# Learning Implicit Generative Models with Theoretical Guarantees

arXiv: https://arxiv.org/abs/2002.02862 · HTML: https://mlsanity.com/paper/2002.02862/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Feb 2020.

## Built on by (most cited first, 1 of 1)

- [GANs as Gradient Flows that Converge](https://mlsanity.com/paper/2205.02910/) (2022)

## Abstract (arXiv)

We propose a unified framework for implicit generative modeling (UnifiGem) with theoretical guarantees by integrating approaches from optimal transport, numerical ODE, density-ratio (density-difference) estimation and deep neural networks. First, the problem of implicit generative learning is formulated as that of finding the optimal transport map between the reference distribution and the target distribution, which is characterized by a totally nonlinear Monge-Amp`ere equation. Interpreting the infinitesimal linearization of the Monge-Amp`ere equation from the perspective of gradient flows in measure spaces leads to the continuity equation or the McKean-Vlasov equation. We then solve the McKean-Vlasov equation numerically using the forward Euler iteration, where the forward Euler map depends on the density ratio (density difference) between the distribution at current iteration and the underlying target distribution. We further estimate the density ratio (density difference) via deep density-ratio (density-difference) fitting and derive explicit upper bounds on the estimation error. Experimental results on both synthetic datasets and real benchmark datasets support our theoretical findings and demonstrate the effectiveness of UnifiGem.