Convolutional Neural Networks are used to obtain state of the art results in the field of image classification. However, they also come with certain drawbacks, such as their vulnerability to adversarial attacks and lacking uncertainty estimation. These drawbacks can be tackled by making use of Bayesian inference, which is the main concept underlying Gaussian Processes for making predictions. A Gaussian Process is a distribution over functions defined by its mean and its covariance or kernel function. Deep Convolutional Networks can be represented as Gaussian Processes in the limit where the number of convolutional filters approaches infinity. The disadvantage of making predictions with Gaussian Processes is the big computational effort coming with each sample of the training set, because the kernel matrix has to be extended for each new sample. Additionally, this large kernel matrix for the training data has to be inverted. In this thesis, it is explained in detail what a Gaussian Process is and how it can be used for Regression. Furthermore, a Gaussian Process is presented that is equivalent to a convolutional neural network in the limit of infinitely many filters. To construct the kernel matrix from given training data, Iterative SVD, Soft Impute, Matrix Factorization and the Nyström method are investigated to increase the efficiency of Gaussian Processes by exploiting the low-rank structure of the kernel matrix. Finally, it can be seen that it is sufficient to compute only 20% of the kernel matrix exactly and approximate the missing elements of it by making use of Iterative SVD or the Nyström method. By that the efficiency for large training sets can be improved significantly and the Gaussian Process model obtains still good classification results.     «

Convolutional Neural Networks are used to obtain state of the art results in the field of image classification. However, they also come with certain drawbacks, such as their vulnerability to adversarial attacks and lacking uncertainty estimation. These drawbacks can be tackled by making use of Bayesian inference, which is the main concept underlying Gaussian Processes for making predictions. A Gaussian Process is a distribution over functions defined by its mean and its covariance or kernel func...     »