# Proper Latent Decomposition

arXiv: https://arxiv.org/abs/2412.00785 · HTML: https://mlsanity.com/paper/2412.00785/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 16 references that are in the map.
- First posted on arXiv 1 Dec 2024.

## Builds on (most cited first, 6 of 6)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Deep Neural Networks for Nonlinear Model Order Reduction of Unsteady Flows](https://mlsanity.com/paper/2007.00936/) (2020)
- [Geometric Autoencoders -- What You See is What You Decode](https://mlsanity.com/paper/2306.17638/) (2023)
- [On interpretability and proper latent decomposition of autoencoders](https://mlsanity.com/paper/2211.08345/) (2022)
- [Computing distances and means on manifolds with a metric-constrained Eikonal approach](https://mlsanity.com/paper/2404.08754/) (2024)

## Abstract (arXiv)

In this paper, we introduce the proper latent decomposition (PLD) as a generalization of the proper orthogonal decomposition (POD) on manifolds. PLD is a nonlinear reduced-order modeling technique for compressing high-dimensional data into nonlinear coordinates. First, we compute a reduced set of intrinsic coordinates (latent space) to accurately describe a flow with fewer degrees of freedom than the numerical discretization. The latent space, which is geometrically a manifold, is inferred by an autoencoder. Second, we leverage tools from differential geometry to develop numerical methods for operating directly on the latent space; namely, a metric-constrained Eikonal solver for distance computations. With this proposed numerical framework, we propose an algorithm to perform PLD on the manifold. Third, we demonstrate results for a laminar flow case and the turbulent Kolmogorov flow. For the laminar flow case, we are able to identify a semi-analytical expression for the solution of Navier-Stokes; in the Kolmogorov flow case, we are able to identify a dominant mode that exhibits physical structures, which are compared with POD. This work opens opportunities for analyzing autoencoders and latent spaces, nonlinear reduced-order modeling and scientific insights into the structure of high-dimensional data.