  The theory of Bayesian learning incorporates the use of Student-t Processes
to model heavy-tailed distributions and datasets with outliers. However,
despite Student-t Processes having a similar computational complexity as
Gaussian Processes, there has been limited emphasis on the sparse
representation of this model. This is mainly due to the increased difficulty in
modeling and computation compared to previous sparse Gaussian Processes. Our
motivation is to address the need for a sparse representation framework that
reduces computational complexity, allowing Student-t Processes to be more
flexible for real-world datasets. To achieve this, we leverage the conditional
distribution of Student-t Processes to introduce sparse inducing points.
Bayesian methods and variational inference are then utilized to derive a
well-defined lower bound, facilitating more efficient optimization of our model
through stochastic gradient descent. We propose two methods for computing the
variational lower bound, one utilizing Monte Carlo sampling and the other
employing Jensen's inequality to compute the KL regularization term in the loss
function. We propose adopting these approaches as viable alternatives to
Gaussian processes when the data might contain outliers or exhibit heavy-tailed
behavior, and we provide specific recommendations for their applicability. We
evaluate the two proposed approaches on various synthetic and real-world
datasets from UCI and Kaggle, demonstrating their effectiveness compared to
baseline methods in terms of computational complexity and accuracy, as well as
their robustness to outliers.
