Graph Laplacians are certain matrices defined in terms of samples of random vectors drawn from a latent, unknown subspace of Euclidean space. The use of graph Laplacians to partially learn the geometry of a latent manifold is one of the dominant paradigms in machine learning. However, graph Laplacians as they are currently used can never completely recover the latent manifold in a non-parametric setting.
The goal of this poster presentation is to show how graph Laplacians can actually be used to obtain a consistent estimator for intrinsic latent manifold distances between sample points, and in particular, a non-parametric but computable method of completely recovering the manifold. There are two main insights behind this method: 1) graph Laplacians can be regarded not just as linear operators but something we might call quadratic operators; and 2) a fundamental result from non-commutative geometry reformulates manifold distance purely in terms of such quadratic operators. This latter reformulation is a special case of the Kontorovich dual reformulation of Wasserstein distances known as Connes' Distance Formula.
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