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.

翻译：高斯过程隐变量模型（GPLVM）是一种用于非线性降维、矩阵分解和状态空间建模的流行概率方法。只有当数据似然为高斯分布时，GPLVM的推理才能在计算上可行。此外，GPLVM的推理通常局限于获取最大后验点估计（这可能导致过拟合）或变分近似（这会错误描述后验不确定性）。本文提出了一种通过马尔可夫链蒙特卡洛（MCMC）方法对广义贝叶斯非线性隐变量建模进行推理的方法。将GPLVM推广到任意观测模型的关键见解在于：我们使用随机傅里叶特征近似高斯过程映射中的核函数，这使得我们能够以闭式形式计算后验相对于隐变量的梯度。我们证明，通过随机特征隐变量模型（RFLVM），可以将GPLVM推广到非高斯观测数据（如泊松分布、负二项分布和多项分布）。在包括运动捕捉、图像和文本数据在内的广泛应用中，我们的广义RFLVM在估计潜在结构及对这些复杂数据集进行缺失数据插补方面，其性能与最先进的隐变量模型相当。