Joanna Marks

We develop interacting particle algorithms for learning latent variable models with energy-based priors. To do so, we leverage recent developments in particle-based methods for solving maximum marginal likelihood estimation (MMLE) problems. Specifically, we provide a continuous-time framework for learning latent energy-based models, by defining stochastic differential equations (SDEs) that provably solve the MMLE problem. We obtain a practical algorithm as a discretisation of these SDEs and provide theoretical guarantees for the convergence of the proposed algorithm. Finally, we demonstrate the empirical effectiveness of our method on synthetic and image datasets.

The Gromov-Wasserstein (GW) distance provides a principled framework for aligning metric measure (mm) spaces based solely on their intrinsic structure. Its ability to identify isomorphic representations of distributions across spaces renders it valuable for comparing data where equality up to isomorphism occurs naturally such as in graphs or, more generally, distributions on graphs. Recently, a type of dual form for the GW distance between Euclidean distributions with the squared Euclidean or inner product costs was derived, spurring the development of new statistical and algorithmic results for this setting. This work furnishes a novel duality result for GW distances with and without entropic regularization that is applicable to all finitely supported mm spaces. Leveraging this result, we derive the sample complexity of empirical GW distances between finite mm spaces, as well as limit distributions under proper centering and scaling. Furthermore, we propose new algorithms for solving the regularized GW problem which are subject to formal convergence guarantees. These statistical and algorithmic advancements give rise to a principled and efficient framework for testing whether two distributions on the set of graphs with a fixed number of nodes are isomorphic based on samples.

The Gromov-Wasserstein (GW) distance measures the discrepancy between metric measure (mm) spaces and identifies optimal alignments between them based solely on their intrinsic structure. Since it identifies isomorphic mm spaces, it provides a natural notion of distance for heterogeneous datasets which may admit isomorphic representations. In order to accelerate computation of GW distances, many practitioners employ entropic regularization to obtain an Entropic GW (EGW) problem. The most popular EGW solver is the Mirror Descent (MD) algorithm, which reduces EGW computations to an iterative process where an entropic optimal transport (EOT) problem is solved at each iteration. Despite its widespread use, the convergence of MD for this problem has only been established for restricted classes of costs. On the other hand, a recently proposed dual gradient method is available for general costs, but requires a choice of step size which depends on the regularization parameter. To address these two issues, we introduce Averaged Mirror Descent (AMD), which averages consecutive MD steps, and prove its convergence for arbitrary costs. Then, we establish that the dual gradient method with a fixed step size also converges for arbitrary costs at the cost of a more complicated iteration. In both cases, we also account for inexact iterations which are inescapable in practice. We compare the empirical performance of these methods across various settings and, in particular, show that AMD and the dual gradient method both converge on an example where classical MD fails.