arXiv subjects

Francesca R. Crucinio

Publications and source records attributed to Francesca R. Crucinio.

12 recordsLinked to original sources

arXiv2026

Solving Fredholm Integral Equations of the Second Kind via Wasserstein Gradient Flows

Motivated by a recent method for approximate solution of Fredholm equations of the first kind, we develop a corresponding method for a class of Fredholm equations of the \emph{second kind}. In particular, we consider the class of equations for which the solution is a probability measure. The approach centres around specifying a functional whose gradient flow admits a minimizer corresponding to a regularized version of the solution of the underlying equation and using a mean-field particle system to approximately simulate that flow. Theoretical support for the method is presented, along with some illustrative numerical results.

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arXiv2026

Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flows

We consider the problem of sampling from a probability distribution $π$. It is well known that this can be written as an optimisation problem over the space of probability distribution in which we aim to minimise the Kullback--Leibler divergence from $π$. We consider several partial differential equations (PDEs) whose solution is a minimiser of the Kullback--Leibler divergence from $π$ and connect them to well-known Monte Carlo algorithms. We focus in particular on PDEs obtained by considering the Wasserstein--Fisher--Rao geometry over the space of probabilities and show that these lead to a natural implementation using importance sampling and sequential Monte Carlo. We propose a novel algorithm to approximate the Wasserstein--Fisher--Rao flow of the Kullback--Leibler divergence and conduct an extensive empirical study to identify when these algorithms outperforms other popular Monte Carlo algorithms.

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arXiv2025

A mirror descent approach to maximum likelihood estimation in latent variable models

We introduce an approach based on mirror descent and sequential Monte Carlo (SMC) to perform joint parameter inference and posterior estimation in latent variable models. This approach is based on minimisation of a functional over the parameter space and the space of probability distributions and, contrary to other popular approaches, can be implemented when the latent variable takes values in discrete spaces. We provide a detailed theoretical analysis of both the mirror descent algorithm and its approximation via SMC. We experimentally show that the proposed algorithm outperforms standard expectation maximisation algorithms and is competitive with other popular methods for real-valued latent variables.

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arXiv2025

Proximal Interacting Particle Langevin Algorithms

We introduce a class of algorithms, termed proximal interacting particle Langevin algorithms (PIPLA), for inference and learning in latent variable models whose joint probability density is non-differentiable. Leveraging proximal Markov chain Monte Carlo techniques and interacting particle Langevin algorithms, we propose three algorithms tailored to the problem of estimating parameters in a non-differentiable statistical model. We prove nonasymptotic bounds for the parameter estimates produced by the different algorithms in the strongly log-concave setting and provide comprehensive numerical experiments on various models to demonstrate the effectiveness of the proposed methods. In particular, we demonstrate the utility of our family of algorithms for sparse Bayesian logistic regression, training of sparse Bayesian neural networks or neural networks with non-differentiable activation functions, image deblurring, and sparse matrix completion. Our theory and experiments together show that PIPLA family can be the de facto choice for parameter estimation problems in non-differentiable latent variable models.

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arXiv2025

Towards a turnkey approach to unbiased Monte Carlo estimation of smooth functions of expectations

Given a smooth function $f$, we develop a general approach to turn Monte Carlo samples with expectation $m$ into an unbiased estimate of $f(m)$. Specifically, we develop estimators that are based on randomly truncating the Taylor series expansion of $f$ and estimating the coefficients of the truncated series. We derive their properties and propose a strategy to set their tuning parameters -- which depend on $m$ -- automatically, with a view to make the whole approach simple to use. We develop our methods for the specific functions $f(x)=\log x$ and $f(x)=1/x$, as they arise in several statistical applications such as maximum likelihood estimation of latent variable models and Bayesian inference for un-normalised models. Detailed numerical studies are performed for a range of applications to determine how competitive and reliable the proposed approach is.

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arXiv2024

Optimal Scaling Results for Moreau-Yosida Metropolis-adjusted Langevin Algorithms

We consider a recently proposed class of MCMC methods which uses proximity maps instead of gradients to build proposal mechanisms which can be employed for both differentiable and non-differentiable targets. These methods have been shown to be stable for a wide class of targets, making them a valuable alternative to Metropolis-adjusted Langevin algorithms (MALA); and have found wide application in imaging contexts. The wider stability properties are obtained by building the Moreau-Yosida envelope for the target of interest, which depends on a parameter $λ$. In this work, we investigate the optimal scaling problem for this class of algorithms, which encompasses MALA, and provide practical guidelines for the implementation of these methods.

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arXiv2024

A connection between Tempering and Entropic Mirror Descent

This paper explores the connections between tempering (for Sequential Monte Carlo; SMC) and entropic mirror descent to sample from a target probability distribution whose unnormalized density is known. We establish that tempering SMC corresponds to entropic mirror descent applied to the reverse Kullback-Leibler (KL) divergence and obtain convergence rates for the tempering iterates. Our result motivates the tempering iterates from an optimization point of view, showing that tempering can be seen as a descent scheme of the KL divergence with respect to the Fisher-Rao geometry, in contrast to Langevin dynamics that perform descent of the KL with respect to the Wasserstein-2 geometry. We exploit the connection between tempering and mirror descent iterates to justify common practices in SMC and derive adaptive tempering rules that improve over other alternative benchmarks in the literature.

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arXiv2024

Solving Fredholm Integral Equations of the First Kind via Wasserstein Gradient Flows

Solving Fredholm equations of the first kind is crucial in many areas of the applied sciences. In this work we adopt a probabilistic and variational point of view by considering a minimization problem in the space of probability measures with an entropic regularization. Contrary to classical approaches which discretize the domain of the solutions, we introduce an algorithm to asymptotically sample from the unique solution of the regularized minimization problem. As a result our estimators do not depend on any underlying grid and have better scalability properties than most existing methods. Our algorithm is based on a particle approximation of the solution of a McKean--Vlasov stochastic differential equation associated with the Wasserstein gradient flow of our variational formulation. We prove the convergence towards a minimizer and provide practical guidelines for its numerical implementation. Finally, our method is compared with other approaches on several examples including density deconvolution and epidemiology.

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arXiv2023

The divide-and-conquer sequential Monte Carlo algorithm: theoretical properties and limit theorems

We provide a comprehensive characterisation of the theoretical properties of the divide-and-conquer sequential Monte Carlo (DaC-SMC) algorithm. We firmly establish it as a well-founded method by showing that it possesses the same basic properties as conventional sequential Monte Carlo (SMC) algorithms do. In particular, we derive pertinent laws of large numbers, $L^p$ inequalities, and central limit theorems; and we characterize the bias in the normalized estimates produced by the algorithm and argue the absence thereof in the unnormalized ones. We further consider its practical implementation and several interesting variants; obtain expressions for its globally and locally optimal intermediate targets, auxiliary measures, and proposal kernels; and show that, in comparable conditions, DaC-SMC proves more statistically efficient than its direct SMC analogue. We close the paper with a discussion of our results, open questions, and future research directions.

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arXiv2023

Properties of Marginal Sequential Monte Carlo Methods

We provide a framework which admits a number of ``marginal'' sequential Monte Carlo (SMC) algorithms as particular cases -- including the marginal particle filter [Klaas et al., 2005, in: Proceedings of Uncertainty in Artificial Intelligence, pp. 308--315], , the independent particle filter [Lin et al., 2005, Journal of the American Statistical Association 100, pp. 1412--1421] and linear-cost Approximate Bayesian Computation SMC [Sisson et al., 2007, Proceedings of the National Academy of Sciences (USA) 104, pp. 1760--1765.]. We provide conditions under which such algorithms obey laws of large numbers and central limit theorems and provide some further asymptotic characterizations. Finally, it is shown that the asymptotic variance of a class of estimators associated with certain marginal SMC algorithms is never greater than that of the estimators provided by a standard SMC algorithm using the same proposal distributions.

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arXiv2022

A divide and conquer sequential Monte Carlo approach to high dimensional filtering

We propose a divide-and-conquer approach to filtering which decomposes the state variable into low-dimensional components to which standard particle filtering tools can be successfully applied and recursively merges them to recover the full filtering distribution. It is less dependent upon factorization of transition densities and observation likelihoods than competing approaches and can be applied to a broader class of models. Performance is compared with state-of-the-art methods on a benchmark problem and it is demonstrated that the proposed method is broadly comparable in settings in which those methods are applicable, and that it can be applied in settings in which they cannot.

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arXiv2021

Product-form estimators: exploiting independence to scale up Monte Carlo

We introduce a class of Monte Carlo estimators that aim to overcome the rapid growth of variance with dimension often observed for standard estimators by exploiting the target's independence structure. We identify the most basic incarnations of these estimators with a class of generalized U-statistics, and thus establish their unbiasedness, consistency, and asymptotic normality. Moreover, we show that they obtain the minimum possible variance amongst a broad class of estimators; and we investigate their computational cost and delineate the settings in which they are most efficient. We exemplify the merger of these estimators with other well-known Monte Carlo estimators so as to better adapt the latter to the target's independence structure and improve their performance. We do this via three simple mergers: one with importance sampling, another with importance sampling squared, and a final one with pseudo-marginal Metropolis-Hasting. In all cases, we show that the resulting estimators are well-founded and achieve lower variances than their standard counterparts. Lastly, we illustrate the various variance reductions through several examples.

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