# LOCA: LOcal Conformal Autoencoder for standardized data coordinates

arXiv: https://arxiv.org/abs/2004.07234 · HTML: https://mlsanity.com/paper/2004.07234/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Apr 2020.

## Built on by (most cited first, 19 of 19)

- [Tractable Density Estimation on Learned Manifolds with Conformal Embedding Flows](https://mlsanity.com/paper/2106.05275/) (2021)
- [Neural Chaos: A Spectral Stochastic Neural Operator](https://mlsanity.com/paper/2502.11835/) (2025)
- [LDLE: Low Distortion Local Eigenmaps](https://mlsanity.com/paper/2101.11055/) (2021)
- [Probabilistic Robust Autoencoders for Outlier Detection](https://mlsanity.com/paper/2110.00494/) (2021)
- [Nonlinear Isometric Manifold Learning for Injective Normalizing Flows](https://mlsanity.com/paper/2203.03934/) (2022)
- [Semi-supervised source localization in reverberant environments with deep generative modeling](https://mlsanity.com/paper/2101.10636/) (2021)
- [Multi-modal Differentiable Unsupervised Feature Selection](https://mlsanity.com/paper/2303.09381/) (2023)
- [Convergent autoencoder approximation of low bending and low distortion manifold embeddings](https://mlsanity.com/paper/2208.10193/) (2022)
- [On Manifold Learning in Plato's Cave: Remarks on Manifold Learning and Physical Phenomena](https://mlsanity.com/paper/2304.14248/) (2023)
- [Gappy local conformal auto-encoders for heterogeneous data fusion: in praise of rigidity](https://mlsanity.com/paper/2312.13155/) (2023)
- [On Learning what to Learn: heterogeneous observations of dynamics and establishing (possibly causal) relations among them](https://mlsanity.com/paper/2406.06812/) (2024)
- [Geometric regularization of autoencoders via observed stochastic dynamics](https://mlsanity.com/paper/2604.16282/) (2026)
- [Multi-Scale Geometric Autoencoder](https://mlsanity.com/paper/2509.24168/) (2025)
- [Unsupervised Acoustic Scene Mapping Based on Acoustic Features and Dimensionality Reduction](https://mlsanity.com/paper/2301.00448/) (2023)
- [LINSCAN -- A Linearity Based Clustering Algorithm](https://mlsanity.com/paper/2406.17952/) (2024)
- [A Resolution Independent Neural Operator](https://mlsanity.com/paper/2407.13010/) (2024)
- [Assessing local deformation and computing scalar curvature with nonlinear conformal regularization of decoders](https://mlsanity.com/paper/2508.20413/) (2025)
- [RANDSMAPs: Random-Feature/multi-Scale Neural Decoders with Mass Preservation](https://mlsanity.com/paper/2601.14794/) (2026)
- [The Degeneracy Distillery](https://mlsanity.com/paper/2606.23838/) (2026)

## Abstract (arXiv)

We propose a deep-learning based method for obtaining standardized data coordinates from scientific measurements.Data observations are modeled as samples from an unknown, non-linear deformation of an underlying Riemannian manifold, which is parametrized by a few normalized latent variables. By leveraging a repeated measurement sampling strategy, we present a method for learning an embedding in ℝd that is isometric to the latent variables of the manifold. These data coordinates, being invariant under smooth changes of variables, enable matching between different instrumental observations of the same phenomenon. Our embedding is obtained using a LOcal Conformal Autoencoder (LOCA), an algorithm that constructs an embedding to rectify deformations by using a local z-scoring procedure while preserving relevant geometric information. We demonstrate the isometric embedding properties of LOCA on various model settings and observe that it exhibits promising interpolation and extrapolation capabilities. Finally, we apply LOCA to single-site Wi-Fi localization data, and to 3-dimensional curved surface estimation based on a 2-dimensional projection.