emphasis on the behaviour for very small values of the scale parameter, which may differ significantly from corresponding results obtained from the fully continuous scale-space theory, as well as between different types of discretization methods.

We consider a convex constrained Gaussian sequence model and characterize necessary and sufficient conditions for the least squares estimator (LSE) to be optimal in a minimax sense. For a closed convex set $K\subset \mathbb{R}^n$ we observe $Y=\mu+\xi$ for $\xi\sim N(0,\sigma^2\mathbb{I}_n)$ and $\mu\in K$ and aim to estimate $\mu$. We characterize the worst case risk of the LSE in multiple ways by analyzing the behavior of the local Gaussian width on $K$. We demonstrate that optimality is equivalent to a Lipschitz property of the local Gaussian width mapping. We also provide theoretical algorithms that search for the worst case risk. We then provide examples showing optimality or suboptimality of the LSE on various sets, including $\ell_p$ balls for $p\in[1,2]$, pyramids, solids of revolution, and multivariate isotonic regression, among others.

The use of ultra-massive multiple-input multiple-output and high-frequency large bandwidth systems is likely in the next-generation wireless communication systems. In such systems, the user moves between near- and far-field regions, and consequently, the channel estimation will need to be carried out in the cross-field scenario. Channel estimation strategies have been proposed for both near- and far-fields, but in the cross-field problem, the first step is to determine whether the near- or far-field is applicable so that an appropriate channel estimation strategy can be employed. In this work, we propose using a hidden Markov model over an ensemble of region estimates to enhance the accuracy of selecting the actual region. The region indicators are calculated using the pair-wise power differences between received signals across the subarrays within an array-of-subarrays architecture. Numerical results show that the proposed method achieves a high success rate in determining the appropriate channel estimation strategy.

Mainly motivated by the problem of modelling directional dependence relationships for multivariate count data in high-dimensional settings, we present a new algorithm, called learnDAG, for learning the structure of directed acyclic graphs (DAGs). In particular, the proposed algorithm tackled the problem of learning DAGs from observational data in two main steps: (i) estimation of candidate parent sets; and (ii) feature selection. We experimentally compare learnDAG to several popular competitors in recovering the true structure of the graphs in situations where relatively moderate sample sizes are available. Furthermore, to make our algorithm is stronger, a validation of the algorithm is presented through the analysis of real datasets.

This paper concerns a class of DC composite optimization problems which, as an extension of convex composite optimization problems and DC programs with nonsmooth components, often arises in robust factorization models of low-rank matrix recovery. For this class of nonconvex and nonsmooth problems, we propose an inexact linearized proximal algorithm (iLPA) by computing in each step an inexact minimizer of a strongly convex majorization constructed with a partial linearization of their objective functions at the current iterate, and establish the convergence of the generated iterate sequence under the Kurdyka-\L\"ojasiewicz (KL) property of a potential function. In particular, by leveraging the composite structure, we provide a verifiable condition for the potential function to have the KL property of exponent $1/2$ at the limit point, so for the iterate sequence to have a local R-linear convergence rate. Finally, we apply the proposed iLPA to a robust factorization model for matrix completions with outliers and non-uniform sampling, and numerical comparison with a proximal alternating minimization (PAM) method confirms iLPA yields the comparable relative errors or NMAEs within less running time, especially for large-scale real data.

The goal of this paper is to provide a simple approach to perform local sensitivity analysis using Physics-informed neural networks (PINN). The main idea lies in adding a new term in the loss function that regularizes the solution in a small neighborhood near the nominal value of the parameter of interest. The added term represents the derivative of the loss function with respect to the parameter of interest. The result of this modification is a solution to the problem along with the derivative of the solution with respect to the parameter of interest (the sensitivity). We call the new technique SA-PNN which stands for sensitivity analysis in PINN. The effectiveness of the technique is shown using four examples: the first one is a simple one-dimensional advection-diffusion problem to show the methodology, the second is a two-dimensional Poisson's problem with nine parameters of interest, and the third and fourth examples are one and two-dimensional transient two-phase flow in porous media problem.

We present a dimension-incremental method for function approximation in bounded orthonormal product bases to learn the solutions of various differential equations. Therefore, we deconstruct the source function of the differential equation into parameters like Fourier or Spline coefficients and treat the solution of the differential equation as a high-dimensional function w.r.t. the spatial variables, these parameters and also further possible parameters from the differential equation itself. Finally, we learn this function in the sense of sparse approximation in a suitable function space by detecting coefficients of the basis expansion with largest absolute value. Investigating the corresponding indices of the basis coefficients yields further insights on the structure of the solution as well as its dependency on the parameters and their interactions and allows for a reasonable generalization to even higher dimensions and therefore better resolutions of the deconstructed source function.

This work aims to improve generalization and interpretability of dynamical systems by recovering the underlying lower-dimensional latent states and their time evolutions. Previous work on disentangled representation learning within the realm of dynamical systems focused on the latent states, possibly with linear transition approximations. As such, they cannot identify nonlinear transition dynamics, and hence fail to reliably predict complex future behavior. Inspired by the advances in nonlinear ICA, we propose a state-space modeling framework in which we can identify not just the latent states but also the unknown transition function that maps the past states to the present. We introduce a practical algorithm based on variational auto-encoders and empirically demonstrate in realistic synthetic settings that we can (i) recover latent state dynamics with high accuracy, (ii) correspondingly achieve high future prediction accuracy, and (iii) adapt fast to new environments.

In this paper we develop a novel neural network model for predicting implied volatility surface. Prior financial domain knowledge is taken into account. A new activation function that incorporates volatility smile is proposed, which is used for the hidden nodes that process the underlying asset price. In addition, financial conditions, such as the absence of arbitrage, the boundaries and the asymptotic slope, are embedded into the loss function. This is one of the very first studies which discuss a methodological framework that incorporates prior financial domain knowledge into neural network architecture design and model training. The proposed model outperforms the benchmarked models with the option data on the S&P 500 index over 20 years. More importantly, the domain knowledge is satisfied empirically, showing the model is consistent with the existing financial theories and conditions related to implied volatility surface.