Multiscale analysis via pseudo-reversing and applications to manifold-valued sequences

Modeling data using manifold values is a powerful concept with numerous advantages, particularly in addressing nonlinear phenomena. This approach captures the intrinsic geometric structure of the data, leading to more accurate descriptors and more efficient computational processes. However, even fundamental tasks like compression and data enhancement present meaningful challenges in the manifold setting. This paper introduces a multiscale transform that aims to represent manifold-valued sequences at different scales, enabling novel data processing tools for various applications. Similar to traditional methods, our construction is based on a refinement operator that acts as an upsampling operator and a corresponding downsampling operator. Inspired by Wiener's lemma, we term the latter as the reverse of the former. It turns out that some upsampling operators, for example, least-squares-based refinement, do not have a practical reverse. Therefore, we introduce the notion of pseudo-reversing and explore its analytical properties and asymptotic behavior. We derive analytical properties of the induced multiscale transform and conclude the paper with numerical illustrations showcasing different aspects of the pseudo-reversing and two data processing applications involving manifolds.