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

This paper introduces the Information-Estimation Metric (IEM), a novel distance function derived from the relationship between information and estimation theories that leverages learned denoisers to adapt to complex data geometries and achieve state-of-the-art performance in predicting human perceptual image quality.

Original authors: Guy Ohayon, Pierre-Etienne H. Fiquet, Florentin Guth, Jona Ballé, Eero P. Simoncelli

Published 2026-02-09

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CC BY 4.0

Original authors: Guy Ohayon, Pierre-Etienne H. Fiquet, Florentin Guth, Jona Ballé, Eero P. Simoncelli

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to measure the "distance" between two things. In the physical world, you might use a ruler to measure how far apart two cities are. But what if you want to measure the distance between two ideas, two images, or two sounds? How do you decide if two pictures of a cat are "close" to each other, or if a picture of a cat and a picture of a toaster are "far" apart?

Usually, computers just count the differences in pixels (like counting how many bricks are different between two walls). But humans don't see it that way. We might think two slightly blurry photos of a cat are very similar, while two sharp photos of a cat and a toaster are very different, even if the pixel counts suggest otherwise.

This paper introduces a new way for computers to measure this "perceptual distance," called the Information-Estimation Metric (IEM). Here is how it works, explained simply:

1. The "Denoising" Detective

Imagine you have a very clear photo of a landscape. Now, imagine someone throws a bucket of snow over it, blurring the image.

The Old Way: A standard computer metric just looks at the blurry photo and the clear photo and counts the pixel differences.
The IEM Way: The IEM asks a different question: "If I give you this blurry photo, how well can you guess what the original clear photo looked like?"

The authors use a special AI (a "denoiser") trained on millions of images to act as this detective. The AI tries to clean up the blurry photo.

If the AI is confident and cleans it up perfectly, the distance is small.
If the AI is confused and makes a bad guess, the distance is large.

2. The "Snowstorm" Analogy

The paper doesn't just test the AI with one level of blur. It tests it with a whole range of "snowstorms," from a light dusting to a heavy blizzard.

The Concept: The IEM measures the distance between two images by comparing how the AI's "guessing strategy" changes as the snow gets heavier.
The Metaphor: Imagine two people standing in a foggy field.
- If they are standing on a solid, familiar path (a common image, like a clear blue sky), the fog doesn't confuse them much. They know where they are.
- If they are standing in a weird, confusing spot (a rare or strange image), the fog makes them very unsure.
- The IEM measures the distance between two people by watching how their "uncertainty" changes as the fog rolls in and out. If their uncertainty patterns are similar, they are "close." If their confusion is totally different, they are "far apart."

3. Why This is Special (The "Shape" of Data)

The paper claims that this method learns the shape of the world without needing a teacher.

Supervised Learning (The Old Way): To teach a computer what humans think is "similar," you usually need thousands of humans to label photos (e.g., "These two look alike," "These two look different"). This is expensive and noisy.
The IEM Way (Unsupervised): The IEM learns purely by looking at the images themselves. It figures out that "cats" usually look a certain way and "clouds" look another way. It builds a map of what is "normal" and what is "weird" just by studying the data.

The authors proved mathematically that this creates a valid "distance" (it follows the rules of geometry) and that for simple data, it acts like a standard ruler. But for complex data (like real photos), it bends and stretches to fit the shape of the data, just like a flexible ruler that adapts to the terrain.

4. Does it Work?

The researchers tested their new "flexible ruler" on a database of images (ImageNet) and compared it to how humans judge image quality.

The Result: Even though the IEM was never shown a single human label saying "this looks good" or "this looks bad," it predicted human opinions just as well as, or better than, the most advanced computer programs that were trained on human labels.
The "Maximum Differentiation" Test: They tried to trick the system. They took a photo and added noise to it, then asked the computer to change the noise to make the image look as different as possible from the original, while keeping the "pixel count" (PSNR) the same.
- Other metrics produced images that looked like abstract art or nonsense.
- The IEM produced images that still looked like the original photo, just with different noise. This suggests the IEM understands the "structure" of the image better than the others.

Summary

The Information-Estimation Metric is a new tool that measures how "far apart" two images are by seeing how hard it is for a smart AI to clean them up. Instead of counting pixels, it looks at the "uncertainty" of the AI. It learns the shape of the world on its own, without needing humans to grade its homework, and it turns out to be surprisingly good at mimicking how humans see similarity.

Technical Summary: Learning a Distance Measure from the Information-Estimation Geometry of Data

Problem Statement

Distance functions are fundamental to data organization and interpretation, yet deriving a meaningful distance metric for natural signals (e.g., images) that aligns with human perception remains a significant challenge. Existing approaches often rely on supervised learning using human-labeled databases, which are costly to annotate, noisy, and difficult to interpret mathematically. Conversely, purely unsupervised methods based on information theory (e.g., mutual information) are often agnostic to the geometric structure of the data distribution. The core problem addressed is how to derive a global distance function and a local Riemannian metric solely from the geometry of an underlying continuous probability density $p_x$ over a signal domain, without relying on human labels.

Methodology: The Information-Estimation Metric (IEM)

The authors introduce the Information-Estimation Metric (IEM), a distance function derived from the relationship between information theory and estimation theory. The methodology rests on the Tweedie–Miyasawa formula, which links the score function (gradient of the log-density) of a blurred signal to the error of an optimal denoiser.

Theoretical Construction

Observation Channel: The authors define a Gaussian observation channel $y_\gamma = \gamma x + w_\gamma$ , where $w_\gamma$ is additive Gaussian noise. As the signal-to-noise ratio (SNR) $\gamma$ increases, the distribution $p_{y_\gamma}$ "zooms in" on the original density $p_x$ .
Pointwise I-MMSE: Building on the I-MMSE relationship, the log-probability of a signal is decomposed into an integral of denoising errors (or equivalently, score vectors) across all SNR levels.
Metric Definition: The IEM between two signals $x_1$ and $x_2$ is defined as the square root of the expected quadratic variation of the log-probability ratio process. Specifically, it is the integrated mean square error (MSE) between the score vector fields of the blurred density around $x_1$ and $x_2$ over a range of SNR levels $[0, \Gamma]$ :
$\text{IEM}(x_1, x_2, \Gamma) := \left( \int_0^\Gamma \mathbb{E} \left[ \| \nabla \log p_{y_\gamma}(\gamma x_1 + w_\gamma) - \nabla \log p_{y_\gamma}(\gamma x_2 + w_\gamma) \|^2 \right] d\gamma \right)^{1/2}$
Generalization: The authors propose a Generalized IEM ( $\text{IEM}_f$ ) by applying a scalar function $f$ to the log-probability ratio process before computing the quadratic variation, allowing for flexibility in weighting different aspects of the distribution.

Local Geometry

The paper derives a local second-order approximation of the squared IEM, yielding a Riemannian metric $G(x, \Gamma)$ . This local metric is shown to be:

Proportional to the expected squared Hessian of the log-density (local curvature).
Related to the posterior covariance of the signal given noisy observations.
Most sensitive in regions of high log-density curvature and to perturbations that induce large changes in signal probability.

Implementation

To compute the IEM in practice, the authors train a neural denoiser (specifically an Hourglass Diffusion Transformer) on the ImageNet dataset. This denoiser approximates the MMSE estimator $E[x|y_\gamma]$ , which is equivalent to the score function via the Tweedie–Miyasawa formula. The distance is then computed by numerically integrating the squared difference of the denoiser's outputs (or derived scores) across noise levels.

Key Contributions

Novel Metric Definition: The introduction of the IEM, a valid global distance metric derived purely from the geometry of a probability density, proven to satisfy symmetry, non-negativity, identity of indiscernibles, and the triangle inequality.
Theoretical Unification: The paper proves that for Gaussian distributions, the IEM coincides with the Mahalanobis distance. For non-Gaussian distributions, it adapts locally and globally to the "data support," behaving like a locally adaptive Mahalanobis distance.
Unsupervised Learning: The framework allows for the learning of a perceptual distance metric from unlabeled data by training a denoiser, bypassing the need for human perceptual labels during the metric definition phase.
Generalized Framework: The proposal of $\text{IEM}_f$ allows for the adaptation of the metric to different types of human data by learning a function $f$ on top of the unsupervised base metric.

Experimental Results

The authors evaluated the IEM on standard full-reference image quality assessment benchmarks (TID2013, LIVE, CSIQ) and a texture-specific dataset (TQD).

Performance on Natural Images: The unsupervised IEM (with $\Gamma=1/4$ ) achieved Spearman's rank correlation coefficients (SRCC) competitive with state-of-the-art supervised metrics (e.g., LPIPS, DISTS, PieAPP) on TID2013, LIVE, and CSIQ, despite being trained without human labels.
Texture Sensitivity: The variant $\text{IEM}_{sq}$ (using a quadratic function $f$ ) outperformed most methods on the TQD texture dataset, a domain where many standard metrics struggle.
Unified Performance: A supervised variant, $\text{IEM}_{f_\omega}$ , which learns only the function $f$ on top of the pre-trained denoiser, achieved strong performance across all datasets simultaneously, demonstrating the flexibility of the framework.
Maximum Differentiation Competition: In experiments minimizing perceptual distance under a fixed PSNR constraint, minimizing the IEM produced images that preserved geometric structure and appeared perceptually natural, whereas minimizing other metrics (like DISTS) resulted in unnatural artifacts. Maximizing the IEM correctly identified unstructured noise as the most perceptually damaging distortion.

Significance and Claims

The paper claims that the IEM provides a principled, mathematically grounded approach to deriving perceptual distances from the statistical geometry of data. Its significance lies in:

Interpretability: Unlike black-box supervised metrics, the IEM is derived from fundamental relationships between probability, information, and estimation, offering a clearer mathematical explanation for perceptual judgments.
Data Efficiency: It demonstrates that high-quality perceptual metrics can be learned from unlabeled data, reducing reliance on expensive and noisy human annotation.
Robustness: The metric's ability to adapt to both natural images and textures, and its performance in optimization tasks (inverse problems), suggests it captures the "data support" more effectively than existing Euclidean or supervised alternatives.

The authors acknowledge limitations, primarily the computational cost of numerical integration compared to standard feed-forward metrics, and the need for a systematic principle to choose the hyperparameter $\Gamma$ and the function $f$ . They suggest future work in applying these principles to other signal types (e.g., audio) and using the metric for unsupervised clustering and generative model evaluation.