# Probabilistic size-and-shape functional mixed models

arXiv: https://arxiv.org/abs/2411.18416 · HTML: https://mlsanity.com/paper/2411.18416/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 42 references that are in the map.
- First posted on arXiv 27 Nov 2024.

## Builds on (most cited first, 9 of 9)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Integrating Random Effects in Deep Neural Networks](https://mlsanity.com/paper/2206.03314/) (2022)
- [Non-linear Functional Modeling using Neural Networks](https://mlsanity.com/paper/2104.09371/) (2021)
- [Modern Non-Linear Function-on-Function Regression](https://mlsanity.com/paper/2107.14151/) (2021)
- [Multivariate Functional Additive Mixed Models](https://mlsanity.com/paper/2103.06606/) (2021)
- [Bayesian Framework for Simultaneous Registration and Estimation of Noisy, Sparse and Fragmented Functional Data](https://mlsanity.com/paper/1912.05125/) (2019)
- [Bayesian registration of functions and curves](https://mlsanity.com/paper/1311.2105/) (2013)
- [Estimating Functional Linear Mixed-Effects Regression Models](https://mlsanity.com/paper/1601.01039/) (2016)
- [Functional Registration and Local Variations: Identifiability, Rank, and Tuning](https://mlsanity.com/paper/1702.03556/) (2017)

## Abstract (arXiv)

The reliable recovery and uncertainty quantification of a fixed effect function μ in a functional mixed model, for modelling population- and object-level variability in noisily observed functional data, is a notoriously challenging task: variations along the x and y axes are confounded with additive measurement error, and cannot in general be disentangled. The question then as to what properties of μ may be reliably recovered becomes important. We demonstrate that it is possible to recover the size-and-shape of a square-integrable μ under a Bayesian functional mixed model. The size-and-shape of μ is a geometric property invariant to a family of space-time unitary transformations, viewed as rotations of the Hilbert space, that jointly transform the x and y axes. A random object-level unitary transformation then captures size-and-shape preserving deviations of μ from an individual function, while a random linear term and measurement error capture size-and-shape altering deviations. The model is regularized by appropriate priors on the unitary transformations, posterior summaries of which may then be suitably interpreted as optimal data-driven rotations of a fixed orthonormal basis for the Hilbert space. Our numerical experiments demonstrate utility of the proposed model, and superiority over the current state-of-the-art.