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How are kernel ridge regression and Gaussian process regression similar?
Click here to download the full example code or to run this example in your browser via Binder Both kernel ridge regression (KRR) and Gaussian process regression (GPR) learn a target function by employing internally the “kernel trick”.
Which is a special case of the Gaussian process?
However, for the special case of having a Gaussian likelihood and prior (those are the ridge regression assumptions), this expression is Gaussian and we can derive its mean and covariance. So, P(y ∗ ∣ D, x) ∼ N(μy ∗ ∣ D, Σy ∗ ∣ D), where μy ∗ ∣ D = KT ∗ (K + σ2I) − 1y and Σy ∗ ∣ D = K ∗ ∗ − KT ∗ (K + σ2I) − 1K ∗.
How is the linear function chosen in KRR?
KRR learns a linear function in the space induced by the respective kernel which corresponds to a non-linear function in the original space. The linear function in the kernel space is chosen based on the mean-squared error loss with ridge regularization.
How are kernels and likelihood functions used in GPR?
GPR uses the kernel to define the covariance of a prior distribution over the target functions and uses the observed training data to define a likelihood function. Based on Bayes theorem, a (Gaussian) posterior distribution over target functions is defined, whose mean is used for prediction.
How is the linear function in the kernel chosen?
The linear function in the kernel space is chosen based on the mean-squared error loss with ridge regularization. GPR uses the kernel to define the covariance of a prior distribution over the target functions and uses the observed training data to define a likelihood function.
How to prove the validity of a Gaussian process?
Here are 3 possibilities for the kernel function: You can prove for yourself that each of these kernel functions is valid i.e. that they construct symmetric positive semi-definite covariance matrices. For example, the covariance matrix associated with the linear kernel is simply σ f 2 X X T, which is indeed symmetric positive semi-definite.
How to do a Gaussian regression with data?
We have some observed data D = [ ( x 1, y 1) … ( x n, y n)] with x ∈ R D and y ∈ R. We assume that each observation y can be related to an underlying function f ( x) through a Gaussian noise model: The aim is to find f ( x), such that given some new test point x ∗, we can accurately estimate the corresponding y ∗.