  Manifold learning techniques have become increasingly valuable as data
continues to grow in size. By discovering a lower-dimensional representation
(embedding) of the structure of a dataset, manifold learning algorithms can
substantially reduce the dimensionality of a dataset while preserving as much
information as possible. However, state-of-the-art manifold learning algorithms
are opaque in how they perform this transformation. Understanding the way in
which the embedding relates to the original high-dimensional space is critical
in exploratory data analysis. We previously proposed a Genetic Programming
method that performed manifold learning by evolving mappings that are
transparent and interpretable. This method required the dimensionality of the
embedding to be known a priori, which makes it hard to use when little is known
about a dataset. In this paper, we substantially extend our previous work, by
introducing a multi-objective approach that automatically balances the
competing objectives of manifold quality and dimensionality. Our proposed
approach is competitive with a range of baseline and state-of-the-art manifold
learning methods, while also providing a range (front) of solutions that give
different trade-offs between quality and dimensionality. Furthermore, the
learned models are shown to often be simple and efficient, utilising only a
small number of features in an interpretable manner.
