DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Canonical polyadic rank

Definition

For $A \in F^{n_{1} \times \dots \times n_{d}}$ , its canonical polyadic rank (CP rank) is the least integer $R \geq 0$ for which $A = \sum_{q = 1}^{R} a_{q}^{(1)} \otimes \dots \otimes a_{q}^{(d)}, a_{q}^{(k)} \in F^{n_{k}} .$ The empty sum is zero, so the zero tensor has CP rank $0$ . The coordinate expansion $A = \sum_{i_{1} = 1}^{n_{1}} \dots \sum_{i_{d} = 1}^{n_{d}} A_{i_{1}, \dots, i_{d}} e_{i_{1}}^{(1)} \otimes \dots \otimes e_{i_{d}}^{(d)}$ shows that the set of admissible finite integers $R$ is nonempty. The well-ordering of $N_{0}$ therefore makes the least such $R$ well-defined.

Depends on

Finite tensor arrays and the Frobenius inner product

Used by

Budgeted CP alternating least squares Algorithm
A best rank-two CP approximation need not exist Counterexample
Exact CP-ALS block updates do not increase the objective Theorem

Dependency tree · one level

1 result within one dependency step of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Tamara G. Kolda and Brett W. Bader, Tensor Decompositions and Applications, SIAM Review 51(3) (2009), §3 (standard reference, not scraped)
ACM 270-1, Complete Lecture Notes, Caltech, Spring 2019, Lecture 13, §4.2 (standard reference, not scraped)