Our world is full of physics-driven data where effective mappings between
data manifolds are desired. There is an increasing demand for understanding
combined model-based and data-driven methods. We propose a nonlinear, learned
singular value decomposition (L-SVD), which combines autoencoders that
simultaneously learn and connect latent codes for desired signals and given
measurements. We provide a convergence analysis for a specifically structured
L-SVD that acts as a regularisation method. In a more general setting, we
investigate the topic of model reduction via data dimensionality reduction to
obtain a regularised inversion. We present a promising direction for solving
inverse problems in cases where the underlying physics are not fully understood
or have very complex behaviour. We show that the building blocks of learned
inversion maps can be obtained automatically, with improved performance upon
classical methods and better interpretability than black-box methods
