Geometric algorithms for interpretable manifold learning

Abstract

This thesis proposes several algorithms in the area of interpretable unsupervised learning.Chapters 3 and 4 introduce a sparse convex regression approach for identifying local diffeomor- phisms from a dictionary of interpretable functions. In Chapter 3, this algorithm makes use of an embedding learned by a manifold learning algorithm, while in Chapter 4, this algorithm is applied without the use of a precomputed embedding. Chapter 5 then introduces a set of alternative algorithms that avoid issues stemming from sparse regression, characterizes the tangent space version of this algorithm as identifying isometries when available, and gives a two-stage algorithm combining this approach with the computational advantages of the algorithms in Chapters 3 and 4. Finally, Chapter 6 gives an alternate tangent space estimator based on a learned embedding, and uses this as an initial estimator to tackle the related gradient estimation problem. Together, these approaches provide a toolbox of methods for computing and associating gradient information to learn descriptive parameterizations of data manifolds.