# Variational Representations of Annealing Paths: Bregman Information under Monotonic Embedding

arXiv: https://arxiv.org/abs/2209.07481 · HTML: https://mlsanity.com/paper/2209.07481/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 81 references that are in the map.
- First posted on arXiv 15 Sep 2022.

## Builds on (most cited first, 13 of 13)

- [On Variational Bounds of Mutual Information](https://mlsanity.com/paper/1905.06922/) (2019)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [A Theory of Regularized Markov Decision Processes](https://mlsanity.com/paper/1901.11275/) (2019)
- [Learning with Fenchel-Young Losses](https://mlsanity.com/paper/1901.02324/) (2019)
- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [Generalized Variational Inference: Three arguments for deriving new Posteriors](https://mlsanity.com/paper/1904.02063/) (2019)
- [The Geometric Foundations of Hamiltonian Monte Carlo](https://mlsanity.com/paper/1410.5110/) (2014)
- [q-Paths: Generalizing the Geometric Annealing Path using Power Means](https://mlsanity.com/paper/2107.00745/) (2021)
- [All in the Exponential Family: Bregman Duality in Thermodynamic Variational Inference](https://mlsanity.com/paper/2007.00642/) (2020)
- [Understanding the bias-variance tradeoff of Bregman divergences](https://mlsanity.com/paper/2202.04167/) (2022)
- [f-GANs in an Information Geometric Nutshell](https://mlsanity.com/paper/1707.04385/) (2017)
- [Bregman Power k-Means for Clustering Exponential Family Data](https://mlsanity.com/paper/2206.10860/) (2022)
- [Likelihood Ratio Exponential Families](https://mlsanity.com/paper/2012.15480/) (2020)

## Built on by (most cited first, 3 of 3)

- [Learning Boltzmann Generators via Constrained Mass Transport](https://mlsanity.com/paper/2510.18460/) (2025)
- [Bias-variance decompositions: the exclusive privilege of Bregman divergences](https://mlsanity.com/paper/2501.18581/) (2025)
- [Divergences induced by dual subtractive and divisive normalizations of exponential families and their convex deformations](https://mlsanity.com/paper/2312.12849/) (2023)

## Abstract (arXiv)

Markov Chain Monte Carlo methods for sampling from complex distributions and estimating normalization constants often simulate samples from a sequence of intermediate distributions along an annealing path, which bridges between a tractable initial distribution and a target density of interest. Prior works have constructed annealing paths using quasi-arithmetic means, and interpreted the resulting intermediate densities as minimizing an expected divergence to the endpoints. To analyze these variational representations of annealing paths, we extend known results showing that the arithmetic mean over arguments minimizes the expected Bregman divergence to a single representative point. In particular, we obtain an analogous result for quasi-arithmetic means, when the inputs to the Bregman divergence are transformed under a monotonic embedding function. Our analysis highlights the interplay between quasi-arithmetic means, parametric families, and divergence functionals using the rho-tau representational Bregman divergence framework, and associates common divergence functionals with intermediate densities along an annealing path.