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LIT-1797: Gromov-Wasserstein Averaging of Kernel and Distance Matrices

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Properties5
StatusActive — read in full 2026-10-29 (NOTE-1343), from the PMLR PDF; worth reading for the explicit form of a Gromov–Wasserstein barycenter, a weighted average of realigned inputs whose size, weights and regularization are chosen in advance, and for the authors' statement of what its algorithm does not guarantee.
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Tagsmathematics
Cited as “paper” byNOTE-1343 — Peyré et al. — Gromov–Wasserstein averaging
Cited as “source” byTHEORY-1121 — With the couplings fixed, the square-loss Gromov–Wasserstein barycenter of several similarity matrices is their realigned average weighted by the barycenter weights, and the KL barycenter its weighted geometric average, so PSD and infinitely divisible inputs give barycenters of the same kind; but the barycenter's size, histogram and weights are chosen beforehand, the couplings are only stationary points of a non-convex problem, and the iteration's convergence at the step size used is not proved, so a structure carried by one input is shrunk by its weight at best and can be lost or redistributed by those choices

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Peyré, G., Cuturi, M., & Solomon, J. (2016). Gromov-Wasserstein averaging of kernel and distance matrices. ICML 2016, PMLR 48, 2664–2672.

Why this is here

A6’s “emergent interlingua”, held as high-risk: a barycenter of several communities’ response structures is a reconstructed common structure, not the meaning common to all languages, and it can average away the distinctions that matter to minorities (CLAIM-348). Its use would be as a test of where common structure fails.

The reading (NOTE-1343) locates the risk: with couplings fixed, one input’s distinction enters the barycenter with that input’s weight, shrunk rather than removed, but the size, weights and regularization fixed in advance, and seed-dependent stationary points, can remove or redistribute it (THEORY-1121).

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