This approach is however difficult to extend to the case in which one wants to compute an OT map (a Monge map) between the original measures.

Conditioning, the central operation in Bayesian statistics, is formalised by the notion of disintegration of measures. However, due to the implicit nature of their definition, constructing disintegrations is often difficult. A folklore result in machine learning conflates the construction of a disintegration with the restriction of probability density functions onto the subset of events that are consistent with a given observation. We provide a comprehensive set of mathematical tools which can be used to construct disintegrations and apply these to find densities of disintegrations on differentiable manifolds. Using our results, we provide a disturbingly simple example in which the restricted density and the disintegration density drastically disagree. Motivated by applications in approximate Bayesian inference and Bayesian inverse problems, we further study the modes of disintegrations. We show that the recently introduced notion of a "conditional mode" does not coincide in general with the modes of the conditional measure obtained through disintegration, but rather the modes of the restricted measure. We also discuss the implications of the discrepancy between the two measures in practice, advocating for the utility of both approaches depending on the modelling context.

Among generative neural models, flow matching techniques stand out for their simple applicability and good scaling properties. Here, velocity fields of curves connecting a simple latent and a target distribution are learned. Then the corresponding ordinary differential equation can be used to sample from a target distribution, starting in samples from the latent one. This paper reviews from a mathematical point of view different techniques to learn the velocity fields of absolutely continuous curves in the Wasserstein geometry. We show how the velocity fields can be characterized and learned via i) transport plans (couplings) between latent and target distributions, ii) Markov kernels and iii) stochastic processes, where the latter two include the coupling approach, but are in general broader. Besides this main goal, we show how flow matching can be used for solving Bayesian inverse problems, where the definition of conditional Wasserstein distances plays a central role. Finally, we briefly address continuous normalizing flows and score matching techniques, which approach the learning of velocity fields of curves from other directions.

We consider the problem of estimating a structured multivariate density, subject to Markov conditions implied by an undirected graph. In the worst case, without Markovian assumptions, this problem suffers from the curse of dimensionality. Our main result shows how the curse of dimensionality can be avoided or greatly alleviated under the Markov property, and applies to arbitrary graphs. While existing results along these lines focus on sparsity or manifold assumptions, we introduce a new graphical quantity called "graph resilience" and show how it controls the sample complexity. Surprisingly, although one might expect the sample complexity of this problem to scale with local graph parameters such as the degree, this turns out not to be the case. Through explicit examples, we compute uniform deviation bounds and illustrate how the curse of dimensionality in density estimation can thus be circumvented. Notable examples where the rate improves substantially include sequential, hierarchical, and spatial data.

In the context of optimal transport methods, the subspace detour approach was recently presented by Muzellec and Cuturi (2019). It consists in building a nearly optimal transport plan in the measures space from an optimal transport plan in a wisely chosen subspace, onto which the original measures are projected. The contribution of this paper is to extend this category of methods to the Gromov-Wasserstein problem, which is a particular type of transport distance involving the inner geometry of the compared distributions. After deriving the associated formalism and properties, we also discuss a specific cost for which we can show connections with the Knothe-Rosenblatt rearrangement. We finally give an experimental illustration on a shape matching problem.

We consider the use of Gaussian process (GP) priors for solving inverse problems in a Bayesian framework. As is well known, the computational complexity of GPs scales cubically in the number of datapoints. We here show that in the context of inverse problems involving integral operators, one faces additional difficulties that hinder inversion on large grids. Furthermore, in that context, covariance matrices can become too large to be stored. By leveraging results about sequential disintegrations of Gaussian measures, we are able to introduce an implicit representation of posterior covariance matrices that reduces the memory footprint by only storing low rank intermediate matrices, while allowing individual elements to be accessed on-the-fly without needing to build full posterior covariance matrices. Moreover, it allows for fast sequential inclusion of new observations. These features are crucial when considering sequential experimental design tasks. We demonstrate our approach by computing sequential data collection plans for excursion set recovery for a gravimetric inverse problem, where the goal is to provide fine resolution estimates of high density regions inside the Stromboli volcano, Italy. Sequential data collection plans are computed by extending the weighted integrated variance reduction (wIVR) criterion to inverse problems. Our results show that this criterion is able to significantly reduce the uncertainty on the excursion volume, reaching close to minimal levels of residual uncertainty. Overall, our techniques allow the advantages of probabilistic models to be brought to bear on large-scale inverse problems arising in the natural sciences.

In clinical trials and other applications, we often see regions of the feature space that appear to exhibit interesting behaviour, but it is unclear whether these observed phenomena are reflected at the population level. Focusing on a regression setting, we consider the subgroup selection challenge of identifying a region of the feature space on which the regression function exceeds a pre-determined threshold. We formulate the problem as one of constrained optimisation, where we seek a low-complexity, data-dependent selection set on which, with a guaranteed probability, the regression function is uniformly at least as large as the threshold; subject to this constraint, we would like the region to contain as much mass under the marginal feature distribution as possible. This leads to a natural notion of regret, and our main contribution is to determine the minimax optimal rate for this regret in both the sample size and the Type I error probability. The rate involves a delicate interplay between parameters that control the smoothness of the regression function, as well as exponents that quantify the extent to which the optimal selection set at the population level can be approximated by families of well-behaved subsets. Finally, we expand the scope of our previous results by illustrating how they may be generalised to a treatment and control setting, where interest lies in the heterogeneous treatment effect.