# Functional additive models on manifolds of planar shapes and forms

arXiv: https://arxiv.org/abs/2109.02624 · HTML: https://mlsanity.com/paper/2109.02624/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 83 references that are in the map.
- First posted on arXiv 6 Sep 2021.

## Builds on (most cited first, 14 of 14)

- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains](https://mlsanity.com/paper/1509.02029/) (2015)
- [Functional Regression](https://mlsanity.com/paper/1406.4068/) (2014)
- [Smoothing parameter and model selection for general smooth models](https://mlsanity.com/paper/1511.03864/) (2015)
- [The Evolution of Boosting Algorithms - From Machine Learning to Statistical Modelling](https://mlsanity.com/paper/1403.1452/) (2014)
- [Stability selection for component-wise gradient boosting in multiple dimensions](https://mlsanity.com/paper/1611.10171/) (2016)
- [Boosting Factor-Specific Functional Historical Models for the Detection of Synchronisation in Bioelectrical Signals](https://mlsanity.com/paper/1609.06070/) (2016)
- [Multivariate Functional Additive Mixed Models](https://mlsanity.com/paper/2103.06606/) (2021)
- [Generalized Functional Additive Mixed Models](https://mlsanity.com/paper/1506.05384/) (2015)
- [Fast symmetric additive covariance smoothing](https://mlsanity.com/paper/1609.07007/) (2016)
- [A Unified Framework of Constrained Regression](https://mlsanity.com/paper/1403.7118/) (2014)
- [Functional Linear Mixed Models for Irregularly or Sparsely Sampled Data](https://mlsanity.com/paper/1508.01686/) (2015)
- [Boosting Functional Response Models for Location, Scale and Shape with an Application to Bacterial Competition](https://mlsanity.com/paper/1809.09881/) (2018)
- [Additive Models for Symmetric Positive-Definite Matrices, Riemannian Manifolds and Lie groups](https://mlsanity.com/paper/2009.08789/) (2020)

## Built on by (most cited first, 2 of 2)

- [A Functional Approach to Curve Alignment and Shape Analysis](https://mlsanity.com/paper/2503.05632/) (2025)
- [Deep Shape Regression for Planar Curves with Multimodal Covariates](https://mlsanity.com/paper/2607.19600/) (2026)

## Abstract (arXiv)

The "shape" of a planar curve and/or landmark configuration is considered its equivalence class under translation, rotation and scaling, its "form" its equivalence class under translation and rotation while scale is preserved. We extend generalized additive regression to models for such shapes/forms as responses respecting the resulting quotient geometry by employing the squared geodesic distance as loss function and a geodesic response function to map the additive predictor to the shape/form space. For fitting the model, we propose a Riemannian L2-Boosting algorithm well suited for a potentially large number of possibly parameter-intensive model terms, which also yields automated model selection. We provide novel intuitively interpretable visualizations for (even non-linear) covariate effects in the shape/form space via suitable tensor-product factorization. The usefulness of the proposed framework is illustrated in an analysis of 1) astragalus shapes of wild and domesticated sheep and 2) cell forms generated in a biophysical model, as well as 3) in a realistic simulation study with response shapes and forms motivated from a dataset on bottle outlines.