# Independent projections of diffusions: Gradient flows for variational inference and optimal mean field approximations

arXiv: https://arxiv.org/abs/2309.13332 · HTML: https://mlsanity.com/paper/2309.13332/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 35 references that are in the map.
- First posted on arXiv 23 Sep 2023.

## Builds on (most cited first, 7 of 7)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Mean-field Variational Inference via Wasserstein Gradient Flow](https://mlsanity.com/paper/2207.08074/) (2022)
- [On the Convergence of Coordinate Ascent Variational Inference](https://mlsanity.com/paper/2306.01122/) (2023)
- [Projected Langevin dynamics and a gradient flow for entropic optimal transport](https://mlsanity.com/paper/2309.08598/) (2023)
- [Particle Mean Field Variational Bayes](https://mlsanity.com/paper/2303.13930/) (2023)
- [Approximately optimal distributed stochastic controls beyond the mean field setting](https://mlsanity.com/paper/2301.02901/) (2023)

## Built on by (most cited first, 12 of 12)

- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Theoretical Guarantees for Variational Inference with Fixed-Variance Mixture of Gaussians](https://mlsanity.com/paper/2406.04012/) (2024)
- [Convergence rate of random scan Coordinate Ascent Variational Inference under log-concavity](https://mlsanity.com/paper/2406.07292/) (2024)
- [Sampling from the Mean-Field Stationary Distribution](https://mlsanity.com/paper/2402.07355/) (2024)
- [Variational Inference with Mixtures of Isotropic Gaussians](https://mlsanity.com/paper/2506.13613/) (2025)
- [A Particle Algorithm for Mean-Field Variational Inference](https://mlsanity.com/paper/2412.20385/) (2024)
- [Beyond Propagation of Chaos: A Stochastic Algorithm for Mean Field Optimization](https://mlsanity.com/paper/2503.13115/) (2025)
- [Rotated Mean-Field Variational Inference and Iterative Gaussianization](https://mlsanity.com/paper/2510.07732/) (2025)
- [Wasserstein Contraction of Coordinate Ascent Variational Inference](https://mlsanity.com/paper/2605.30253/) (2026)
- [Gaussian mixture layers for neural networks](https://mlsanity.com/paper/2508.04883/) (2025)
- [Mode Collapse of Mean-Field Variational Inference](https://mlsanity.com/paper/2510.17063/) (2025)
- [Stability of Finite-Batch Particle Mean-Field Variational Inference Beyond Strong Convexity](https://mlsanity.com/paper/2608.11486/) (2026)

## Abstract (arXiv)

What is the optimal way to approximate a high-dimensional diffusion process by one in which the coordinates are independent? This paper presents a construction, called the independent projection, which is optimal for two natural criteria. First, when the original diffusion is reversible with invariant measure ρ_*, the independent projection serves as the Wasserstein gradient flow for the relative entropy H(·,|,ρ_*) constrained to the space of product measures. This is related to recent Langevin-based sampling schemes proposed in the statistical literature on mean field variational inference. In addition, we provide both qualitative and quantitative results on the long-time convergence of the independent projection, with quantitative results in the log-concave case derived via a new variant of the logarithmic Sobolev inequality. Second, among all processes with independent coordinates, the independent projection is shown to exhibit the slowest growth rate of path-space entropy relative to the original diffusion. This sheds new light on the classical McKean-Vlasov equation and recent variants proposed for non-exchangeable systems, which can be viewed as special cases of the independent projection.