# Deep Q-Exponential Processes

arXiv: https://arxiv.org/abs/2410.22119 · HTML: https://mlsanity.com/paper/2410.22119/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 41 references that are in the map.
- First posted on arXiv 29 Oct 2024.

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

- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Scalable Variational Gaussian Process Classification](https://mlsanity.com/paper/1411.2005/) (2014)
- [Doubly Stochastic Variational Inference for Deep Gaussian Processes](https://mlsanity.com/paper/1705.08933/) (2017)
- [A Haar Wavelet-Based Perceptual Similarity Index for Image Quality Assessment](https://mlsanity.com/paper/1607.06140/) (2016)
- [Deep kernel processes](https://mlsanity.com/paper/2010.01590/) (2020)
- [Deep convolutional Gaussian processes](https://mlsanity.com/paper/1810.03052/) (2018)
- [Bayesian Image Classification with Deep Convolutional Gaussian Processes](https://mlsanity.com/paper/1902.05888/) (2019)
- [Deep Sigma Point Processes](https://mlsanity.com/paper/2002.09112/) (2020)
- [Deep Bayesian Gaussian Processes for Uncertainty Estimation in Electronic Health Records](https://mlsanity.com/paper/2003.10170/) (2020)
- [Deep Variational Implicit Processes](https://mlsanity.com/paper/2206.06720/) (2022)
- [Inverse Reinforcement Learning via Deep Gaussian Process](https://mlsanity.com/paper/1512.08065/) (2015)
- [Deep Horseshoe Gaussian Processes](https://mlsanity.com/paper/2403.01737/) (2024)
- [Bayesian Learning via Q-Exponential Process](https://mlsanity.com/paper/2210.07987/) (2022)
- [Spatiotemporal Besov Priors for Bayesian Inverse Problems](https://mlsanity.com/paper/2306.16378/) (2023)

## Abstract (arXiv)

Motivated by deep neural networks, the deep Gaussian process (DGP) generalizes the standard GP by stacking multiple layers of GPs. Despite the enhanced expressiveness, GP, as an L2 regularization prior, tends to be over-smooth and sub-optimal for inhomogeneous subjects, such as images with edges. Recently, Q-exponential process (Q-EP) has been proposed as an Lq relaxation to GP and demonstrated with more desirable regularization properties through a parameter q>0 with q=2 corresponding to GP. Sharing the similar tractability of posterior and predictive distributions with GP, Q-EP can also be stacked to improve its modeling flexibility. In this paper, we generalize Q-EP to deep Q-EP to enjoy both proper regularization and improved expressiveness. The generalization is realized by introducing shallow Q-EP as a latent variable model and then building a hierarchy of the shallow Q-EP layers. Sparse approximation by inducing points and scalable variational strategy are applied to facilitate the inference. We demonstrate the numerical advantages of the proposed deep Q-EP model by comparing with multiple state-of-the-art deep probabilistic models.