The isometry group of Wasserstein spaces: the Hilbertian case

Abstract

Motivated by Kloeckner's result on the isometry group of the quadratic Wasserstein space W2(R n ), we describe the isometry group Isom(Wp(E)) for all parameters 0 < p < ∞ and for all separable real Hilbert spaces E. In particular, we show that Wp(X) is isometrically rigid for all Polish space X whenever 0 < p < 1. This is a consequence of our more general result: we prove that W1(X) is isometrically rigid if X is a complete separable metric space that satisfies the strict triangle inequality. Furthermore, we show that this latter rigidity result does not generalise to parameters p > 1, by solving Kloeckner's problem affirmatively on the existence of mass-splitting isometries. As a byproduct of our methods we also obtain the isometric rigidity of Wp(X) for all complete and separable ultrametric spaces X and parameters 0 < p < ∞.

Isometric rigidity of Wasserstein tori and spheres

“…In fact, this method works in the $$0&lt;p&lt;1$$ case as well, but we decided to not include it in the main body. On the one hand, we have already proved in [10] that p ‐Wasserstein spaces are all isometrically rigid if $$0&lt;p&lt;1$$, regardless of what the underlying space is. On the other hand, as the definition of the p ‐Wasserstein distance is slightly different in the $$0&lt;p&lt;1$$ case, we should add one more branch to all proofs, without any serious novelty.…”

“…In this case, $$(X,\varrho )$$ is a complete separable metric space and $$\mathcal {S}=\mathcal {P}(X)$$. ‐The quadratic Wasserstein metric $$d_{\mathcal {W}_2}$$ turned out to be very effective in a wide range of AI applications including pattern recognition and image processing problems. In these applications, $$(X,r)$$ is typically the n ‐dimensional Euclidean space and $$\mathcal {S}$$ is the collection of all Borel probability measures with finite second moment. In recent years, there has been a considerable interest in the characterisation of the above‐mentioned (and many other) metric spaces of measures, see, for example, [1, 3, 4, 6–12, 15, 17, 19]. In most cases, it turned out that isometries of $$\mathcal {S}$$ are strongly related to self‐maps of the underlying space X .…”

Quantum Wasserstein isometries on the qubit state space

“…[1, 3, 18, 19, 25, 26, 30-37, 46, 50-52, 57, 64]. We highlight three papers which deals with the structure of Wasserstein isometries over Euclidean spaces [33,35,46]. In [46] Kloeckner described the isometry group of the quadratic Wasserstein space W 2 (R n ).…”

“…In [46] Kloeckner described the isometry group of the quadratic Wasserstein space W 2 (R n ). Later in [33] and [35] we gave a complete characterisation of isometries of p-Wasserstein spaces over real and separable Hilbert spaces for all parameters 1 ≤ p < ∞.…”