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  1. Article ; Online: A Particle Method for Solving Fredholm Equations of the First Kind

    Crucinio, Francesca R. / Doucet, Arnaud / Johansen, Adam M.

    Journal of the American Statistical Association. 2023 Apr. 3, v. 118, no. 542 p.937-947

    2023  

    Abstract: Fredholm integral equations of the first kind are the prototypical example of ill-posed linear inverse problems. They model, among other things, reconstruction of distorted noisy observations and indirect density estimation and also appear in ... ...

    Abstract Fredholm integral equations of the first kind are the prototypical example of ill-posed linear inverse problems. They model, among other things, reconstruction of distorted noisy observations and indirect density estimation and also appear in instrumental variable regression. However, their numerical solution remains a challenging problem. Many techniques currently available require a preliminary discretization of the domain of the solution and make strong assumptions about its regularity. For example, the popular expectation maximization smoothing (EMS) scheme requires the assumption of piecewise constant solutions which is inappropriate for most applications. We propose here a novel particle method that circumvents these two issues. This algorithm can be thought of as a Monte Carlo approximation of the EMS scheme which not only performs an adaptive stochastic discretization of the domain but also results in smooth approximate solutions. We analyze the theoretical properties of the EMS iteration and of the corresponding particle algorithm. Compared to standard EMS, we show experimentally that our novel particle method provides state-of-the-art performance for realistic systems, including motion deblurring and reconstruction of cross-section images of the brain from positron emission tomography.
    Keywords algorithms ; brain ; positron-emission tomography ; Expectation maximization ; Indirect density estimation ; Inverse problems ; Monte Carlo methods ; Positron emission tomography
    Language English
    Dates of publication 2023-0403
    Size p. 937-947.
    Publishing place Taylor & Francis
    Document type Article ; Online
    ZDB-ID 2064981-2
    ISSN 1537-274X
    ISSN 1537-274X
    DOI 10.1080/01621459.2021.1962328
    Database NAL-Catalogue (AGRICOLA)

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  2. Article ; Online: Differentiable samplers for deep latent variable models.

    Doucet, Arnaud / Moulines, Eric / Thin, Achille

    Philosophical transactions. Series A, Mathematical, physical, and engineering sciences

    2023  Volume 381, Issue 2247, Page(s) 20220147

    Abstract: Latent variable models are a popular class of models in statistics. Combined with neural networks to improve their expressivity, the resulting deep latent variable models have also found numerous applications in machine learning. A drawback of these ... ...

    Abstract Latent variable models are a popular class of models in statistics. Combined with neural networks to improve their expressivity, the resulting deep latent variable models have also found numerous applications in machine learning. A drawback of these models is that their likelihood function is intractable so approximations have to be carried out to perform inference. A standard approach consists of maximizing instead an evidence lower bound (ELBO) obtained based on a variational approximation of the posterior distribution of the latent variables. The standard ELBO can, however, be a very loose bound if the variational family is not rich enough. A generic strategy to tighten such bounds is to rely on an unbiased low-variance Monte Carlo estimate of the evidence. We review here some recent importance sampling, Markov chain Monte Carlo and sequential Monte Carlo strategies that have been proposed to achieve this. This article is part of the theme issue 'Bayesian inference: challenges, perspectives, and prospects'.
    Language English
    Publishing date 2023-03-27
    Publishing country England
    Document type Journal Article ; Review
    ZDB-ID 208381-4
    ISSN 1471-2962 ; 0080-4614 ; 0264-3820 ; 0264-3952 ; 1364-503X
    ISSN (online) 1471-2962
    ISSN 0080-4614 ; 0264-3820 ; 0264-3952 ; 1364-503X
    DOI 10.1098/rsta.2022.0147
    Database MEDical Literature Analysis and Retrieval System OnLINE

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    In stock of ZB MED Bonn / Germany

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  3. Book ; Online: Denoising Diffusion Samplers

    Vargas, Francisco / Grathwohl, Will / Doucet, Arnaud

    2023  

    Abstract: Denoising diffusion models are a popular class of generative models providing state-of-the-art results in many domains. One adds gradually noise to data using a diffusion to transform the data distribution into a Gaussian distribution. Samples from the ... ...

    Abstract Denoising diffusion models are a popular class of generative models providing state-of-the-art results in many domains. One adds gradually noise to data using a diffusion to transform the data distribution into a Gaussian distribution. Samples from the generative model are then obtained by simulating an approximation of the time-reversal of this diffusion initialized by Gaussian samples. Practically, the intractable score terms appearing in the time-reversed process are approximated using score matching techniques. We explore here a similar idea to sample approximately from unnormalized probability density functions and estimate their normalizing constants. We consider a process where the target density diffuses towards a Gaussian. Denoising Diffusion Samplers (DDS) are obtained by approximating the corresponding time-reversal. While score matching is not applicable in this context, we can leverage many of the ideas introduced in generative modeling for Monte Carlo sampling. Existing theoretical results from denoising diffusion models also provide theoretical guarantees for DDS. We discuss the connections between DDS, optimal control and Schr\"odinger bridges and finally demonstrate DDS experimentally on a variety of challenging sampling tasks.

    Comment: In The Eleventh International Conference on Learning Representations, 2023
    Keywords Computer Science - Machine Learning ; Statistics - Machine Learning
    Subject code 541 ; 519
    Publishing date 2023-02-27
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  4. Book ; Online: Error Bounds for Flow Matching Methods

    Benton, Joe / Deligiannidis, George / Doucet, Arnaud

    2023  

    Abstract: Score-based generative models are a popular class of generative modelling techniques relying on stochastic differential equations (SDE). From their inception, it was realized that it was also possible to perform generation using ordinary differential ... ...

    Abstract Score-based generative models are a popular class of generative modelling techniques relying on stochastic differential equations (SDE). From their inception, it was realized that it was also possible to perform generation using ordinary differential equations (ODE) rather than SDE. This led to the introduction of the probability flow ODE approach and denoising diffusion implicit models. Flow matching methods have recently further extended these ODE-based approaches and approximate a flow between two arbitrary probability distributions. Previous work derived bounds on the approximation error of diffusion models under the stochastic sampling regime, given assumptions on the $L^2$ loss. We present error bounds for the flow matching procedure using fully deterministic sampling, assuming an $L^2$ bound on the approximation error and a certain regularity condition on the data distributions.
    Keywords Statistics - Machine Learning ; Computer Science - Machine Learning
    Subject code 518
    Publishing date 2023-05-26
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  5. Book ; Online: Causal Falsification of Digital Twins

    Cornish, Rob / Taufiq, Muhammad Faaiz / Doucet, Arnaud / Holmes, Chris

    2023  

    Abstract: Digital twins are virtual systems designed to predict how a real-world process will evolve in response to interventions. This modelling paradigm holds substantial promise in many applications, but rigorous procedures for assessing their accuracy are ... ...

    Abstract Digital twins are virtual systems designed to predict how a real-world process will evolve in response to interventions. This modelling paradigm holds substantial promise in many applications, but rigorous procedures for assessing their accuracy are essential for safety-critical settings. We consider how to assess the accuracy of a digital twin using real-world data. We formulate this as causal inference problem, which leads to a precise definition of what it means for a twin to be "correct" appropriate for many applications. Unfortunately, fundamental results from causal inference mean observational data cannot be used to certify that a twin is correct in this sense unless potentially tenuous assumptions are made, such as that the data are unconfounded. To avoid these assumptions, we propose instead to find situations in which the twin is not correct, and present a general-purpose statistical procedure for doing so. Our approach yields reliable and actionable information about the twin under only the assumption of an i.i.d. dataset of observational trajectories, and remains sound even if the data are confounded. We apply our methodology to a large-scale, real-world case study involving sepsis modelling within the Pulse Physiology Engine, which we assess using the MIMIC-III dataset of ICU patients.
    Keywords Statistics - Methodology ; Computer Science - Computational Engineering ; Finance ; and Science ; Computer Science - Machine Learning ; Statistics - Applications
    Subject code 006
    Publishing date 2023-01-17
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  6. Book ; Online: Tree-Based Diffusion Schr\"odinger Bridge with Applications to Wasserstein Barycenters

    Noble, Maxence / De Bortoli, Valentin / Doucet, Arnaud / Durmus, Alain

    2023  

    Abstract: Multi-marginal Optimal Transport (mOT), a generalization of OT, aims at minimizing the integral of a cost function with respect to a distribution with some prescribed marginals. In this paper, we consider an entropic version of mOT with a tree-structured ...

    Abstract Multi-marginal Optimal Transport (mOT), a generalization of OT, aims at minimizing the integral of a cost function with respect to a distribution with some prescribed marginals. In this paper, we consider an entropic version of mOT with a tree-structured quadratic cost, i.e., a function that can be written as a sum of pairwise cost functions between the nodes of a tree. To address this problem, we develop Tree-based Diffusion Schr\"odinger Bridge (TreeDSB), an extension of the Diffusion Schr\"odinger Bridge (DSB) algorithm. TreeDSB corresponds to a dynamic and continuous state-space counterpart of the multimarginal Sinkhorn algorithm. A notable use case of our methodology is to compute Wasserstein barycenters which can be recast as the solution of a mOT problem on a star-shaped tree. We demonstrate that our methodology can be applied in high-dimensional settings such as image interpolation and Bayesian fusion.
    Keywords Statistics - Machine Learning ; Computer Science - Machine Learning ; Mathematics - Probability
    Publishing date 2023-05-25
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  7. Book ; Online: Diffusion Schr\"odinger Bridge Matching

    Shi, Yuyang / De Bortoli, Valentin / Campbell, Andrew / Doucet, Arnaud

    2023  

    Abstract: Solving transport problems, i.e. finding a map transporting one given distribution to another, has numerous applications in machine learning. Novel mass transport methods motivated by generative modeling have recently been proposed, e.g. Denoising ... ...

    Abstract Solving transport problems, i.e. finding a map transporting one given distribution to another, has numerous applications in machine learning. Novel mass transport methods motivated by generative modeling have recently been proposed, e.g. Denoising Diffusion Models (DDMs) and Flow Matching Models (FMMs) implement such a transport through a Stochastic Differential Equation (SDE) or an Ordinary Differential Equation (ODE). However, while it is desirable in many applications to approximate the deterministic dynamic Optimal Transport (OT) map which admits attractive properties, DDMs and FMMs are not guaranteed to provide transports close to the OT map. In contrast, Schr\"odinger bridges (SBs) compute stochastic dynamic mappings which recover entropy-regularized versions of OT. Unfortunately, existing numerical methods approximating SBs either scale poorly with dimension or accumulate errors across iterations. In this work, we introduce Iterative Markovian Fitting (IMF), a new methodology for solving SB problems, and Diffusion Schr\"odinger Bridge Matching (DSBM), a novel numerical algorithm for computing IMF iterates. DSBM significantly improves over previous SB numerics and recovers as special/limiting cases various recent transport methods. We demonstrate the performance of DSBM on a variety of problems.
    Keywords Statistics - Machine Learning ; Computer Science - Machine Learning
    Subject code 518
    Publishing date 2023-03-29
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  8. Book ; Online: Marginal Density Ratio for Off-Policy Evaluation in Contextual Bandits

    Taufiq, Muhammad Faaiz / Doucet, Arnaud / Cornish, Rob / Ton, Jean-Francois

    2023  

    Abstract: Off-Policy Evaluation (OPE) in contextual bandits is crucial for assessing new policies using existing data without costly experimentation. However, current OPE methods, such as Inverse Probability Weighting (IPW) and Doubly Robust (DR) estimators, ... ...

    Abstract Off-Policy Evaluation (OPE) in contextual bandits is crucial for assessing new policies using existing data without costly experimentation. However, current OPE methods, such as Inverse Probability Weighting (IPW) and Doubly Robust (DR) estimators, suffer from high variance, particularly in cases of low overlap between target and behavior policies or large action and context spaces. In this paper, we introduce a new OPE estimator for contextual bandits, the Marginal Ratio (MR) estimator, which focuses on the shift in the marginal distribution of outcomes $Y$ instead of the policies themselves. Through rigorous theoretical analysis, we demonstrate the benefits of the MR estimator compared to conventional methods like IPW and DR in terms of variance reduction. Additionally, we establish a connection between the MR estimator and the state-of-the-art Marginalized Inverse Propensity Score (MIPS) estimator, proving that MR achieves lower variance among a generalized family of MIPS estimators. We further illustrate the utility of the MR estimator in causal inference settings, where it exhibits enhanced performance in estimating Average Treatment Effects (ATE). Our experiments on synthetic and real-world datasets corroborate our theoretical findings and highlight the practical advantages of the MR estimator in OPE for contextual bandits.

    Comment: Conference on Neural Information Processing Systems (NeurIPS 2023)
    Keywords Statistics - Machine Learning ; Computer Science - Machine Learning ; Statistics - Methodology
    Subject code 310
    Publishing date 2023-12-03
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  9. Book ; Online: Nearly $d$-Linear Convergence Bounds for Diffusion Models via Stochastic Localization

    Benton, Joe / De Bortoli, Valentin / Doucet, Arnaud / Deligiannidis, George

    2023  

    Abstract: Denoising diffusions are a powerful method to generate approximate samples from high-dimensional data distributions. Recent results provide polynomial bounds on their convergence rate, assuming $L^2$-accurate scores. Until now, the tightest bounds were ... ...

    Abstract Denoising diffusions are a powerful method to generate approximate samples from high-dimensional data distributions. Recent results provide polynomial bounds on their convergence rate, assuming $L^2$-accurate scores. Until now, the tightest bounds were either superlinear in the data dimension or required strong smoothness assumptions. We provide the first convergence bounds which are linear in the data dimension (up to logarithmic factors) assuming only finite second moments of the data distribution. We show that diffusion models require at most $\tilde O(\frac{d \log^2(1/\delta)}{\varepsilon^2})$ steps to approximate an arbitrary distribution on $\mathbb{R}^d$ corrupted with Gaussian noise of variance $\delta$ to within $\varepsilon^2$ in KL divergence. Our proof extends the Girsanov-based methods of previous works. We introduce a refined treatment of the error from discretizing the reverse SDE inspired by stochastic localization.
    Keywords Statistics - Machine Learning ; Computer Science - Machine Learning
    Subject code 519 ; 518
    Publishing date 2023-08-07
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  10. Book ; Online: Importance Weighting Approach in Kernel Bayes' Rule

    Xu, Liyuan / Chen, Yutian / Doucet, Arnaud / Gretton, Arthur

    2022  

    Abstract: We study a nonparametric approach to Bayesian computation via feature means, where the expectation of prior features is updated to yield expected kernel posterior features, based on regression from learned neural net or kernel features of the ... ...

    Abstract We study a nonparametric approach to Bayesian computation via feature means, where the expectation of prior features is updated to yield expected kernel posterior features, based on regression from learned neural net or kernel features of the observations. All quantities involved in the Bayesian update are learned from observed data, making the method entirely model-free. The resulting algorithm is a novel instance of a kernel Bayes' rule (KBR), based on importance weighting. This results in superior numerical stability to the original approach to KBR, which requires operator inversion. We show the convergence of the estimator using a novel consistency analysis on the importance weighting estimator in the infinity norm. We evaluate KBR on challenging synthetic benchmarks, including a filtering problem with a state-space model involving high dimensional image observations. Importance weighted KBR yields uniformly better empirical performance than the original KBR, and competitive performance with other competing methods.
    Keywords Statistics - Machine Learning ; Computer Science - Machine Learning
    Subject code 310
    Publishing date 2022-02-04
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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