# Log-Euclidean Signatures for Intrinsic Distances Between Unaligned Datasets

arXiv: https://arxiv.org/abs/2202.01671 · HTML: https://mlsanity.com/paper/2202.01671/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 72 references that are in the map.
- First posted on arXiv 3 Feb 2022.

## Builds on (most cited first, 25 of 25)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [WILDS: A Benchmark of in-the-Wild Distribution Shifts](https://mlsanity.com/paper/2012.07421/) (2020)
- [Striving for Simplicity: The All Convolutional Net](https://mlsanity.com/paper/1412.6806/) (2014)
- [Meta-Learning with Differentiable Convex Optimization](https://mlsanity.com/paper/1904.03758/) (2019)
- [TADAM: Task dependent adaptive metric for improved few-shot learning](https://mlsanity.com/paper/1805.10123/) (2018)
- [Meta-learning with differentiable closed-form solvers](https://mlsanity.com/paper/1805.08136/) (2018)
- [Task2Vec: Task Embedding for Meta-Learning](https://mlsanity.com/paper/1902.03545/) (2019)
- [BREEDS: Benchmarks for Subpopulation Shift](https://mlsanity.com/paper/2008.04859/) (2020)
- [Geometry Score: A Method For Comparing Generative Adversarial Networks](https://mlsanity.com/paper/1802.02664/) (2018)
- [NetLSD: Hearing the Shape of a Graph](https://mlsanity.com/paper/1805.10712/) (2018)
- [The Shape of Data: Intrinsic Distance for Data Distributions](https://mlsanity.com/paper/1905.11141/) (2019)
- [Generalization of Model-Agnostic Meta-Learning Algorithms: Recurring and Unseen Tasks](https://mlsanity.com/paper/2102.03832/) (2021)
- [Manifold Topology Divergence: a Framework for Comparing Data Manifolds](https://mlsanity.com/paper/2106.04024/) (2021)
- [Fixed-Rank Approximation of a Positive-Semidefinite Matrix from Streaming Data](https://mlsanity.com/paper/1706.05736/) (2017)
- [Fairness Warnings and Fair-MAML: Learning Fairly with Minimal Data](https://mlsanity.com/paper/1908.09092/) (2019)
- [Isospectralization, or how to hear shape, style, and correspondence](https://mlsanity.com/paper/1811.11465/) (2018)
- [On sensitivity of meta-learning to support data](https://mlsanity.com/paper/2110.13953/) (2021)
- [Impact of signal-to-noise ratio and bandwidth on graph Laplacian spectrum from high-dimensional noisy point cloud](https://mlsanity.com/paper/2011.10725/) (2020)
- [Similarity of Classification Tasks](https://mlsanity.com/paper/2101.11201/) (2021)
- [Spatiotemporal Analysis Using Riemannian Composition of Diffusion Operators](https://mlsanity.com/paper/2201.08530/) (2022)
- [Quality Evaluation of GANs Using Cross Local Intrinsic Dimensionality](https://mlsanity.com/paper/1905.00643/) (2019)
- [Spectral Flow on the Manifold of SPD Matrices for Multimodal Data Processing](https://mlsanity.com/paper/2009.08062/) (2020)
- [Task-similarity Aware Meta-learning through Nonparametric Kernel Regression](https://mlsanity.com/paper/2006.07212/) (2020)

## Built on by (most cited first, 5 of 5)

- [Hyperbolic Diffusion Embedding and Distance for Hierarchical Representation Learning](https://mlsanity.com/paper/2305.18962/) (2023)
- [Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy](https://mlsanity.com/paper/2410.21107/) (2024)
- [Measuring Time-Series Dataset Similarity using Wasserstein Distance](https://mlsanity.com/paper/2507.22189/) (2025)
- [Changes from Classical Statistics to Modern Statistics and Data Science](https://mlsanity.com/paper/2211.03756/) (2022)
- [Generalized infinite dimensional Alpha-Procrustes based geometries](https://mlsanity.com/paper/2511.09801/) (2025)

## Abstract (arXiv)

The need for efficiently comparing and representing datasets with unknown alignment spans various fields, from model analysis and comparison in machine learning to trend discovery in collections of medical datasets. We use manifold learning to compare the intrinsic geometric structures of different datasets by comparing their diffusion operators, symmetric positive-definite (SPD) matrices that relate to approximations of the continuous Laplace-Beltrami operator from discrete samples. Existing methods typically assume known data alignment and compare such operators in a pointwise manner. Instead, we exploit the Riemannian geometry of SPD matrices to compare these operators and define a new theoretically-motivated distance based on a lower bound of the log-Euclidean metric. Our framework facilitates comparison of data manifolds expressed in datasets with different sizes, numbers of features, and measurement modalities. Our log-Euclidean signature (LES) distance recovers meaningful structural differences, outperforming competing methods in various application domains.