# Diffusion Bridge Variational Inference for Deep Gaussian Processes

arXiv: https://arxiv.org/abs/2509.19078 · HTML: https://mlsanity.com/paper/2509.19078/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 33 references that are in the map.
- First posted on arXiv 23 Sep 2025.

## Builds on (most cited first, 22 of 22)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [BBDM: Image-to-image Translation with Brownian Bridge Diffusion Models](https://mlsanity.com/paper/2205.07680/) (2022)
- [Denoising Diffusion Bridge Models](https://mlsanity.com/paper/2309.16948/) (2023)
- [Doubly Stochastic Variational Inference for Deep Gaussian Processes](https://mlsanity.com/paper/1705.08933/) (2017)
- [Semi-Amortized Variational Autoencoders](https://mlsanity.com/paper/1802.02550/) (2018)
- [Inference in Deep Gaussian Processes using Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1806.05490/) (2018)
- [Amortized Variational Inference: When and Why?](https://mlsanity.com/paper/2307.11018/) (2023)
- [Consistency Diffusion Bridge Models](https://mlsanity.com/paper/2410.22637/) (2024)
- [Amortized Variational Inference for Simple Hierarchical Models](https://mlsanity.com/paper/2111.03144/) (2021)
- [Implicit Posterior Variational Inference for Deep Gaussian Processes](https://mlsanity.com/paper/1910.11998/) (2019)
- [Amortized Variational Inference: A Systematic Review](https://mlsanity.com/paper/2209.10888/) (2022)
- [Sparse Inducing Points in Deep Gaussian Processes: Enhancing Modeling with Denoising Diffusion Variational Inference](https://mlsanity.com/paper/2407.17033/) (2024)
- [Dequantified Diffusion-Schr"odinger Bridge for Density Ratio Estimation](https://mlsanity.com/paper/2505.05034/) (2025)
- [SciRE-Solver: Accelerating Diffusion Models Sampling by Score-integrand Solver with Recursive Difference](https://mlsanity.com/paper/2308.07896/) (2023)
- [Sparse Variational Student-t Processes](https://mlsanity.com/paper/2312.05568/) (2023)
- [Bayesian Gaussian Process ODEs via Double Normalizing Flows](https://mlsanity.com/paper/2309.09222/) (2023)
- [Neural Operator Variational Inference based on Regularized Stein Discrepancy for Deep Gaussian Processes](https://mlsanity.com/paper/2309.12658/) (2023)
- [Fully Bayesian Differential Gaussian Processes through Stochastic Differential Equations](https://mlsanity.com/paper/2408.06069/) (2024)
- [Variational Learning of Gaussian Process Latent Variable Models through Stochastic Gradient Annealed Importance Sampling](https://mlsanity.com/paper/2408.06710/) (2024)

## Built on by (most cited first, 4 of 4)

- [Onsager-Machlup Posterior Transport for Deep Gaussian Processes](https://mlsanity.com/paper/2605.23434/) (2026)
- [Rethinking the Flow-Based Gradual Domain Adaptation: A Semi-Dual Optimal Transport Perspective](https://mlsanity.com/paper/2602.01179/) (2026)
- [Implicit Variational Rejection Sampling](https://mlsanity.com/paper/2606.14235/) (2026)
- [Mode Collapse Is Cheap to Detect: A Ground-Truth-Free Pre-Flight Check for Neural Samplers](https://mlsanity.com/paper/2609.26272/) (2026)

## Abstract (arXiv)

Deep Gaussian processes (DGPs) enable expressive hierarchical Bayesian modeling but pose substantial challenges for posterior inference, especially over inducing variables. Denoising diffusion variational inference (DDVI) addresses this by modeling the posterior as a time-reversed diffusion from a simple Gaussian prior. However, DDVI's fixed unconditional starting distribution remains far from the complex true posterior, resulting in inefficient inference trajectories and slow convergence. In this work, we propose Diffusion Bridge Variational Inference (DBVI), a principled extension of DDVI that initiates the reverse diffusion from a learnable, data-dependent initial distribution. This initialization is parameterized via an amortized neural network and progressively adapted using gradients from the ELBO objective, reducing the posterior gap and improving sample efficiency. To enable scalable amortization, we design the network to operate on the inducing inputs, which serve as structured, low-dimensional summaries of the dataset and naturally align with the inducing variables' shape. DBVI retains the mathematical elegance of DDVI, including Girsanov-based ELBOs and reverse-time SDEs,while reinterpreting the prior via a Doob-bridged diffusion process. We derive a tractable training objective under this formulation and implement DBVI for scalable inference in large-scale DGPs. Across regression, classification, and image reconstruction tasks, DBVI consistently outperforms DDVI and other variational baselines in predictive accuracy, convergence speed, and posterior quality.