# Riemannian Archetypal Analysis: Interpretable non-linear data analysis on deformed star distributions

arXiv: https://arxiv.org/abs/2605.24113 · HTML: https://mlsanity.com/paper/2605.24113/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 68 references that are in the map.
- First posted on arXiv 22 May 2026.

## Builds on (most cited first, 30 of 40)

- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [ReduNet: A White-box Deep Network from the Principle of Maximizing Rate Reduction](https://mlsanity.com/paper/2105.10446/) (2021)
- [Phase Retrieval Under a Generative Prior](https://mlsanity.com/paper/1807.04261/) (2018)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [Intermediate Layer Optimization for Inverse Problems using Deep Generative Models](https://mlsanity.com/paper/2102.07364/) (2021)
- [Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions](https://mlsanity.com/paper/2409.02426/) (2024)
- [Global Guarantees for Enforcing Deep Generative Priors by Empirical Risk](https://mlsanity.com/paper/1705.07576/) (2017)
- [Inverting Deep Generative models, One layer at a time](https://mlsanity.com/paper/1906.07437/) (2019)
- [Statistical exploration of the Manifold Hypothesis](https://mlsanity.com/paper/2208.11665/) (2022)
- [A Locally Adaptive Normal Distribution](https://mlsanity.com/paper/1606.02518/) (2016)
- [Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithms](https://mlsanity.com/paper/2007.11446/) (2020)
- [Riemannian Metric Learning via Optimal Transport](https://mlsanity.com/paper/2205.09244/) (2022)
- [Maximum Volume Inscribed Ellipsoid: A New Simplex-Structured Matrix Factorization Framework via Facet Enumeration and Convex Optimization](https://mlsanity.com/paper/1708.02883/) (2017)
- [Manifold Learning by Mixture Models of VAEs for Inverse Problems](https://mlsanity.com/paper/2303.15244/) (2023)
- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [Hyperbolic VAE via Latent Gaussian Distributions](https://mlsanity.com/paper/2209.15217/) (2022)
- [A Review on Riemannian Metric Learning: Closer to You than You Imagine](https://mlsanity.com/paper/2503.05321/) (2025)
- [LDLE: Low Distortion Local Eigenmaps](https://mlsanity.com/paper/2101.11055/) (2021)
- [Representation Learning via Manifold Flattening and Reconstruction](https://mlsanity.com/paper/2305.01777/) (2023)
- [Learning Distances from Data with Normalizing Flows and Score Matching](https://mlsanity.com/paper/2407.09297/) (2024)
- [Learning Extremal Representations with Deep Archetypal Analysis](https://mlsanity.com/paper/2002.00815/) (2020)
- [K-Deep Simplex: Deep Manifold Learning via Local Dictionaries](https://mlsanity.com/paper/2012.02134/) (2020)
- [Pulling back symmetric Riemannian geometry for data analysis](https://mlsanity.com/paper/2403.06612/) (2024)
- [Finding Archetypal Spaces Using Neural Networks](https://mlsanity.com/paper/1901.09078/) (2019)
- [Optimal Regularization for a Data Source](https://mlsanity.com/paper/2212.13597/) (2022)
- [Curvature corrected tangent space-based approximation of manifold-valued data](https://mlsanity.com/paper/2306.00507/) (2023)
- [Geometry-Preserving Encoder/Decoder in Latent Generative Models](https://mlsanity.com/paper/2501.09876/) (2025)
- [A Survey on Archetypal Analysis](https://mlsanity.com/paper/2504.12392/) (2025)

## Built on by (most cited first, 1 of 1)

- [Beyond Unimodal Bases: Pullback Geometry for Multimodal Data](https://mlsanity.com/paper/2610.00708/) (2026)

## Abstract (arXiv)

Classical archetypal analysis is appealing for its interpretability, but its linear geometry can limit performance on data with strongly non-linear structure; at the same time, existing neural extensions improve flexibility while often weakening the geometric meaning of archetypes and interpolations. In this work, we develop a Riemannian version of archetypal analysis based on data-driven pullback geometry for real-valued data, with the goal of combining the interpretability of classical archetypal analysis with the expressive power of modern non-linear models. We introduce a class of deformed star distributions together with associated pullback Riemannian geometry to provide a statistical interpretation of the resulting manifold mappings, define the Riemannian archetypal mapping (RAM) as a projection onto the manifold of geodesically convex combinations of archetypes, and propose a practical optimization scheme based on convex relaxation followed by non-convex refinement. We further propose a learning scheme that yields reasonable, albeit generally suboptimal, deformed star distributions from data. Experiments on synthetic examples and MNIST show that the resulting framework produces meaningful geodesics, useful denoising projections, and geometry-aware classifications, while also clarifying where current optimization limitations remain.