Bayesian Non-linear Latent Variable Modeling via Random Fourier Features
Michael Minyi Zhang Gregory W. Gundersen Barbara E. Engelhardt
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.