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.

Tucker format, multilinear rank, and mode products

Definition

For a tensor $A \in F^{n_{1} \times \dots \times n_{d}}$ , its multilinear rank is $mlrank (A) = (rank A_{(1)}, \dots, rank A_{(d)}) .$ If $U \in F^{m \times n_{k}}$ , the mode- $k$ product $A \times_{k} U$ is the tensor with mode- $k$ size $m$ and entries $(A \times_{k} U)_{i_{1}, \dots, i_{k - 1}, j, i_{k + 1}, \dots, i_{d}} = \sum_{i_{k} = 1}^{n_{k}} A_{i_{1}, \dots, i_{d}} U_{j, i_{k}} .$ The product is bilinear in the entries; no conjugation is used in this tensor contraction.

A Tucker representation with mode ranks bounded by $r = (r_{1}, \dots, r_{d})$ is a factorization $A = G \times_{1} U_{1} \times_{2} \dots \times_{d} U_{d},$ where $U_{k} \in F^{n_{k} \times r_{k}}$ and $G \in F^{r_{1} \times \dots \times r_{d}}$ . Separately, write $T_{r} = {B \in F^{n_{1} \times \dots \times n_{d}} : rank B_{(k)} \leq r_{k} for each k}$ for the multilinear-rank-bounded set used in approximation statements.

Depends on

Finite tensor arrays and the Frobenius inner product

Used by

Higher-order SVD truncation Algorithm
HOSVD quasioptimality for multilinear rank 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), §§2.4, 4 (standard reference, not scraped)
Lieven De Lathauwer, Bart De Moor, and Joos Vandewalle, A Multilinear Singular Value Decomposition, SIAM Journal on Matrix Analysis and Applications 21(4) (2000), §§2–3 (standard reference, not scraped)
ACM 270-1, Complete Lecture Notes, Caltech, Spring 2019, Lecture 14, §3 (standard reference, not scraped)