Triple
T13035724
| Position | Surface form | Disambiguated ID | Type / Status |
|---|---|---|---|
| Subject | Monge problem in optimal transport |
E326555
|
entity |
| Predicate | contrastedWith |
P278
|
FINISHED |
| Object |
Kantorovich problem in optimal transport
The Kantorovich problem in optimal transport is a relaxed, linear-programming formulation of transporting mass between probability distributions that allows splitting mass and guarantees existence of optimal transport plans.
|
E1020368
|
NE FINISHED |
How this triple was built (4 steps)
Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.
NER
Named-entity recognition
gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Kantorovich problem in optimal transport | Statement: [Monge problem in optimal transport, contrastedWith, Kantorovich problem in optimal transport]
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Kantorovich problem in optimal transport Context triple: [Monge problem in optimal transport, contrastedWith, Kantorovich problem in optimal transport]
-
A.
Monge problem in optimal transport
The Monge problem in optimal transport is a foundational mathematical formulation that seeks the most efficient way to move mass from one distribution to another, minimizing a given transportation cost.
-
B.
Optimal Transport: Old and New
"Optimal Transport: Old and New" is a comprehensive monograph by Cédric Villani that develops the theory of optimal transport and its applications across analysis, geometry, and probability.
-
C.
Kantorovich duality
Kantorovich duality is a fundamental result in optimal transport theory that characterizes the optimal transport cost as the supremum of a dual variational problem over suitable test functions.
-
D.
Brenier map
The Brenier map is the unique gradient of a convex function that provides the optimal transport between probability measures under a quadratic cost, playing a central role in modern optimal transport theory.
-
E.
Monge–Ampère equation
The Monge–Ampère equation is a fully nonlinear partial differential equation central to differential geometry, optimal transport, and several complex variables, often used to study curvature and geometric structures.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg
Description generation
gpt-5.1
Instruction
Generate a one-sentence description of the target entity. You are given a context triple in the form (subject, predicate, object), where the object is the target entity. # Instructions Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. Avoid repeating the information from the triple, unless really essential. # Response Format Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Kantorovich problem in optimal transport Triple: [Monge problem in optimal transport, contrastedWith, Kantorovich problem in optimal transport]
Generated description
The Kantorovich problem in optimal transport is a relaxed, linear-programming formulation of transporting mass between probability distributions that allows splitting mass and guarantees existence of optimal transport plans.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Kantorovich problem in optimal transport Target entity description: The Kantorovich problem in optimal transport is a relaxed, linear-programming formulation of transporting mass between probability distributions that allows splitting mass and guarantees existence of optimal transport plans.
-
A.
Monge problem in optimal transport
The Monge problem in optimal transport is a foundational mathematical formulation that seeks the most efficient way to move mass from one distribution to another, minimizing a given transportation cost.
-
B.
Optimal Transport: Old and New
"Optimal Transport: Old and New" is a comprehensive monograph by Cédric Villani that develops the theory of optimal transport and its applications across analysis, geometry, and probability.
-
C.
Kantorovich duality
Kantorovich duality is a fundamental result in optimal transport theory that characterizes the optimal transport cost as the supremum of a dual variational problem over suitable test functions.
-
D.
Brenier map
The Brenier map is the unique gradient of a convex function that provides the optimal transport between probability measures under a quadratic cost, playing a central role in modern optimal transport theory.
-
E.
Monge–Ampère equation
The Monge–Ampère equation is a fully nonlinear partial differential equation central to differential geometry, optimal transport, and several complex variables, often used to study curvature and geometric structures.
- F. None of above. chosen
Provenance (5 batches)
The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.
| Step | Stage | Batch ID | Status | When |
|---|---|---|---|---|
| creating | Elicitation | batch_69d8076cc45c81908123123f43e69266 |
completed | April 9, 2026, 8:09 p.m. |
| NER | Named-entity recognition | batch_69d97f2a71a0819098bb6cf8a4b2208a |
completed | April 10, 2026, 10:52 p.m. |
| NED1 | Entity disambiguation (via context triple) | batch_69f6d5fa10fc81908a37b85894f8f849 |
completed | May 3, 2026, 4:58 a.m. |
| NEDg | Description generation | batch_69f6db65879c819092387e028b792ab9 |
completed | May 3, 2026, 5:21 a.m. |
| NED2 | Entity disambiguation (via description) | batch_69f6dbec7e808190a7cee821ba193bb8 |
completed | May 3, 2026, 5:23 a.m. |
Created at: April 9, 2026, 8:55 p.m.