Learning Subspaces by Pieces and Algorithms for Ordinal Embedding

Speaker: Daniel Pimentel-Alarcon
Title: Learning subspaces by pieces

Abstract: We love subspaces. We observe a phenomenon and try to find a line that explains it. We get our hands on some data, and try to find a subspace that fits it. But in many relevant applications, data is missing, and all we can observe are small “pieces” of the subspace. Examples of these pieces include rows of different bases of the subspace, or canonical projections of the subspace, or incomplete vectors in the subspace. Fortunately, if we observe “the right” pieces, we can recover the whole subspace. In this talk I will explain which are the right pieces, and how to reconstruct the whole subspace from them. Depending on the pieces we get, some cases may be more challenging than others. For instance, some cases only require to find the null-space of a sparse matrix, while others require to solve complex systems of polynomial equations. I will also discuss some of the practical applications where these scenarios arise (like background segmentation, networks inference and recommender systems) and the remaining challenges of estimating subspaces from incomplete data.

Speaker: Lalit Jain
Title: Algorithms for Ordinal Embedding

Abstract: The standard problem of metric ordinal embedding concerns learning the embedding of n objects into a d dimensional Euclidean space by asking questions of the form “Is object i closer to object j than object k.” In this talk, we discuss some algebraic questions that arise from this problem, various algorithms and their learning rates, and connections to standard matrix completion problems.