# Gaussian process : function approximation properties

I'm learning about Gaussian Process and have heard only bits and pieces. Would really appreciate comments and answers.

For any set of data, is it true that a Gaussian Process function approximation would give zero or negligible fitting error at the data points ? In another place I also heard that Gaussian Process is particularly good for noisy data. This seems to be in conflict with the low fitting error for any observed data?

Additionally, further away from the data points there seem to be more uncertainty (larger covariance). If so, does it behave like local models (RBF etc)?

Finally, is there any universal approximation property?

Suppose data sample is $D = (X, \mathbf{y}) = \{\mathbf{x}_i, y_i = y(x_i)\}_{i = 1}^N$. Also suppose, that we have a covariance function $k(\mathbf{x}_1, \mathbf{x}_2)$ and zero mean specified for a Gussian process. Distribution for a new point $\mathbf{x}$ will be Gaussian with mean $$m(\mathbf{x}) = \mathbf{k} K^{-1} \mathbf{y}$$ and variance $$V(\mathbf{x}) = k(\mathbf{x}, \mathbf{x}) - \mathbf{k} K^{-1} \mathbf{k}^T.$$ Vector $\mathbf{k} = \{k(\mathbf{x}, \mathbf{x}_1), \ldots, k(\mathbf{x}, \mathbf{x}_N)\}$ is a vector of covariances, matrix $K = \{k(\mathbf{x}_i, \mathbf{x}_j)\}_{i, j = 1}^N$ is a matrix of sample covariances. In case we make prediction using mean value of posterior distribution for sample interpolation property holds. Really, $$m(X) = K K^{-1} \mathbf{y} = \mathbf{y}.$$ But, it isn't the case if we use regularization i.e. incorporate white noise term. in this case covariance matrix for sample has form $K + \sigma I$, but for covariances with real function values we have covariance matrix $K$, and posterior mean is $$m(X) = K (K + \sigma I)^{-1} \mathbf{y} \neq \mathbf{y}.$$ In addition, regularization makes problem more computationally stable.
Choosing noise variance $\sigma$ we can select if we want interpolation ($\sigma = 0$) or we want to handle noisy observations ($\sigma$ is big).
Also, the Gaussian processes regression is local method because variance of predictions grows with distance to learning sample, but we can select appropriate covariance function $k$ and handle more complex problems, than with RBF. Another nice property is small number of parameters. Usually it equals $O(n)$, where $n$ is data dimension.