# Bures-Wasserstein Importance-Weighted Evidence Lower Bound: Exposition and Applications

arXiv: https://arxiv.org/abs/2602.04272 · HTML: https://mlsanity.com/paper/2602.04272/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 32 references that are in the map.
- First posted on arXiv 4 Feb 2026.

## Builds on (most cited first, 16 of 16)

- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [R'enyi Divergence Variational Inference](https://mlsanity.com/paper/1602.02311/) (2016)
- [Filtering Variational Objectives](https://mlsanity.com/paper/1705.09279/) (2017)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Variational inference for Monte Carlo objectives](https://mlsanity.com/paper/1602.06725/) (2016)
- [Tighter Variational Bounds are Not Necessarily Better](https://mlsanity.com/paper/1802.04537/) (2018)
- [Importance Weighting and Variational Inference](https://mlsanity.com/paper/1808.09034/) (2018)
- [Doubly Reparameterized Gradient Estimators for Monte Carlo Objectives](https://mlsanity.com/paper/1810.04152/) (2018)
- [Reinterpreting Importance-Weighted Autoencoders](https://mlsanity.com/paper/1704.02916/) (2017)
- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [Wasserstein Regression](https://mlsanity.com/paper/2006.09660/) (2020)
- [Challenges and Opportunities in High-dimensional Variational Inference](https://mlsanity.com/paper/2103.01085/) (2021)
- [Distribution-on-Distribution Regression via Optimal Transport Maps](https://mlsanity.com/paper/2104.09418/) (2021)
- [Alpha-divergence Variational Inference Meets Importance Weighted Auto-Encoders: Methodology and Asymptotics](https://mlsanity.com/paper/2210.06226/) (2022)
- [Importance Weighted Variational Inference without the Reparameterization Trick](https://mlsanity.com/paper/2602.01412/) (2026)

## Abstract (arXiv)

The Importance-Weighted Evidence Lower Bound (IW-ELBO) has emerged as an effective objective for variational inference (VI), tightening the standard ELBO and mitigating the mode-seeking behaviour. However, optimizing the IW-ELBO in Euclidean space is often inefficient, as its gradient estimators suffer from a vanishing signal-to-noise ratio (SNR). This paper formulates the optimisation of the IW-ELBO in Bures-Wasserstein space, a manifold of Gaussian distributions equipped with the 2-Wasserstein metric. We derive the Wasserstein gradient of the IW-ELBO and project it onto the Bures-Wasserstein space to yield a tractable algorithm for Gaussian VI. A pivotal contribution of our analysis concerns the stability of the gradient estimator. While the SNR of the standard Euclidean gradient estimator is known to vanish as the number of importance samples K increases, we prove that the SNR of the Wasserstein gradient scales favourably as Omega(√K), ensuring optimisation efficiency even for large K. We further extend this geometric analysis to the Variational R'enyi Importance-Weighted Autoencoder bound, establishing analogous stability guarantees. Experiments demonstrate that the proposed framework achieves superior approximation performance compared to other baselines.