We present a simple, yet effective, approach to Semi-Supervised Learning. Our
approach is based on estimating density-based distances (DBD) using a shortest
path calculation on a graph. These Graph-DBD estimates can then be used in any
distance-based supervised learning method, such as Nearest Neighbor methods and
SVMs with RBF kernels. In order to apply the method to very large data sets, we
also present a novel algorithm which integrates nearest neighbor computations
into the shortest path search and can find exact shortest paths even in
extremely large dense graphs.

We consider the problem of approximately reconstructing a partially-observed,
approximately low-rank matrix. This problem has received much attention lately,
mostly using the trace-norm as a surrogate to the rank. Here we study low-rank
matrix reconstruction using both the trace-norm, as well as the less-studied
max-norm, and present reconstruction guarantees based on existing analysis on
the Rademacher complexity of the unit balls of these norms. We show how these
are superior in several ways to recently published guarantees based on
specialized analysis.