Signed Wasserstein Distance Overview

• Signed Wasserstein distance is a formulation that extends optimal transport to signed measures by decomposing data into positive and negative components.
• For p=1, it defines a true metric via Kantorovich duality, while for p>1 it is treated as a transport cost with practical stability and robustness features.
• Empirical studies show its noise robustness in imaging and cryo-EM applications, revealing phenomena like decreasing transport cost with added noise.

Searching arXiv for the cited work and closely related signed-Wasserstein formulations. {"query":"(Lager et al., 1 Oct 2025) signed Wasserstein distance images Mainini Piccoli Rossi Tournus virtual persistence diagrams", "max_results": 10} The signed Wasserstein distance is a family of optimal-transport constructions extending Wasserstein geometry from nonnegative measures to signed measures, real-valued images, and algebraic objects such as virtual persistence diagrams. In the image setting, a standard formulation decomposes each signed measure into positive and negative parts and transports the aggregate positive content of one object against the aggregate negative content of the other; in the $p=1$ case this yields a metric, whereas for $p>1$ it is generally interpreted as a transport cost rather than a metric. Closely related arXiv literature also develops a signed $1$-Wasserstein distance via weak transport plans and a Kantorovich norm based on generalized Wasserstein distances with mass creation and annihilation. Taken together, these works place signed Wasserstein constructions at the intersection of optimal transport, signed-measure geometry, image analysis, PDEs with source terms, and topological data analysis (Lager et al., 1 Oct 2025, Piccoli et al., 2019, Bubenik et al., 2020).

1. Classical and signed formulations

For a metric space $(\mathcal X,d)$ and two nonnegative measures $\mu,\nu$ of equal mass, the classical $p$-Wasserstein distance is

$$W_p(\mu,\nu) := \biggl( \inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal X\times\mathcal X} d(x,y)^p\, d\pi(x,y) \biggr)^{1/p},$$

where $\Gamma(\mu,\nu)$ is the set of couplings of $\mu$ and $\nu$. In image applications, $p>1$0 and $p>1$1 are often histograms on a finite grid $p>1$2 with total mass $p>1$3 (Lager et al., 1 Oct 2025).

For $p>1$4, Kantorovich–Rubinstein duality gives

$p>1$5

with the supremum taken over $p>1$6-Lipschitz test functions. This dual form remains central when extending Wasserstein geometry to signed objects (Lager et al., 1 Oct 2025).

A widely used signed extension in imaging, following Mainini (2012), starts from the Jordan decompositions

$p>1$7

and defines

$p>1$8

The signed $p>1$9-Wasserstein cost is then

$1$0

The construction is tailored to real-valued data, including images after additive noise, where negative pixel values are unavoidable. In the formulation summarized for image analysis, $1$1 is a metric on signed measures, while for $1$2 the triangle inequality fails, so the object is retained as a useful transport cost rather than a genuine metric (Lager et al., 1 Oct 2025).

2. Metric status, duality, and normed-space viewpoints

The $1$3 case admits several structurally important extensions. Piccoli, Rossi, and Tournus define a generalized Wasserstein distance $1$4 for nonnegative Radon measures of possibly different masses, where one may first trim mass and then transport the remainder: $1$5 For signed measures, they apply the “AMS trick” in the $1$6 case: $1$7 and define the Kantorovich norm of a signed measure by

$1$8

For $1$9, this norm coincides with the Hanin–Fortet–Mourier norm, equivalently the bounded–Lipschitz distance (Piccoli et al., 2019).

The same work establishes that $(\mathcal X,d)$0 is a normed vector space, although not Banach in general; completeness is recovered on Cauchy sequences with uniformly bounded mass and tightness. The norm also supports stability estimates for Lipschitz flows, which is the basis for existence and uniqueness results for a nonlocal transport equation with source term (Piccoli et al., 2019).

A different but compatible $(\mathcal X,d)$1-Wasserstein extension appears in the theory of virtual persistence diagrams. Bubenik and Elchesen define a signed $(\mathcal X,d)$2-Wasserstein distance on finite signed Radon measures $(\mathcal X,d)$3 via weak transport plans $(\mathcal X,d)$4 with cost

$(\mathcal X,d)$5

and

$(\mathcal X,d)$6

They show that this is equivalent to the transshipment form

$(\mathcal X,d)$7

and prove the dual identity

$(\mathcal X,d)$8

This suggests that, at least for $(\mathcal X,d)$9, several signed-Wasserstein constructions converge on a common transport principle built from the interaction of positive and negative parts (Bubenik et al., 2020).

3. Noise sensitivity for images

A recent image-analysis study examines the sensitivity of signed Wasserstein costs to pixel-wise additive noise on an $\mu,\nu$0 grid with cyclic boundary conditions. The noise model is a zero-sum Gaussian field $\mu,\nu$1 satisfying

$\mu,\nu$2

so that $\mu,\nu$3 almost surely and each $\mu,\nu$4 (Lager et al., 1 Oct 2025).

Under this model, the paper states a non-asymptotic upper bound for any integer $\mu,\nu$5: $\mu,\nu$6 hence by Jensen’s inequality

$\mu,\nu$7

The same source also states, in its abstract and summary discussion, that the error in the signed $\mu,\nu$8-Wasserstein distance scales with the square root of the noise standard deviation, in contrast with the Euclidean norm, which scales linearly in $\mu,\nu$9. The Euclidean comparison is

$p$0

so the reported qualitative contrast is $p$1 behavior for $p$2 versus square-root behavior for $p$3 (Lager et al., 1 Oct 2025).

The proof strategy summarized in the same work relies on a multiscale dyadic upper bound due to Weed and Bach: $p$4 applied to $p$5 and $p$6. The argument exploits the scaling of block variances across dyadic partitions and the balance condition $p$7 (Lager et al., 1 Oct 2025).

4. Empirical behavior and the “decreasing-distance” phenomenon

The image paper reports exact computations using POT Python on $p$8 random $p$9 microscopy images from DOTMark, with zero-sum Gaussian noise on $$W_p(\mu,\nu) := \biggl( \inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal X\times\mathcal X} d(x,y)^p\, d\pi(x,y) \biggr)^{1/p},$$0 and $$W_p(\mu,\nu) := \biggl( \inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal X\times\mathcal X} d(x,y)^p\, d\pi(x,y) \biggr)^{1/p},$$1 ranging from $$W_p(\mu,\nu) := \biggl( \inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal X\times\mathcal X} d(x,y)^p\, d\pi(x,y) \biggr)^{1/p},$$2 to $$W_p(\mu,\nu) := \biggl( \inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal X\times\mathcal X} d(x,y)^p\, d\pi(x,y) \biggr)^{1/p},$$3. Empirical log-log slopes from least-squares fits are approximately $$W_p(\mu,\nu) := \biggl( \inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal X\times\mathcal X} d(x,y)^p\, d\pi(x,y) \biggr)^{1/p},$$4 for $$W_p(\mu,\nu) := \biggl( \inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal X\times\mathcal X} d(x,y)^p\, d\pi(x,y) \biggr)^{1/p},$$5, $$W_p(\mu,\nu) := \biggl( \inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal X\times\mathcal X} d(x,y)^p\, d\pi(x,y) \biggr)^{1/p},$$6 for $$W_p(\mu,\nu) := \biggl( \inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal X\times\mathcal X} d(x,y)^p\, d\pi(x,y) \biggr)^{1/p},$$7, and $$W_p(\mu,\nu) := \biggl( \inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal X\times\mathcal X} d(x,y)^p\, d\pi(x,y) \biggr)^{1/p},$$8 for $$W_p(\mu,\nu) := \biggl( \inf_{\pi\in\Gamma(\mu,\nu)} \int_{\mathcal X\times\mathcal X} d(x,y)^p\, d\pi(x,y) \biggr)^{1/p},$$9. The reported $\Gamma(\mu,\nu)$0 slope of $\Gamma(\mu,\nu)$1 is presented as agreement with the square-root scaling interpretation of the theoretical analysis (Lager et al., 1 Oct 2025).

The same study evaluates robustness of inter-image distances by corrupting two fixed distinct DOTMark images with independent zero-sum noise for $\Gamma(\mu,\nu)$2 and considering the distance ratio

$\Gamma(\mu,\nu)$3

Averaged over $\Gamma(\mu,\nu)$4 trials, the $\Gamma(\mu,\nu)$5 ratio diverges from $\Gamma(\mu,\nu)$6 earliest, $\Gamma(\mu,\nu)$7 next, and $\Gamma(\mu,\nu)$8 remains within approximately $\Gamma(\mu,\nu)$9 over the largest $\mu$0. Within the scope of these experiments, the signed $\mu$1-Wasserstein cost is therefore the most stable of the three reported quantities (Lager et al., 1 Oct 2025).

A particularly notable observation is the “decreasing-distance” phenomenon. For two one-pixel images located at opposite corners, $\mu$2 initially decreases as $\mu$3 increases. The paper’s interpretation is that noise can “bridge” the gap by creating intermediate mass that makes transport cheaper; optimal couplings visualized as mass-flow maps show noisy pixels mediating transport. This directly contradicts the common intuition that additive noise must monotonically increase a transport-based discrepancy (Lager et al., 1 Oct 2025).

5. Cryo-EM and other application domains

The most detailed application in the cited literature is a cryo-electron microscopy case study involving $\mu$4 projection images of bacterial Hsp90 protein in different orientations, generated via CryoJAX, with random $\mu$5D shifts and heavy noise $\mu$6. For each of $\mu$7, $\mu$8, and $\mu$9, the paper computes $\nu$0 pairwise distance matrices on clean images and on noisy versions averaged over $\nu$1 trials. Under noise, the $\nu$2 distance matrix loses the banded or diagonal structure associated with small rotations and shifts, whereas $\nu$3 preserves the main diagonal structure. The reported interpretation is that Wasserstein geometry remains sensitive to geometric similarity even at low SNR, making $\nu$4 better suited than $\nu$5 for aligning and clustering extremely noisy cryo-EM projections (Lager et al., 1 Oct 2025).

The same source lists broader practical implications. In cryo-EM single-particle analysis, where angular and translational heterogeneity coexist with very low SNR, $\nu$6 can serve as a robust distance for classification, alignment, and clustering. In generative modeling, including WGANs and WAEs, a noise-robust metric is described as yielding a smoother loss landscape and preventing overfitting to spurious pixel-level noise. In inverse problems such as seismic inversion and tomography, reduced sensitivity to high-frequency noise is described as potentially improving stability and convergence (Lager et al., 1 Oct 2025).

A broader mathematical implication follows from the signed-measure literature. Because generalized signed-Wasserstein constructions encode both transport and cancellation, they are naturally adapted to settings where source terms, negative components, or algebraic subtraction are intrinsic rather than pathological. This is explicit in the PDE framework of Piccoli–Rossi–Tournus and in the virtual-diagram framework of Bubenik–Elchesen (Piccoli et al., 2019, Bubenik et al., 2020).

6. Related constructions, misconceptions, and open problems

The current literature indicates that “signed Wasserstein distance” is not a single universally fixed object. One family, used for noisy images, is Mainini’s rearrangement $\nu$7. Another is the generalized Wasserstein/Kantorovich norm $\nu$8, which explicitly prices mass removal and creation. A third is the weak-coupling formulation of signed $\nu$9 for virtual persistence diagrams and signed Radon measures. This suggests a common theme—transport after accounting for positive and negative parts—together with substantial variation in admissible mass imbalance, topology, and ambient category (Lager et al., 1 Oct 2025, Piccoli et al., 2019, Bubenik et al., 2020).

A recurrent misconception is that every signed-Wasserstein extension is a metric for every $p>1$00. In the image formulation summarized above, only $p>1$01 is metric; for $p>1$02, the triangle inequality fails. Another misconception is that adding noise must increase Wasserstein discrepancy. The reported decreasing-distance effect shows that, once signed mass is allowed, noise may generate intermediate mass configurations that lower transport cost. A third misconception is that robustness properties are automatic. The available non-asymptotic results are model-dependent, being stated for a specific zero-sum Gaussian field on a cyclic grid (Lager et al., 1 Oct 2025).

Several open directions are explicitly identified. For $p>1$03, sharp non-asymptotic bounds for

$p>1$04

remain open. The current multiscale bounds capture the correct polynomial dependence in $p>1$05 and $p>1$06 but are not necessarily tight in constants. Higher $p>1$07 appears even more robust, with $p>1$08 decay, but computational cost grows. Extensions to unbalanced optimal transport, entropic regularization, or sliced-Wasserstein are described as possible routes to combining robustness with computational efficiency (Lager et al., 1 Oct 2025).

In parallel, the signed $p>1$09-Wasserstein theory has continued structural consequences beyond imaging. The Kantorovich norm yields existence and uniqueness for a nonlocal transport equation with source term under assumptions of Lipschitz dependence on the measure, while the virtual-diagram formulation embeds persistence-diagram geometry isometrically into the Arens–Eells or Lipschitz-free Banach space. A plausible implication is that signed Wasserstein geometry is most mature where $p>1$10, duality is available, and cancellation can be integrated directly into the ambient linear or metric structure (Piccoli et al., 2019, Bubenik et al., 2020).