ManiFactor: Factor analysis for maps into manifolds

A central topic in data science is constructing a data-driven surrogate
model that approximates an unknown multivariate target function f.
The key goal of a surrogate model is to predict the outputs of the
target function both accurately and efficiently. A practical data-driven
surrogate model of f can be constructed numerically from
approximate input-output pairs by a learning algorithm, is
computationally cheap to evaluate, can approximate a wide range of
target functions, and can reach a target approximation quality by
increasing its model complexity. A surrogate model can be evaluated
efficiently at every point in the domain, hereby enabling applications
to tackle larger, previously infeasible, and more realistic scientific and
engineering problems. Examples of surrogate models include
polynomial, spline, and rational function approximation, deep
learning, and low-rank matrix and tensor decompositions, among
others. A fundamental shortcoming of state-of-the-art surrogate
models is that target functions are limited to flat codomains. In this
project, we will develop general, application-independent,
mathematical and algorithmic foundations for constructing surrogate
models of maps into Riemannian submanifolds based on a functorial
version of factor analysis or the tensor rank decomposition.