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Abstract:Bayesian Experimental Design (BED), which aims to find the optimal experimental conditions for Bayesian inference, is usually posed as to optimize the expected information gain (EIG). The gradient information is often needed for efficient EIG optimization, and as a result the ability to estimate the gradient of EIG is essential for BED problems. The primary goal of this work is to develop methods for estimating the gradient of EIG, which, combined with the stochastic gradient descent algorithms, result in efficient optimization of EIG. Specifically, we first introduce a posterior expected representation of the EIG gradient with respect to the design variables. Based on this, we propose two methods for estimating the EIG gradient, UEEG-MCMC that leverages posterior samples generated through Markov Chain Monte Carlo (MCMC) to estimate the EIG gradient, and BEEG-AP that focuses on achieving high simulation efficiency by repeatedly using parameter samples. Theoretical analysis and numerical studies illustrate that UEEG-MCMC is robust agains the actual EIG value, while BEEG-AP is more efficient when the EIG value to be optimized is small. Moreover, both methods show superior performance compared to several popular benchmarks in our numerical experiments.

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Abstract:Combining the strengths of model-based iterative algorithms and data-driven deep learning solutions, deep unrolling networks (DuNets) have become a popular tool to solve inverse imaging problems. While DuNets have been successfully applied to many linear inverse problems, nonlinear problems tend to impair the performance of the method. Inspired by momentum acceleration techniques that are often used in optimization algorithms, we propose a recurrent momentum acceleration (RMA) framework that uses a long short-term memory recurrent neural network (LSTM-RNN) to simulate the momentum acceleration process. The RMA module leverages the ability of the LSTM-RNN to learn and retain knowledge from the previous gradients. We apply RMA to two popular DuNets -- the learned proximal gradient descent (LPGD) and the learned primal-dual (LPD) methods, resulting in LPGD-RMA and LPD-RMA respectively. We provide experimental results on two nonlinear inverse problems: a nonlinear deconvolution problem, and an electrical impedance tomography problem with limited boundary measurements. In the first experiment we have observed that the improvement due to RMA largely increases with respect to the nonlinearity of the problem. The results of the second example further demonstrate that the RMA schemes can significantly improve the performance of DuNets in strongly ill-posed problems.

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Authors:Ziruo Cai, Junqi Tang, Subhadip Mukherjee, Jinglai Li, Carola Bibiane Schönlieb, Xiaoqun Zhang

Abstract:Bayesian methods for solving inverse problems are a powerful alternative to classical methods since the Bayesian approach gives a probabilistic description of the problems and offers the ability to quantify the uncertainty in the solution. Meanwhile, solving inverse problems by data-driven techniques also proves to be successful, due to the increasing representation ability of data-based models. In this work, we try to incorporate the data-based models into a class of Langevin-based sampling algorithms in Bayesian inference. Loosely speaking, we introduce NF-ULA (Unadjusted Langevin algorithms by Normalizing Flows), which involves learning a normalizing flow as the prior. In particular, our algorithm only requires a pre-trained normalizing flow, which is independent of the considered inverse problem and the forward operator. We perform theoretical analysis by investigating the well-posedness of the Bayesian solution and the non-asymptotic convergence of the NF-ULA algorithm. The efficacy of the proposed NF-ULA algorithm is demonstrated in various imaging problems, including image deblurring, image inpainting, and limited-angle X-ray computed tomography (CT) reconstruction.

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Abstract:Solving high-dimensional Bayesian inverse problems (BIPs) with the variational inference (VI) method is promising but still challenging. The main difficulties arise from two aspects. First, VI methods approximate the posterior distribution using a simple and analytic variational distribution, which makes it difficult to estimate complex spatially-varying parameters in practice. Second, VI methods typically rely on gradient-based optimization, which can be computationally expensive or intractable when applied to BIPs involving partial differential equations (PDEs). To address these challenges, we propose a novel approximation method for estimating the high-dimensional posterior distribution. This approach leverages a deep generative model to learn a prior model capable of generating spatially-varying parameters. This enables posterior approximation over the latent variable instead of the complex parameters, thus improving estimation accuracy. Moreover, to accelerate gradient computation, we employ a differentiable physics-constrained surrogate model to replace the adjoint method. The proposed method can be fully implemented in an automatic differentiation manner. Numerical examples demonstrate two types of log-permeability estimation for flow in heterogeneous media. The results show the validity, accuracy, and high efficiency of the proposed method.

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Abstract:Predicting the behaviors of pedestrian crowds is of critical importance for a variety of real-world problems. Data driven modeling, which aims to learn the mathematical models from observed data, is a promising tool to construct models that can make accurate predictions of such systems. In this work, we present a data-driven modeling approach based on the ODE-Net framework, for constructing continuous-time models of crowd dynamics. We discuss some challenging issues in applying the ODE-Net method to such problems, which are primarily associated with the dimensionality of the underlying crowd system, and we propose to address these issues by incorporating the social-force concept in the ODE-Net framework. Finally application examples are provided to demonstrate the performance of the proposed method.

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Abstract:We propose a linear-mapping based variational Ensemble Kalman filter for sequential Bayesian filtering problems with generic observation models. Specifically, the proposed method is formulated as to construct a linear mapping from the prior ensemble to the posterior one, and the linear mapping is computed via a variational Bayesian formulation, i.e., by minimizing the Kullback-Leibler divergence between the transformed distribution by the linear mapping and the actual posterior. A gradient descent scheme is proposed to solve the resulting optimization problem. With numerical examples we demonstrate that the method has competitive performance against existing methods.

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Abstract:Data collection is a critical step in statistical inference and data science, and the goal of statistical experimental design (ED) is to find the data collection setup that can provide most information for the inference. In this work we consider a special type of ED problems where the likelihoods are not available in a closed form. In this case, the popular information-theoretic Kullback-Leibler divergence (KLD) based design criterion can not be used directly, as it requires to evaluate the likelihood function. To address the issue, we derive a new utility function, which is a lower bound of the original KLD utility. This lower bound is expressed in terms of the summation of two or more entropies in the data space, and thus can be evaluated efficiently via entropy estimation methods. We provide several numerical examples to demonstrate the performance of the proposed method.

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Abstract:Global optimization finds applications in a wide range of real world problems. The multi-start methods are a popular class of global optimization techniques, which are based on the ideas of conducting local searches at multiple starting points, and then sequentially determine the starting points according to some prescribed rules. In this work we propose a new multi-start algorithm where the starting points are determined in a Bayesian optimization framework. Specifically, the method can be understood as to construct a new function by conducting local searches of the original objective function, where the new function attains the same global optima as the original one. Bayesian optimization is then applied to find the global optima of the new local search based function.

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Abstract:We consider supervised dimension reduction problems, namely to identify a low dimensional projection of the predictors $\-x$ which can retain the statistical relationship between $\-x$ and the response variable $y$. We follow the idea of the sliced inverse regression (SIR) class of methods, which is to use the statistical information of the conditional distribution $\pi(\-x|y)$ to identify the dimension reduction (DR) space and in particular we focus on the task of computing this conditional distribution. We propose a Bayesian framework to compute the conditional distribution where the likelihood function is obtained using the Gaussian process regression model. The conditional distribution $\pi(\-x|y)$ can then be obtained directly by assigning weights to the original data points. We then can perform DR by considering certain moment functions (e.g. the first moment) of the samples of the posterior distribution. With numerical examples, we demonstrate that the proposed method is especially effective for small data problems.

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Abstract:We consider Bayesian inference problems with computationally intensive likelihood functions. We propose a Gaussian process (GP) based method to approximate the joint distribution of the unknown parameters and the data. In particular, we write the joint density approximately as a product of an approximate posterior density and an exponentiated GP surrogate. We then provide an adaptive algorithm to construct such an approximation, where an active learning method is used to choose the design points. With numerical examples, we illustrate that the proposed method has competitive performance against existing approaches for Bayesian computation.

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