# Deep Geometric Learning with Monotonicity Constraints for Alzheimer's Disease Progression

arXiv: https://arxiv.org/abs/2310.03353 · HTML: https://mlsanity.com/paper/2310.03353/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 53 references that are in the map.
- First posted on arXiv 5 Oct 2023.

## Builds on (most cited first, 21 of 21)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling](https://mlsanity.com/paper/1803.01271/) (2018)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [SAITS: Self-Attention-based Imputation for Time Series](https://mlsanity.com/paper/2202.08516/) (2022)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Estimating Missing Data in Temporal Data Streams Using Multi-directional Recurrent Neural Networks](https://mlsanity.com/paper/1711.08742/) (2017)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/1805.11204/) (2018)
- [Training recurrent neural networks robust to incomplete data: application to Alzheimer's disease progression modeling](https://mlsanity.com/paper/1903.07173/) (2019)
- [Morphological feature visualization of Alzheimer's disease via Multidirectional Perception GAN](https://mlsanity.com/paper/2111.12886/) (2021)
- [Dilated Convolutional Neural Networks for Sequential Manifold-valued Data](https://mlsanity.com/paper/1910.02206/) (2019)
- [MinimalRNN: Toward More Interpretable and Trainable Recurrent Neural Networks](https://mlsanity.com/paper/1711.06788/) (2017)
- [Deep Efficient Continuous Manifold Learning for Time Series Modeling](https://mlsanity.com/paper/2112.03379/) (2021)
- [BrainPainter: A software for the visualisation of brain structures, biomarkers and associated pathological processes](https://mlsanity.com/paper/1905.08627/) (2019)
- [Mixed Effects Neural ODE: A Variational Approximation for Analyzing the Dynamics of Panel Data](https://mlsanity.com/paper/2202.09463/) (2022)

## Built on by (most cited first, 4 of 4)

- [SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges](https://mlsanity.com/paper/2504.18882/) (2025)
- [Alzheimers Disease Progression Prediction Based on Manifold Mapping of Irregularly Sampled Longitudinal Data](https://mlsanity.com/paper/2511.20154/) (2025)
- [Permutation-Equivariant 2D State Space Models: Theory and Canonical Architecture for Multivariate Time Series](https://mlsanity.com/paper/2603.08753/) (2026)
- [Reconstructing and forecasting disease trajectories of patients with Alzheimer's disease using routine data in resource-constrained settings](https://mlsanity.com/paper/2606.07798/) (2026)

## Abstract (arXiv)

Alzheimer's disease (AD) is a devastating neurodegenerative condition that precedes progressive and irreversible dementia; thus, predicting its progression over time is vital for clinical diagnosis and treatment. Numerous studies have implemented structural magnetic resonance imaging (MRI) to model AD progression, focusing on three integral aspects: (i) temporal variability, (ii) incomplete observations, and (iii) temporal geometric characteristics. However, deep learning-based approaches regarding data variability and sparsity have yet to consider inherent geometrical properties sufficiently. The ordinary differential equation-based geometric modeling method (ODE-RGRU) has recently emerged as a promising strategy for modeling time-series data by intertwining a recurrent neural network and an ODE in Riemannian space. Despite its achievements, ODE-RGRU encounters limitations when extrapolating positive definite symmetric metrics from incomplete samples, leading to feature reverse occurrences that are particularly problematic, especially within the clinical facet. Therefore, this study proposes a novel geometric learning approach that models longitudinal MRI biomarkers and cognitive scores by combining three modules: topological space shift, ODE-RGRU, and trajectory estimation. We have also developed a training algorithm that integrates manifold mapping with monotonicity constraints to reflect measurement transition irreversibility. We verify our proposed method's efficacy by predicting clinical labels and cognitive scores over time in regular and irregular settings. Furthermore, we thoroughly analyze our proposed framework through an ablation study.