⚡ electrical engineering

Quantifying the noise sensitivity of the Wasserstein metric for images

This paper establishes theoretical bounds demonstrating that the Wasserstein metric's sensitivity to additive Gaussian noise scales with the square root of the noise standard deviation—outperforming the linear scaling of the Euclidean metric—and validates these findings through experiments showing its superior ability to capture data geometry in high-noise settings like cryo-electron microscopy.

Original authors: Erik Lager, Gilles Mordant, Amit Moscovich

Published 2026-08-12

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CC BY 4.0

Original authors: Erik Lager, Gilles Mordant, Amit Moscovich

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 compare two pictures, maybe two photos of a cat or two maps of a city. In the world of science and computers, we often need to measure how "different" two images are. The most common way to do this is like a simple game of "spot the difference" where you just look at every single pixel (the tiny dots that make up the image) and add up how much the colors don't match. This is called the Euclidean distance. It's fast and easy, but it's a bit like judging a painting by only counting the number of paint splatters, ignoring the actual shapes and the story the picture tells.

Enter a smarter, more geometric way of measuring difference called the Wasserstein metric (or "Earth Mover's Distance"). Imagine the pixels in an image are piles of dirt. To turn one image into another, you have to move the dirt from the piles in the first image to the piles in the second. The Wasserstein metric calculates the minimum amount of "work" (distance $\times$ amount of dirt) needed to do this. It cares about the shape and structure of the image, not just the raw numbers. This is incredibly useful in fields like cryo-electron microscopy, where scientists try to build 3D models of tiny proteins from blurry, noisy 2D snapshots. But here's the catch: real-world images are messy. They are covered in "noise," which is like static on an old TV or grain in a photo. Scientists have always felt that the smart "dirt-moving" method was better at ignoring this noise than the simple "spot the difference" method, but nobody had a solid mathematical proof to explain exactly why or how much better it was.

This paper steps in to put those feelings to the test with hard math and computer simulations. The authors, Erik Lager, Gilles Mordant, and Amit Moscovich, wanted to quantify exactly how sensitive the Wasserstein metric is to noise. They treated images as grids of pixels and added random "static" (Gaussian noise) to them, then measured how much the distance between the images changed.

Their main discovery is a mathematical rule that explains why the Wasserstein metric is a superhero in noisy environments. They proved that when you add noise to an image, the error in the "dirt-moving" distance grows very slowly. Specifically, for the most common version of this metric (called the 2-Wasserstein distance), the error scales with the square root of the noise level. To put that in perspective: if you double the amount of noise, the error in the Wasserstein distance only goes up by about 1.4 times (the square root of 2). In contrast, the old-fashioned "spot the difference" method (Euclidean distance) is much more fragile; its error grows linearly. If you double the noise, the error doubles. This means the Wasserstein metric is mathematically proven to be much more robust, or "noise-tolerant," than the standard method.

The researchers didn't just stop at the math; they ran simulations to see this in action. They found that as noise increases, the standard distance metric quickly goes haywire, losing all sense of the original image's structure. The Wasserstein metric, however, stays calm and keeps the original distance between images recognizable for much longer. They even found a weird and wonderful phenomenon: sometimes, adding more noise actually makes two images look closer to each other in the Wasserstein sense. They explain this with a clever analogy: imagine you are moving a heavy box from point A to point B. If the floor is smooth, you have to drag it the whole way. But if you sprinkle some random "pebbles" (noise) on the floor between A and B, you can use those pebbles as stepping stones to move the box in smaller, easier jumps. The "noise" accidentally creates a shortcut, making the total work required to move the mass less than it was in the clean, smooth case.

Finally, the team tested this on real-world data from cryo-electron microscopy, a field where images are notoriously noisy. They showed that when the noise is heavy, the standard method fails completely, turning the data into a random mess. The Wasserstein metric, however, successfully preserved the geometric relationships between the images, allowing scientists to see the underlying structure of the proteins even when the pictures were barely recognizable. The paper concludes that while the math is complex, the takeaway is clear: for noisy images, the "dirt-moving" metric is not just a fancy alternative; it is a theoretically superior tool that handles real-world messiness far better than the traditional methods.

Technical Summary: Quantifying the Noise Sensitivity of the Wasserstein Metric for Images

Problem Statement

Optimal Transport (OT) and Wasserstein metrics are increasingly utilized as similarity scores in imaging applications, particularly in fields like single-particle cryo-electron microscopy (cryo-EM) characterized by extremely high noise levels. Empirical evidence suggests that Wasserstein metrics are more robust to noise than standard Euclidean ( $L_2$ ) distances. However, there is a lack of theoretical understanding regarding how these metrics behave under pixel-wise additive noise when images are treated as discrete measures on a fixed grid. Specifically, the relationship between the noise standard deviation ( $\sigma$ ) and the resulting error in the Wasserstein distance remains unquantified. This paper addresses the gap by deriving finite-sample expectation bounds for the sensitivity of signed Wasserstein metrics to Gaussian noise.

Methodology

Theoretical Framework

The authors model images as real-valued signals on an $n \times n$ square grid ( $G_n$ ), identified with signed discrete measures. To handle noise that may introduce negative pixel values, the study extends the Wasserstein metric to signed measures.

Signed Wasserstein Discrepancy: For two signed measures $\mu$ and $\nu$ with Jordan decompositions $\mu = \mu^+ - \mu^-$ and $\nu = \nu^+ - \nu^-$ , the signed Wasserstein discrepancy is defined as $W_p^\pm(\mu, \nu) := W_p(S_{\mu,\nu}, T_{\mu,\nu})$ , where $S_{\mu,\nu} = \mu^+ + \nu^-$ and $T_{\mu,\nu} = \nu^+ + \mu^-$ .
Noise Model: The paper considers a specific Gaussian noise model where the noise vector $N$ is drawn from a multivariate normal distribution $\mathcal{N}(0, \Sigma)$ . The covariance matrix $\Sigma$ is constructed such that the marginal distribution of each pixel is $\mathcal{N}(0, \sigma^2)$ , but the sum of all pixels is exactly zero ( $\sum N_i = 0$ ). This "zero-sum" constraint avoids the need for rescaling the total mass of the images, isolating the geometric impact of the noise.
Analytical Tools: The theoretical bounds are derived using a multiscale argument based on dyadic partitions of the pixel grid. This approach recursively matches mass imbalances across quadrants at various scales, leveraging the Kantorovich-Rubinstein duality for $p=1$ and specific coupling constructions for $p > 1$ .

Experimental Validation

The theoretical findings are corroborated through numerical experiments using the DOTMark 1.0 MicroscopyImages dataset and synthetic data.

Scaling Laws: The authors measure how the distance between an image and its noisy counterpart scales with increasing noise variance ( $\sigma^2$ ).
Robustness Comparison: They compare the stability of $L_2$ , $W_1$ , and $W_2$ metrics when two distinct images are corrupted by noise, evaluating the ratio of the noisy distance to the clean distance.
Case Study: A specific analysis is performed on cryo-EM projections of the E. coli Hsp90 protein to test the recovery of geometric relationships (rotations and translations) under high noise.

Key Contributions and Results

1. Theoretical Bounds on Noise Sensitivity

The paper establishes finite-sample expectation bounds for the signed Wasserstein discrepancy between a clean image and its noisy version (or between two noisy images).

For $p=1$ (Signed 1-Wasserstein):
The expected error between a clean image $\mu$ and a noisy version $\mu + \varepsilon$ scales as:
$\mathbb{E}[W_1^\pm(\mu, \mu+\varepsilon)] = \Theta(n \sigma \log_2 n)$
When comparing two different noisy images, the deviation from the clean distance is bounded by a term of order $O(n \sigma \log_2 n)$ .
For $p > 1$ (Signed $p$ -Wasserstein):
The behavior differs significantly. The expected $p$ -th power of the discrepancy scales linearly with $\sigma$ :
$\mathbb{E}[(W_p^\pm(\mu+\varepsilon_1, \mu+\varepsilon_2))^p] = \Theta(n \sigma)$
Consequently, by Jensen's inequality, the expected distance scales as:
$\mathbb{E}[W_p^\pm(\mu+\varepsilon_1, \mu+\varepsilon_2)] = \Theta((n \sigma)^{1/p})$
Specifically for $p=2$ , the error scales with the square root of the noise standard deviation ( $\sqrt{\sigma}$ ), whereas the Euclidean metric scales linearly with $\sigma$ . This provides a theoretical basis for the superior noise robustness of $W_2$ .

2. The "Decreasing Distance" Phenomenon

The authors identify and explain a counter-intuitive phenomenon where increasing the noise level can decrease the Wasserstein distance between two sparse images.

Mechanism: In sparse images (e.g., two distinct structures on a dark background), additive noise creates "stepping stones" (noise peaks) between the structures. The optimal transport plan can utilize these noise peaks to bridge the gap, effectively shortening the transport cost compared to the clean case where mass must be transported directly across the full distance.
Implication: This non-monotonic behavior highlights the complex interaction between noise geometry and transport costs, particularly for $p > 1$ .

3. Empirical Validation

Scaling Confirmation: Experiments confirm that $W_2$ cost scales with an exponent of approximately 0.5 relative to noise variance, matching the theoretical $\sigma^{1/2}$ prediction, while $L_2$ scales linearly.
Robustness: In pairwise image comparisons, $W_2$ maintains the original distance ratio closer to 1 over a wider range of noise levels compared to $W_1$ and $L_2$ .
Cryo-EM Application: In the cryo-EM case study, $L_2$ distance matrices degrade into random patterns under heavy noise, losing geometric structure. In contrast, $W_1$ and $W_2$ matrices preserve the global diagonal structure of the ground-truth transformations. Furthermore, $W_2$ is shown to tolerate approximately 3.5 times more noise than $W_1$ before confusing additive noise with genuine spatial shifts.

Significance and Claims

The paper claims to be the first to study the noise robustness of Wasserstein metrics for measures on a fixed grid. Its primary significance lies in providing a theoretical basis for the empirical observation that optimal transport distances are superior to Euclidean norms in noisy imaging settings.

Theoretical Insight: By proving that the error in the signed 2-Wasserstein discrepancy scales with $\sqrt{\sigma}$ (favorable compared to the linear scaling of $L_2$ ), the work explains why OT-based methods succeed in high-noise domains like cryo-EM.
Practical Guidance: The results suggest that for applications involving high noise and geometric structure preservation, $W_2$ is a more robust metric than $W_1$ or $L_2$ .
Limitations and Future Work: The authors modestly note that while their bounds capture the correct scaling behavior, they may not be tight (particularly for $p>1$ ). They also highlight that the lack of a triangle inequality for signed $W_p$ ( $p>1$ ) complicates the derivation of sharp bounds for the difference between noisy and clean expectations. Future work is suggested to investigate the practical utility of higher $p$ values and to refine bounds for unbalanced transport scenarios.

The authors conclude that this theoretical understanding is a necessary step for further methodological developments in high-noise imaging modalities, such as cryo-electron microscopy and medical imaging.