# Learning a distance measure from the information-estimation geometry of data

arXiv: https://arxiv.org/abs/2510.02514 · HTML: https://mlsanity.com/paper/2510.02514/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 70 references that are in the map.
- First posted on arXiv 2 Oct 2025.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Image Quality Assessment: Unifying Structure and Texture Similarity](https://mlsanity.com/paper/2004.07728/) (2020)
- [TOPIQ: A Top-down Approach from Semantics to Distortions for Image Quality Assessment](https://mlsanity.com/paper/2308.03060/) (2023)
- [Gradient Magnitude Similarity Deviation: A Highly Efficient Perceptual Image Quality Index](https://mlsanity.com/paper/1308.3052/) (2013)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [PieAPP: Perceptual Image-Error Assessment through Pairwise Preference](https://mlsanity.com/paper/1806.02067/) (2018)
- [Scalable High-Resolution Pixel-Space Image Synthesis with Hourglass Diffusion Transformers](https://mlsanity.com/paper/2401.11605/) (2024)
- [Robust and interpretable blind image denoising via bias-free convolutional neural networks](https://mlsanity.com/paper/1906.05478/) (2019)
- [Eigen-Distortions of Hierarchical Representations](https://mlsanity.com/paper/1710.02266/) (2017)
- [Learning normalized image densities via dual score matching](https://mlsanity.com/paper/2506.05310/) (2025)
- [What's Inside Your Diffusion Model? A Score-Based Riemannian Metric to Explore the Data Manifold](https://mlsanity.com/paper/2505.11128/) (2025)
- [Image Interpolation with Score-based Riemannian Metrics of Diffusion Models](https://mlsanity.com/paper/2504.20288/) (2025)
- [Diffusion Maps : Using the Semigroup Property for Parameter Tuning](https://mlsanity.com/paper/2203.02867/) (2022)

## Built on by (most cited first, 3 of 3)

- [Energy-Tweedie: Score meets Score, Energy meets Energy](https://mlsanity.com/paper/2512.23818/) (2025)
- [JLD: Perceptual Distance Through A Jacobian Lens](https://mlsanity.com/paper/2610.05967/) (2026)
- [Texture Interpolation as Texton Matching](https://mlsanity.com/paper/2610.18655/) (2026)

## Abstract (arXiv)

We introduce the Information-Estimation Metric (IEM), a novel form of distance function derived from an underlying continuous probability density over a domain of signals. The IEM is rooted in a fundamental relationship between information theory and estimation theory, which links the log-probability of a signal with the errors of an optimal denoiser, applied to noisy observations of the signal. In particular, the IEM between a pair of signals is obtained by comparing their denoising error vectors over a range of noise amplitudes. Geometrically, this amounts to comparing the score vector fields of the blurred density around the signals over a range of blur levels. We prove that the IEM is a valid global distance metric and derive a closed-form expression for its local second-order approximation, which yields a Riemannian metric. For Gaussian-distributed signals, the IEM coincides with the Mahalanobis distance. But for more complex distributions, it adapts, both locally and globally, to the geometry of the distribution. In practice, the IEM can be computed using a learned denoiser (analogous to generative diffusion models) and solving a one-dimensional integral. To demonstrate the value of our framework, we learn an IEM on the ImageNet database. Experiments show that this IEM is competitive with or outperforms state-of-the-art supervised image quality metrics in predicting human perceptual judgments.