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A fundamental problem in manifold learning is to approximate a functionalrelationship in a data chosen randomly from a probability distributionsupported on a low dimensional sub-manifold of a high dimensional ambientEuclidean space. The manifold is essentially defined by the data set itselfand, typically, designed so that the data is dense on the manifold in somesense. The notion of a data space is an abstraction of a manifold encapsulatingthe essential properties that allow for function approximation. The problem oftransfer learning (meta-learning) is to use the learning of a function on onedata set to learn a similar function on a new data set. In terms of functionapproximation, this means lifting a function on one data space (the base dataspace) to another (the target data space). This viewpoint enables us to connectsome inverse problems in applied mathematics (such as inverse Radon transform)with transfer learning. In this paper we examine the question of such liftingwhen the data is assumed to be known only on a part of the base data space. Weare interested in determining subsets of the target data space on which thelifting can be defined, and how the local smoothness of the function and itslifting are related.

Local transfer learning from one data space to another