Inverse parameter estimation via learned rectification of model manifolds

In this contribution, we address the challenge of non-convex optimization for parameter estimation and propose a general theoretical framework to improve global convergence and its practical implementation based on neural networks. Such inverse problems, as discussed here, are typically formulated in terms of a forward model. We interpret the set of model predictions as a Riemannian manifold and identify the origin of non-convexity in the twisted and entangled surface of the model manifold. Under assumptions of regularity, we aim to unfold and rectify the manifold using an interpretable residual autoencoder to facilitate global optimization. Crucially, this transformation must preserve the orthogonality of noisy measurements below a specified signal-to-noise ratio to support unique identification. In the theoretical limit, perfect rectification of the model manifold and its tubular neighborhood reduces the inverse problem to linear least squares. We validate our approach in a proof of concept and demonstrate its practical feasibility using a synthetic model based on a parametrized cosine series with two, three, and four parameters, confirming significant acceleration in convergence and robust global identification of parameter estimates.