# Measuring dissimilarity with diffeomorphism invariance

arXiv: https://arxiv.org/abs/2202.05614 · HTML: https://mlsanity.com/paper/2202.05614/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 49 references that are in the map.
- First posted on arXiv 11 Feb 2022.

## Builds on (most cited first, 11 of 11)

- [Soft-DTW: a Differentiable Loss Function for Time-Series](https://mlsanity.com/paper/1703.01541/) (2017)
- [Less is More: Nystr"om Computational Regularization](https://mlsanity.com/paper/1507.04717/) (2015)
- [Convolutional Kernel Networks](https://mlsanity.com/paper/1406.3332/) (2014)
- [Differentiable Divergences Between Time Series](https://mlsanity.com/paper/2010.08354/) (2020)
- [Finding Global Minima via Kernel Approximations](https://mlsanity.com/paper/2012.11978/) (2020)
- [PSD Representations for Effective Probability Models](https://mlsanity.com/paper/2106.16116/) (2021)
- [A Dimension-free Computational Upper-bound for Smooth Optimal Transport Estimation](https://mlsanity.com/paper/2101.05380/) (2021)
- [Aligning Time Series on Incomparable Spaces](https://mlsanity.com/paper/2006.12648/) (2020)
- [Learning with Group Invariant Features: A Kernel Perspective](https://mlsanity.com/paper/1506.02544/) (2015)
- [Relative stability toward diffeomorphisms indicates performance in deep nets](https://mlsanity.com/paper/2105.02468/) (2021)
- [Time Series Alignment with Global Invariances](https://mlsanity.com/paper/2002.03848/) (2020)

## Abstract (arXiv)

Measures of similarity (or dissimilarity) are a key ingredient to many machine learning algorithms. We introduce DID, a pairwise dissimilarity measure applicable to a wide range of data spaces, which leverages the data's internal structure to be invariant to diffeomorphisms. We prove that DID enjoys properties which make it relevant for theoretical study and practical use. By representing each datum as a function, DID is defined as the solution to an optimization problem in a Reproducing Kernel Hilbert Space and can be expressed in closed-form. In practice, it can be efficiently approximated via Nystr"om sampling. Empirical experiments support the merits of DID.