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

TL;DR

This work presents a probabilistic and variational framework for solving Fredholm integral equations of the first kind by minimizing an entropically regularized functional over probability measures. It derives a Wasserstein gradient flow and a corresponding McKean–Vlasov SDE, whose particle system provides a grid-free, scalable means to approximate the minimizer of the surrogate functional, with convergence guarantees and stability under empirical observations. The authors establish existence/uniqueness of minimizers for positive regularization, connect the method to maximum entropy, generalized Bayesian inference, and Tikhonov regularization, and supply thorough numerical guidelines. Empirical results in density deconvolution, epidemiology, and CT demonstrate competitive or superior performance to state-of-the-art methods, especially in higher dimensions or under model misspecification, and the approach enables leveraging problem-specific information via a reference measure $\pi_0$.

Abstract

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.

Figures (9)

• Figure 1: Average accuracy and runtime for FE-WGF, SMC-EMS and DKDE with number of particles $N$ ranging between $10^2$ and $10^4$. The shaded regions represent a interval of two standard deviations over 100 repetitions centred at the average $\mathop{\mathrm{ISE}}\nolimits$.
• Figure 2: Distribution of $\mathop{\mathrm{MSE}}\nolimits$ as a function of runtime (in $\log$ seconds) for FE-WGF, SMC-EMS and DKDE. The number of particles $N$ ranges between 100 and 10,000.
• Figure 3: Example fit of the reconstructions of the synthetic incidence curve \ref{['eq:incidence']} and of the corresponding reconstruction of the number of cases.
• Figure 4: Reconstruction of a lung CT scan via FE-WGF and FBP. FBP provides reconstructions which preserve sharp edges but present speckle noise, while the reconstructions obtained with FE-WGF are smooth but with blurry edges.
• Figure 5: Distribution of $\mathop{\mathrm{ISE}}\nolimits$ ratios ($\mathop{\mathrm{ISE}}\nolimits$ of OSL-EM divided by $\mathop{\mathrm{ISE}}\nolimits$ of FE-WGF) over 100 repetitions. The runtime of FE-WGF is 1.3 times that of OSL-EM on average, while the average gain ranges from $1.3$ for $d=1$ to $\approx 30$ for $d=3, 4$.

Theorems & Definitions (47)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof