# FineMorphs: Affine-diffeomorphic sequences for regression

arXiv: https://arxiv.org/abs/2305.17255 · HTML: https://mlsanity.com/paper/2305.17255/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 53 references that are in the map.
- First posted on arXiv 26 May 2023.

## Builds on (most cited first, 26 of 26)

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- [NeurEPDiff: Neural Operators to Predict Geodesics in Deformation Spaces](https://mlsanity.com/paper/2303.07115/) (2023)
- [Shape deformation analysis from the optimal control viewpoint](https://mlsanity.com/paper/1401.0661/) (2014)
- [Diffeomorphic Learning](https://mlsanity.com/paper/1806.01240/) (2018)
- [Deep Learning: Hydrodynamics, and Lie-Poisson Hamilton-Jacobi Theory](https://mlsanity.com/paper/2105.09542/) (2021)
- [Sub-Riemannian Landmark Matching and its interpretation as residual neural networks](https://mlsanity.com/paper/2204.09351/) (2022)

## Abstract (arXiv)

A multivariate regression model of affine and diffeomorphic transformation sequences - FineMorphs - is presented. Leveraging concepts from shape analysis, model states are optimally "reshaped" by diffeomorphisms generated by smooth vector fields during learning. Affine transformations and vector fields are optimized within an optimal control setting, and the model can naturally reduce (or increase) dimensionality and adapt to large datasets via suboptimal vector fields. An existence proof of solution and necessary conditions for optimality for the model are derived. Experimental results on real datasets from the UCI repository are presented, with favorable results in comparison with state-of-the-art in the literature and densely-connected neural networks in TensorFlow.