Spectrum Gaussian Processes Based On Tunable Basis Functions

Spectral approximation and variational inducing learning for the Gaussian process are two popular methods to reduce computational complexity. However, in previous research, those methods always tend to adopt the orthonormal basis functions, such as eigenvectors in the Hilbert space, in the spectrum method, or decoupled orthogonal components in the variational framework. In this paper, inspired by quantum physics, we introduce a novel basis function, which is tunable, local and bounded, to approximate the kernel function in the Gaussian process. There are two adjustable parameters in these functions, which control their orthogonality to each other and limit their boundedness. And we conduct extensive experiments on open-source datasets to testify its performance. Compared to several state-of-the-art methods, it turns out that the proposed method can obtain satisfactory or even better results, especially with poorly chosen kernel functions.

Sparse Gaussian Process Based On Hat Basis Functions

Gaussian process is one of the most popular non-parametric Bayesian methodologies for modeling the regression problem. It is completely determined by its mean and covariance functions. And its linear property makes it relatively straightforward to solve the prediction problem. Although Gaussian process has been successfully applied in many fields, it is still not enough to deal with physical systems that satisfy inequality constraints. This issue has been addressed by the so-called constrained Gaussian process in recent years. In this paper, we extend the core ideas of constrained Gaussian process. According to the range of training or test data, we redefine the hat basis functions mentioned in the constrained Gaussian process. Based on hat basis functions, we propose a new sparse Gaussian process method to solve the unconstrained regression problem. Similar to the exact Gaussian process and Gaussian process with Fully Independent Training Conditional approximation, our method obtains satisfactory approximate results on open-source datasets or analytical functions. In terms of performance, the proposed method reduces the overall computational complexity from $O(n^{3})$ computation in exact Gaussian process to $O(nm^{2})$ with $m$ hat basis functions and $n$ training data points.