During the seminar we will consider two separate problems. First, we study learning problems in which we would like to learn intrinsic values of objects based on pairwise comparisons. We suggest an algorithm, determine a minimax rate and show that both the upper and the lower error bounds are connected to the trace of the Moore-Penrose inverse of the weighted Laplacian of the comparison graph. In the second part of the seminar, we consider feedforward neural networks and investigate how the tangent space of the network can be exploited in case of ReLU (Rectified Linear Unit) activations by forming a similarity function with a Riemannian metric. Finally, we connect the two problems.
