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Manifold learning by spectral methods

Research Area: Learning Theory and Algorithms
Status: In progress  
Faculty: Ernesto De Vito, Alessandro Verri Participants: Paolo Albini, Lorenzo Rosasco, Nicola Rebagliati, Veronica Umanità

We are interested in the estimation of the geometrical properties of a subset of a Euclidean space from a finite sample of points affected by noise. Our approach is based on spectral regularization techniques from the theory of inverse problem and results from harmonic analysis as well as some concentration inequalities for random variables in Banach spaces.

We address questions like:

  • Support density estimation: the data are sampled according to some unknown probability distribution and we wish to construct a real function from the empirical data such that one of its level sets is the support of the probability distribution
  • Clustering: we aim at finding a partition of a set of unlabeled points starting from a distance function or a similarity measure such that points in the same partition can be given the same label
  • Dimensionality reduction: our goal is to both estimate the intrinsic dimension of the subset under study and find a suitable set of coordinates for the underlying submanifold.
  • Riemannian distance estimation: if the points belong to a Riemannian submanifold, we would like to estimate the Riemannian distance by means of graph operators defined on the empirical data, like the empirical graph Laplacian.