# Bayesian Non-linear Latent Variable Modeling via Random Fourier Features

arXiv: https://arxiv.org/abs/2306.08352 · HTML: https://mlsanity.com/paper/2306.08352/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 90 references that are in the map.
- First posted on arXiv 14 Jun 2023.

## Builds on (most cited first, 15 of 15)

- [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)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Gaussian Process Kernels for Pattern Discovery and Extrapolation](https://mlsanity.com/paper/1302.4245/) (2013)
- [Variational Fourier features for Gaussian processes](https://mlsanity.com/paper/1611.06740/) (2016)
- [How to Construct Deep Recurrent Neural Networks](https://mlsanity.com/paper/1312.6026/) (2013)
- [Active Learning for Deep Gaussian Process Surrogates](https://mlsanity.com/paper/2012.08015/) (2020)
- [Dependent Multinomial Models Made Easy: Stick Breaking with the P'olya-Gamma Augmentation](https://mlsanity.com/paper/1506.05843/) (2015)
- [A la Carte - Learning Fast Kernels](https://mlsanity.com/paper/1412.6493/) (2014)
- [Neural Dynamics Discovery via Gaussian Process Recurrent Neural Networks](https://mlsanity.com/paper/1907.00650/) (2019)
- [Bayesian Nonparametric Kernel-Learning](https://mlsanity.com/paper/1506.08776/) (2015)
- [Latent variable modeling with random features](https://mlsanity.com/paper/2006.11145/) (2020)
- [Gaussian Process Latent Variable Alignment Learning](https://mlsanity.com/paper/1803.02603/) (2018)
- [Latent Gaussian Processes for Distribution Estimation of Multivariate Categorical Data](https://mlsanity.com/paper/1503.02182/) (2015)
- [Sparse Infinite Random Feature Latent Variable Modeling](https://mlsanity.com/paper/2205.09909/) (2022)

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

- [Preventing Model Collapse in Gaussian Process Latent Variable Models](https://mlsanity.com/paper/2404.01697/) (2024)
- [Trigonometric Quadrature Fourier Features for Scalable Gaussian Process Regression](https://mlsanity.com/paper/2310.14544/) (2023)
- [Multi-View Oriented GPLVM: Expressiveness and Efficiency](https://mlsanity.com/paper/2502.08253/) (2025)
- [Scalable Unsupervised Segmentation via Random Fourier Feature-based Gaussian Process](https://mlsanity.com/paper/2507.10632/) (2025)

## Abstract (arXiv)

The Gaussian process latent variable model (GPLVM) is a popular probabilistic method used for nonlinear dimension reduction, matrix factorization, and state-space modeling. Inference for GPLVMs is computationally tractable only when the data likelihood is Gaussian. Moreover, inference for GPLVMs has typically been restricted to obtaining maximum a posteriori point estimates, which can lead to overfitting, or variational approximations, which mischaracterize the posterior uncertainty. Here, we present a method to perform Markov chain Monte Carlo (MCMC) inference for generalized Bayesian nonlinear latent variable modeling. The crucial insight necessary to generalize GPLVMs to arbitrary observation models is that we approximate the kernel function in the Gaussian process mappings with random Fourier features; this allows us to compute the gradient of the posterior in closed form with respect to the latent variables. We show that we can generalize GPLVMs to non-Gaussian observations, such as Poisson, negative binomial, and multinomial distributions, using our random feature latent variable model (RFLVM). Our generalized RFLVMs perform on par with state-of-the-art latent variable models on a wide range of applications, including motion capture, images, and text data for the purpose of estimating the latent structure and imputing the missing data of these complex data sets.