Geometric Algorithms for Metric and Graph Learning

Graph and metric space representations are currently popular due to their multiple applications in modeling complex data sets from social networks to human genome sequences. In this dissertation, we examine various problems on metrics and graphs through the lens of geometric algorithms. The work in this dissertation can be categorized into three parts: Metric learning: In a metric space representation, each element is considered a point, and the similarity or dissimilarity between two objects is encoded by their pairwise distance. Our goal is to find a unique mapping from the initial metric to the host metric. The problem of learning a target underlying distance function can be cast as an optimization problem, where the objective function quantifies the extent to which a solution satisfies the input constraints. In particular, we explore the problem of learning lines with ordinal constraints and propose a solution by leveraging the geometric properties of metric spaces. Stability of metric data: Using Bilu-Linial stability of metrics is a relatively new perspective that can expose interesting structural properties that can motivate a re-exploration of some of the famous NP-hard problems. Bilu-Linial stability introduced a new point of view on complexity, where instead of focusing on worst-case elements of a problem, we instead focus on particular classes of inputs. We study the Steiner tree problem, one of the famous NP problems, under Bilu-Linial stability, and we give strong geometric structural properties that need to be satisfied by stable instances. Then by strengthening and using these geometric properties we show that $1.59$-stable instances of Euclidean Steiner trees are polynomial-time solvable. Graph learning and verification: In a graph learning problem, the goal is to learn or verify a hidden graph or its properties by having query access to the graphs. We study various queries (edge detection, edge counting), for both directed and undirected graphs but we focus mainly on bipartite edge counting queries and on undirected graphs. We give a randomized algorithm for learning graphs using $O(m \log n)$ bipartite edge counting queries as well as a randomized constant-query graph verification algorithm.