nucleation

LIT-1804: CO-Optimal Transport

1 min read

Properties5
StatusActive — read in full 2026-10-29 (NOTE-1335), from arXiv v3; worth reading for the exact relation between co-optimal transport and Gromov–Wasserstein (a lower bound, equal for squared Euclidean distances) and for what its zero theorem implies: joint sample and feature plans are fixed only up to the matrix's joint row–column symmetries.
Version2
Tagsmathematics, representation-learning, anthology-candidate
Cited as “paper” byNOTE-1335 — Redko et al. — CO-Optimal Transport
Cited as “source” byTHEORY-1149 — For symmetric matrices compared under squared loss, co-optimal transport with independent row and column couplings is a lower bound on the Gromov–Wasserstein value, and for squared Euclidean distance matrices the two are equal and share optimal couplings, because GW is then a concave quadratic program whose bilinear relaxation is tight; for other symmetric matrices the bound can be strict, THEORY-1155 — Co-optimal transport between two data matrices of equal shape with uniform weights vanishes exactly when one is the other with its rows and its columns separately permuted, and its objective is unchanged when a pair of optimal plans is composed with any joint relabelling of rows and columns that leaves a matrix unchanged; its sample and feature couplings are therefore identified at best up to that joint symmetry group, which can exchange a block of samples together with its block of features though neither exchange alone is a symmetry, and the paper proves no uniqueness beyond it

Graph View

Redko, I., Vayer, T., Flamary, R., & Courty, N. (2020). CO-optimal transport. NeurIPS 2020.

Why this is here

A6’s highest-priority extension: with utterances as samples and pragmatic probes as features, it matches what two communities say and the questions through which they read it, so the correspondence of probes is fitted, not assumed. That is CLAIM-257’s change of cover and TERM-092’s frontier in fittable form. A6’s warning is kept: utterances and probes can be swapped together at equal cost, so anchors must be independent. An arm of CASE-185.

The reading (NOTE-1335) makes the warning exact. COOT never matches an utterance with a probe; the tie is a joint relabelling, within each side, of a block of utterances together with its block of probes (THEORY-1155). Its fixed-marginal probe coupling also cannot leave a probe unmatched, so CLAIM-257’s change of cover would be absorbed rather than reported unless an unbalanced variant is used.

Backlinks