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LIT-1805: Towards Optimal Transport with Global Invariances

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Properties5
StatusActive — read in full 2026-10-29 (NOTE-1333); worth reading for the exact statement of what a global-invariance alignment treats as arbitrary, one transformation from a stated class acting on the whole space, and for Lemma 4.3, which shows that for one class this is the same problem as Gromov–Wasserstein (THEORY-1135).
Version2
Tagsmathematics, representation-learning, anthology-candidate
Cited as “paper” byNOTE-1333 — Alvarez-Melis et al. — OT with global invariances
Cited as “source” byTHEORY-1133 — The Gromov–Wasserstein objective between two spaces depends on each only through its within-space similarity matrix and weights, so every transformation of a space that preserves them, a single orthogonal map or per-vector rescaling under cosine similarity, leaves every coupling's cost unchanged, and any coupling composed with a symmetry of either structure costs exactly what it did; GW can therefore identify a correspondence only up to the automorphisms of the two similarity structures, and on symmetric data it cannot choose between branches or categories that the symmetry exchanges, THEORY-1135 — Optimal transport between two point sets related by one unknown global linear map from a Schatten ℓp ball reduces, when every admissible map preserves angles or the target is whitened, to choosing the coupling Γ that maximises the dual Schatten-q norm of the transported cross-covariance XΓYᵀ: the nuclear norm when the maps are orthogonal, and the Frobenius norm when they lie on a Frobenius sphere, which is exactly the Gromov–Wasserstein problem with cosine similarity and squared loss; in every case one transformation acts on the whole space

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Alvarez-Melis, D., Jegelka, S., & Jaakkola, T. S. (2019). Towards optimal transport with global invariances. AISTATS 2019, PMLR 89, 1870–1879.

Read in full in NOTE-1333.

Why this is here

A6’s foundational distinction for QUESTION-130: GW treats the registration of whole metric spaces as arbitrary, this treats one global transformation as arbitrary, and lattice gauge theory (LIT-1820) a separate frame at every node. Keeping the three apart stops an arbitrary orientation being read as path dependence. The reading confirms the middle term: the paper assumes “an unknown function f in a pre-specified class F which characterizes the global correspondence between spaces” (p. 4), one map for every point. It also shows that the first two terms meet: for maps on a Frobenius sphere, with whitened target vectors of equal norm, this problem is GW with cosine similarity (Lemma 4.3; THEORY-1135). Neither has a frame per node, so neither can carry holonomy.

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