Weak isomorphism of measured kernels #
Two measured kernels are weakly isomorphic when a coupling of their measures makes their pullbacks equal almost everywhere on the square of the coupling. For probability measures this is equivalently a common probability parametrization, in either of two regimes: a.e. measurable kernels into a measurable target with measurable equality, or a.e. strongly measurable kernels into a metrizable target. The coordinate projections give a canonical parametrizing space, and two measure-preserving maps with equal pulled-back kernels give an exact coupling.
The relation is defined for raw measures and kernels with arbitrary target. Symmetry and invariance under a.e. changes require only a sigma-finite first measure. The exact coupling then is sigma-finite, although the second measure need not be. Common-space introduction, reflexivity and transitivity are available in both regimes above. The measurable-target API requires no topology or essentially separable range; the metrizable API requires no measurable structure on the target. Transitivity uses gluing: an explicit gluing hypothesis needs only a sigma-finite middle measure, while a finite middle measure and a standard Borel last carrier ensure such a gluing without topology on the other carriers.
For a weak extended metric target and every nonzero exponent, an exact coupling is precisely a coupling of zero distortion. Weakly isomorphic kernels have Gromov–Wasserstein distance zero whenever their target discrepancy vanishes on the diagonal; no topology, symmetry or triangle inequality is needed for this implication. The converse requires an attainment theorem and is not asserted here.
Raw kernels admit pullback along a transport map whose target carrier has measurable equality. The identity case gives reflexivity under measurable carrier equality. These laws require no structure on the kernel values and no finiteness of the measures. The metrizable pullback and reflexivity laws use a.e. strong measurability without measurable carrier equality.
References #
- M. Bauer, F. Mémoli, T. Needham and M. Nishino, The Z-Gromov–Wasserstein distance, J. Mach. Learn. Res. 26 (2025), Definition 28, for weak isomorphism via a common probability parametrization. The coupling formulation here uses the paired pushforward and its coordinate projections. No Lean code is taken from this source.
Two measured kernels are weakly isomorphic if some coupling identifies their kernels almost everywhere on its square. For probability measures the coupling itself is a common probability parametrizing space, with the two coordinate projections as parametrizing maps.
Instances For
A coupling that identifies the pulled-back kernels almost everywhere gives a weak isomorphism. This constructs weak isomorphisms without unfolding the predicate downstream.
A raw kernel and its pullback along a transport map are weakly isomorphic when the target carrier has measurable equality. The graph plan supplies the exact coupling, so neither structure on the kernel target, measurability of the kernel nor finiteness of the source measure is required. In particular this permits duplication of a measured carrier.
Every raw kernel on a carrier with measurable equality is weakly isomorphic to itself, over any measure. No structure or measurability is required of its values.
Extract an exact coupling from a weak isomorphism, without unfolding the predicate in an importing module. Neither construction nor extraction requires structure on the target.
A weak isomorphism of probability kernels admits a common probability parametrization on
the canonical carrier X × Y.
Exchanging the two kernels preserves weak isomorphism when the first measure is sigma-finite. No measurability of the target is needed: the coordinate swap is a measurable equivalence.
Almost-everywhere changes of the kernels preserve weak isomorphism when the first measure is sigma-finite.
Exact couplings that admit a gluing give a common sigma-finite parametrization of the outer kernels. No structure or measurability is required of any kernel or its target.
A common s-finite parametrization with equal pulled-back a.e. measurable kernels gives a weak isomorphism. A measurable diagonal in the target suffices; no topology is needed.
Pullback along a measure-preserving map preserves weak isomorphism for a.e. measurable kernels whose target has measurable equality.
Every a.e. measurable kernel into a target with measurable equality is weakly isomorphic to itself over an s-finite measure.
Weak isomorphisms compose when their exact couplings admit a gluing and their outer kernels are a.e. measurable into a target with measurable equality.
Weak isomorphisms of a.e. measurable kernels are transitive when the target has measurable equality, the last carrier is standard Borel, and the middle measure is finite.
Weakly isomorphic kernels have zero Gromov–Wasserstein distance when self-discrepancies vanish. No topology, symmetry or triangle inequality is needed, and the exponent may be zero.
Weak isomorphism is equivalent to the existence of a zero-distortion coupling at any
nonzero exponent, including ∞. This does not assert attainment of the distance infimum.
A common s-finite parametrizing measure with equal pulled-back kernels gives a weak isomorphism. The witness coupling is the pushforward along the paired parametrizing maps.
A measured kernel and its pullback along a measure-preserving map are weakly isomorphic. In particular this permits duplication of a measured carrier without changing its kernel.
Every a.e. strongly measurable kernel on an s-finite measure is weakly isomorphic to itself.
Weak isomorphisms compose whenever their exact couplings admit a gluing. Only the outer kernels need a.e. strong measurability; equality of their pullbacks passes through the middle kernel on the common glued space. The middle measure need only be sigma-finite.
Weak isomorphisms are transitive when the last carrier is standard Borel and the middle measure is finite. The other carriers need no topology or standard Borel structure.