# On interpretability and proper latent decomposition of autoencoders

arXiv: https://arxiv.org/abs/2211.08345 · HTML: https://mlsanity.com/paper/2211.08345/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 4 references that are in the map.
- First posted on arXiv 15 Nov 2022.

## Built on by (most cited first, 6 of 6)

- [Stability analysis of chaotic systems in latent spaces](https://mlsanity.com/paper/2410.00480/) (2024)
- [Decoder Decomposition for the Analysis of the Latent Space of Nonlinear Autoencoders With Wind-Tunnel Experimental Data](https://mlsanity.com/paper/2404.19660/) (2024)
- [Short and Straight: Geodesics on Differentiable Manifolds](https://mlsanity.com/paper/2305.15228/) (2023)
- [Proper Latent Decomposition](https://mlsanity.com/paper/2412.00785/) (2024)
- [Enhancing material behavior discovery using embedding-oriented Physically-Guided Neural Networks with Internal Variables](https://mlsanity.com/paper/2508.00959/) (2025)
- [AI Emulation of Stochastic Sudden Stratospheric Warming with Interpretable Latent Structure](https://mlsanity.com/paper/2610.02069/) (2026)

## Abstract (arXiv)

The dynamics of a turbulent flow tend to occupy only a portion of the phase space at a statistically stationary regime. From a dynamical systems point of view, this portion is the attractor. The knowledge of the turbulent attractor is useful for two purposes, at least: (i) We can gain physical insight into turbulence (what is the shape and geometry of the attractor?), and (ii) it provides the minimal number of degrees of freedom to accurately describe the turbulent dynamics. Autoencoders enable the computation of an optimal latent space, which is a low-order representation of the dynamics. If properly trained and correctly designed, autoencoders can learn an approximation of the turbulent attractor, as shown by Doan, Racca and Magri (2022). In this paper, we theoretically interpret the transformations of an autoencoder. First, we remark that the latent space is a curved manifold with curvilinear coordinates, which can be analyzed with simple tools from Riemann geometry. Second, we characterize the geometrical properties of the latent space. We mathematically derive the metric tensor, which provides a mathematical description of the manifold. Third, we propose a method -- proper latent decomposition (PLD) -- that generalizes proper orthogonal decomposition of turbulent flows on the autoencoder latent space. This decomposition finds the dominant directions in the curved latent space. This theoretical work opens up computational opportunities for interpreting autoencoders and creating reduced-order models of turbulent flows.