arXiv subjects

Bamdad Hosseini

Publications and source records attributed to Bamdad Hosseini.

At least 37 records · Page 2Linked to original sources

arXiv2023

Conditional Sampling with Monotone GANs: from Generative Models to Likelihood-Free Inference

We present a novel framework for conditional sampling of probability measures, using block triangular transport maps. We develop the theoretical foundations of block triangular transport in a Banach space setting, establishing general conditions under which conditional sampling can be achieved and drawing connections between monotone block triangular maps and optimal transport. Based on this theory, we then introduce a computational approach, called monotone generative adversarial networks (M-GANs), to learn suitable block triangular maps. Our algorithm uses only samples from the underlying joint probability measure and is hence likelihood-free. Numerical experiments with M-GAN demonstrate accurate sampling of conditional measures in synthetic examples, Bayesian inverse problems involving ordinary and partial differential equations, and probabilistic image in-painting.

stat.ML↗

arXiv2023

Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs

We introduce a priori Sobolev-space error estimates for the solution of nonlinear, and possibly parametric, PDEs using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

math.NA↗

arXiv2023

Optimal Transport Particle Filters

This paper is concerned with the theoretical and computational development of a new class of nonlinear filtering algorithms called the optimal transport particle filters (OTPF). The algorithm is based on a recently introduced variational formulation of the Bayes' rule, which aims to find the Brenier optimal transport map between the prior and the posterior distributions as the solution to a stochastic optimization problem. On the theoretical side, the existing methods for the error analysis of particle filters and stability results for optimal transport map estimation are combined to obtain uniform error bounds for the filter's performance in terms of the optimization gap in solving the variational problem. The error analysis reveals a bias-variance trade-off that can ultimately be used to understand if/when the curse of dimensionality can be avoided in these filters. On the computational side, the proposed algorithm is evaluated on a nonlinear filtering example in comparison with the ensemble Kalman filter (EnKF) and the sequential importance resampling (SIR) particle filter.

math.OC↗

arXiv2023

A Kernel Approach for PDE Discovery and Operator Learning

This article presents a three-step framework for learning and solving partial differential equations (PDEs) using kernel methods. Given a training set consisting of pairs of noisy PDE solutions and source/boundary terms on a mesh, kernel smoothing is utilized to denoise the data and approximate derivatives of the solution. This information is then used in a kernel regression model to learn the algebraic form of the PDE. The learned PDE is then used within a kernel based solver to approximate the solution of the PDE with a new source/boundary term, thereby constituting an operator learning framework. Numerical experiments compare the method to state-of-the-art algorithms and demonstrate its competitive performance.

stat.ML↗

arXiv2023

Bayesian Posterior Perturbation Analysis with Integral Probability Metrics

In recent years, Bayesian inference in large-scale inverse problems found in science, engineering and machine learning has gained significant attention. This paper examines the robustness of the Bayesian approach by analyzing the stability of posterior measures in relation to perturbations in the likelihood potential and the prior measure. We present new stability results using a family of integral probability metrics (divergences) akin to dual problems that arise in optimal transport. Our results stand out from previous works in three directions: (1) We construct new families of integral probability metrics that are adapted to the problem at hand; (2) These new metrics allow us to study both likelihood and prior perturbations in a convenient way; and (3) our analysis accommodates likelihood potentials that are only locally Lipschitz, making them applicable to a wide range of nonlinear inverse problems. Our theoretical findings are further reinforced through specific and novel examples where the approximation rates of posterior measures are obtained for different types of perturbations and provide a path towards the convergence analysis of recently adapted machine learning techniques for Bayesian inverse problems such as data-driven priors and neural network surrogates.

stat.ML↗

arXiv2022

An Optimal Transport Formulation of Bayes' Law for Nonlinear Filtering Algorithms

This paper presents a variational representation of the Bayes' law using optimal transportation theory. The variational representation is in terms of the optimal transportation between the joint distribution of the (state, observation) and their independent coupling. By imposing certain structure on the transport map, the solution to the variational problem is used to construct a Brenier-type map that transports the prior distribution to the posterior distribution for any value of the observation signal. The new formulation is used to derive the optimal transport form of the Ensemble Kalman filter (EnKF) for the discrete-time filtering problem and propose a novel extension of EnKF to the non-Gaussian setting utilizing input convex neural networks. Finally, the proposed methodology is used to derive the optimal transport form of the feedback particle filler (FPF) in the continuous-time limit, which constitutes its first variational construction without explicitly using the nonlinear filtering equation or Bayes' law.

math.OC↗

arXiv2022

Intrinsic Sparsity of Kantorovich Solutions

Let $X,Y$ be two finite sets of points having $\#X = m$ and $\#Y = n$ points with $μ= (1/m) \sum_{i=1}^{m} δ_{x_i}$ and $ν= (1/n) \sum_{j=1}^{n} δ_{y_j}$ being the associated uniform probability measures. A result of Birkhoff implies that if $m = n$, then the Kantorovich problem has a solution which also solves the Monge problem: optimal transport can be realized with a bijection $π: X \rightarrow Y$. This is impossible when $m \neq n$. We observe that when $m \neq n$, there exists a solution of the Kantorovich problem such that the mass of each point in $X$ is moved to at most $n/\gcd(m,n)$ different points in $Y$ and that, conversely, each point in $Y$ receives mass from at most $m/\gcd(m,n)$ points in $X$.

math.OC↗

arXiv2022

Spectral gaps and error estimates for infinite-dimensional Metropolis-Hastings with non-Gaussian priors

We study a class of Metropolis-Hastings algorithms for target measures that are absolutely continuous with respect to a large class of non-Gaussian prior measures on Banach spaces. The algorithm is shown to have a spectral gap in a Wasserstein-like semimetric weighted by a Lyapunov function. A number of error bounds are given for computationally tractable approximations of the algorithm including bounds on the closeness of Cesáro averages and other pathwise quantities via perturbation theory. Several applications illustrate the breadth of problems to which the results apply such as various likelihood approximations and perturbations of prior measures.

math.PR↗

arXiv2021

Solving and Learning Nonlinear PDEs with Gaussian Processes

We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian processes. The proposed approach: (1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and IPs; (2) has guaranteed convergence for a very general class of PDEs, and comes equipped with a path to compute error bounds for specific PDE approximations; (3) inherits the state-of-the-art computational complexity of linear solvers for dense kernel matrices. The main idea of our method is to approximate the solution of a given PDE as the maximum a posteriori (MAP) estimator of a Gaussian process conditioned on solving the PDE at a finite number of collocation points. Although this optimization problem is infinite-dimensional, it can be reduced to a finite-dimensional one by introducing additional variables corresponding to the values of the derivatives of the solution at collocation points; this generalizes the representer theorem arising in Gaussian process regression. The reduced optimization problem has the form of a quadratic objective function subject to nonlinear constraints; it is solved with a variant of the Gauss--Newton method. The resulting algorithm (a) can be interpreted as solving successive linearizations of the nonlinear PDE, and (b) in practice is found to converge in a small number of iterations (2 to 10), for a wide range of PDEs. Most traditional approaches to IPs interleave parameter updates with numerical solution of the PDE; our algorithm solves for both parameter and PDE solution simultaneously. Experiments on nonlinear elliptic PDEs, Burgers' equation, a regularized Eikonal equation, and an IP for permeability identification in Darcy flow illustrate the efficacy and scope of our framework.

math.NA↗

arXiv2021

Model Reduction and Neural Networks for Parametric PDEs

We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology. We also include numerical experiments which demonstrate the effectiveness of the method, showing convergence and robustness of the approximation scheme with respect to the size of the discretization, and compare it with existing algorithms from the literature; our examples include the mapping from coefficient to solution in a divergence form elliptic partial differential equation (PDE) problem, and the solution operator for viscous Burgers' equation.

math.NA↗

arXiv2021

Posterior Consistency of Semi-Supervised Regression on Graphs

Graph-based semi-supervised regression (SSR) is the problem of estimating the value of a function on a weighted graph from its values (labels) on a small subset of the vertices. This paper is concerned with the consistency of SSR in the context of classification, in the setting where the labels have small noise and the underlying graph weighting is consistent with well-clustered nodes. We present a Bayesian formulation of SSR in which the weighted graph defines a Gaussian prior, using a graph Laplacian, and the labeled data defines a likelihood. We analyze the rate of contraction of the posterior measure around the ground truth in terms of parameters that quantify the small label error and inherent clustering in the graph. We obtain bounds on the rates of contraction and illustrate their sharpness through numerical experiments. The analysis also gives insight into the choice of hyperparameters that enter the definition of the prior.

stat.ML↗

arXiv2020

Spectral Analysis Of Weighted Laplacians Arising In Data Clustering

Graph Laplacians computed from weighted adjacency matrices are widely used to identify geometric structure in data, and clusters in particular; their spectral properties play a central role in a number of unsupervised and semi-supervised learning algorithms. When suitably scaled, graph Laplacians approach limiting continuum operators in the large data limit. Studying these limiting operators, therefore, sheds light on learning algorithms. This paper is devoted to the study of a parameterized family of divergence form elliptic operators that arise as the large data limit of graph Laplacians. The link between a three-parameter family of graph Laplacians and a three-parameter family of differential operators is explained. The spectral properties of these differential operators are analyzed in the situation where the data comprises two nearly separated clusters, in a sense which is made precise. In particular, we investigate how the spectral gap depends on the three parameters entering the graph Laplacian, and on a parameter measuring the size of the perturbation from the perfectly clustered case. Numerical results are presented which exemplify and extend the analysis: the computations study situations in which there are two nearly separated clusters, but which violate the assumptions used in our theory; situations in which more than two clusters are present, also going beyond our theory; and situations which demonstrate the relevance of our studies of differential operators for the understanding of finite data problems via the graph Laplacian. The findings provide insight into parameter choices made in learning algorithms which are based on weighted adjacency matrices; they also provide the basis for analysis of the consistency of various unsupervised and semi-supervised learning algorithms, in the large data limit.

math.SP↗

arXiv2020

Consistency of semi-supervised learning algorithms on graphs: Probit and one-hot methods

Graph-based semi-supervised learning is the problem of propagating labels from a small number of labelled data points to a larger set of unlabelled data. This paper is concerned with the consistency of optimization-based techniques for such problems, in the limit where the labels have small noise and the underlying unlabelled data is well clustered. We study graph-based probit for binary classification, and a natural generalization of this method to multi-class classification using one-hot encoding. The resulting objective function to be optimized comprises the sum of a quadratic form defined through a rational function of the graph Laplacian, involving only the unlabelled data, and a fidelity term involving only the labelled data. The consistency analysis sheds light on the choice of the rational function defining the optimization.

stat.ML↗

arXiv2019

Simultaneous model calibration and source inversion in atmospheric dispersion models

We present a cost-effective method for model calibration and solution of source inversion problems in atmospheric dispersion modelling. We use Gaussian process emulations of atmospheric dispersion models within a Bayesian framework for solution of inverse problems. The model and source parameters are treated as unknowns and we obtain point estimates and approximation of uncertainties for sources while simultaneously calibrating the forward model. The method is validated in the context of an industrial case study involving emissions from a smelting operation for which cumulative monthly measurements of zinc particulate depositions are available.

math.NA↗

arXiv2019

Two Metropolis-Hastings algorithms for posterior measures with non-Gaussian priors in infinite dimensions

We introduce two classes of Metropolis-Hastings algorithms for sampling target measures that are absolutely continuous with respect to non-Gaussian prior measures on infinite-dimensional Hilbert spaces. In particular, we focus on certain classes of prior measures for which prior-reversible proposal kernels of the autoregressive type can be designed. We then use these proposal kernels to design algorithms that satisfy detailed balance with respect to the target measures. Afterwards, we introduce a new class of prior measures, called the Bessel-K priors, as a generalization of the gamma distribution to measures in infinite dimensions. The Bessel-K priors interpolate between well-known priors such as the gamma distribution and Besov priors and can model sparse or compressible parameters. We present concrete instances of our algorithms for the Bessel-K priors in the context of numerical examples in density estimation, finite-dimensional denoising and deconvolution on the circle.

stat.CO↗

arXiv2019

Geometric structure of graph Laplacian embeddings

We analyze the spectral clustering procedure for identifying coarse structure in a data set $x_1, \dots, x_n$, and in particular study the geometry of graph Laplacian embeddings which form the basis for spectral clustering algorithms. More precisely, we assume that the data is sampled from a mixture model supported on a manifold $\mathcal{M}$ embedded in $\mathbb{R}^d$, and pick a connectivity length-scale $\varepsilon>0$ to construct a kernelized graph Laplacian. We introduce a notion of a well-separated mixture model which only depends on the model itself, and prove that when the model is well separated, with high probability the embedded data set concentrates on cones that are centered around orthogonal vectors. Our results are meaningful in the regime where $\varepsilon = \varepsilon(n)$ is allowed to decay to zero at a slow enough rate as the number of data points grows. This rate depends on the intrinsic dimension of the manifold on which the data is supported.

math.SP↗

arXiv2018

A Bayesian approach for energy-based estimation of acoustic aberrations in high intensity focused ultrasound treatment

High intensity focused ultrasound is a non-invasive method for treatment of diseased tissue that uses a beam of ultrasound to generate heat within a small volume. A common challenge in application of this technique is that heterogeneity of the biological medium can defocus the ultrasound beam. Here we reduce the problem of refocusing the beam to the inverse problem of estimating the acoustic aberration due to the biological tissue from acoustic radiative force imaging data. We solve this inverse problem using a Bayesian framework with a hierarchical prior and solve the inverse problem using a Metropolis-within-Gibbs algorithm. The framework is tested using both synthetic and experimental datasets. We demonstrate that our approach has the ability to estimate the aberrations using small datasets, as little as 32 sonication tests, which can lead to significant speedup in the treatment process. Furthermore, our approach is compatible with a wide range of sonication tests and can be applied to other energy-based measurement techniques.

physics.med-ph↗

arXiv2017

Well-posed Bayesian Inverse Problems: Priors with Exponential Tails

We consider the well-posedness of Bayesian inverse problems when the prior measure has exponential tails. In particular, we consider the class of convex (log-concave) probability measures which include the Gaussian and Besov measures as well as certain classes of hierarchical priors. We identify appropriate conditions on the likelihood distribution and the prior measure which guarantee existence, uniqueness and stability of the posterior measure with respect to perturbations of the data. We also consider consistent approximations of the posterior such as discretization by projection. Finally, we present a general recipe for construction of convex priors on Banach spaces which will be of interest in practical applications where one often works with spaces such as $L^2$ or the continuous functions.

math.PR↗