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The Gaussian process (GP) model, which has been extensively applied as priors of functions, has demonstrated excellent performance. The specification of a large number of parameters affects the computational efficiency and the feasibility…
We study the close connection between rational functions that approximate a given Boolean function, and quantum algorithms that compute the same function using postselection. We show that the minimal degree of the former equals (up to a…
We propose a novel method for maximum likelihood-based parameter inference in nonlinear and/or non-Gaussian state space models. The method is an iterative procedure with three steps. At each iteration a particle filter is used to estimate…
In this paper we address the complexity of solving linear programming problems with a set of differential equations that converge to a fixed point that represents the optimal solution. Assuming a probabilistic model, where the inputs are…
Approximate Bayesian inference methods provide a powerful suite of tools for finding approximations to intractable posterior distributions. However, machine learning applications typically involve selecting actions, which -- in a Bayesian…
The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric…
Particle filters for data assimilation in nonlinear problems use "particles" (replicas of the underlying system) to generate a sequence of probability density functions (pdfs) through a Bayesian process. This can be expensive because a…
Hard optimization problems are often approached by finding approximate solutions. Here, we highlight the concept of proportional sampling and discuss how it can be used to improve the performance of stochastic algorithms for optimization.…
An approximation method is presented for probabilistic inference with continuous random variables. These problems can arise in many practical problems, in particular where there are "second order" probabilities. The approximation, based on…
The filtering equations govern the evolution of the conditional distribution of a signal process given partial, and possibly noisy, observations arriving sequentially in time. Their numerical approximation plays a central role in many…
Gradient flow in the 2-Wasserstein space is widely used to optimize functionals over probability distributions and is typically implemented using an interacting particle system with $n$ particles. Analyzing these algorithms requires showing…
Approximate Counting refers to the problem where we are given query access to a function $f : [N] \to \{0,1\}$, and we wish to estimate $K = #\{x : f(x) = 1\}$ to within a factor of $1+\epsilon$ (with high probability), while minimizing the…
In this article, we study Bayesian inverse problems with multi-layered Gaussian priors. We first describe the conditionally Gaussian layers in terms of a system of stochastic partial differential equations. We build the computational…
We present a randomized distributed approximation algorithm for the metric uncapacitated facility location problem. The algorithm is executed on a bipartite graph in the Congest model yielding a (1.861 + epsilon) approximation factor, where…
Context. Whenever correlation functions are used for inference about cosmological parameters in the context of a Bayesian analysis, the likelihood function of correlation functions needs to be known. Usually, it is approximated as a…
Near-optimal computational complexity of an adaptive stochastic Galerkin method with independently refined spatial meshes for elliptic partial differential equations is shown. The method takes advantage of multilevel structure in expansions…
We investigate a new sampling scheme aimed at improving the performance of particle filters whenever (a) there is a significant mismatch between the assumed model dynamics and the actual system, or (b) the posterior probability tends to…
Approximate Bayesian computation methods are useful for generative models with intractable likelihoods. These methods are however sensitive to the dimension of the parameter space, requiring exponentially increasing resources as this…
We study the performance of adaptive Fourier-Galerkin methods in a periodic box in $\mathbb{R}^d$ with dimension $d\ge 1$. These methods offer unlimited approximation power only restricted by solution and data regularity. They are of…
Given a decision process based on the approximate probability density function returned by a data assimilation algorithm, an interaction level between the decision making level and the data assimilation level is designed to incorporate the…