Transport-based density estimation methods are receiving growing interest because of their ability to efficiently generate samples from the approximated density. We further invertigate the sequential transport maps framework proposed from arXiv:2106.04170 arXiv:2303.02554, which builds on a sequence of composed Knothe-Rosenblatt (KR) maps. Each of those maps are built by first estimating an intermediate density of moderate complexity, and then by computing the exact KR map from a reference density to the precomputed approximate density. In our work, we explore the use of Sum-of-Squares (SoS) densities and $\alpha$-divergences for approximating the intermediate densities. Combining SoS densities with $\alpha$-divergence interestingly yields convex optimization problems which can be efficiently solved using semidefinite programming. The main advantage of $\alpha$-divergences is to enable working with unnormalized densities, which provides benefits both numerically and theoretically. In particular, we provide two new convergence analyses of the sequential transport maps: one based on a triangle-like inequality and the second on information geometric properties of $\alpha$-divergences for unnormalizied densities. The choice of intermediate densities is also crucial for the efficiency of the method. While tempered (or annealed) densities are the state-of-the-art, we introduce diffusion-based intermediate densities which permits to approximate densities known from samples only. Such intermediate densities are well-established in machine learning for generative modeling. Finally we propose and try different low-dimensional maps (or lazy maps) for dealing with high-dimensional problems and numerically demonstrate our methods on several benchmarks, including Bayesian inference problems and unsupervised learning task.

基于传输的密度估计方法结合了序贯传输映射、平方和密度以及α分散度来近似密度分布，为解决高维问题提供了高效的解决方案。

使用SoS密度估计和$α$-散度的序列传输映射