Abstract: We will discuss variational problems arising in machine learning and their limits as the number of data points goes to infinity. Consider point clouds obtained as random samples of an underlying "ground-truth" measure. Graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points. Many machine learning tasks, such as clustering and semi-supervised learning, can be posed as minimizing  functionals on such graphs. We consider functionals involving graph cuts, graph laplacians…

Abstract: We give a solution to the following question from manifold learning. Suppose data belonging to a high dimensional Euclidean space is drawn independently, identically distributed from a measure supported on a low dimensional twice differentiable embedded compact manifold M, and is corrupted by a small amount of i.i.d gaussian noise. How can we produce a manifold M whose Hausdorff distance to M is small and whose reach (normal injectivity radius) is not much smaller than the reach of M? This…