Tensorunder

Tensorunder is a term used in mathematical and computational discussions to describe a generalized tensor operation that emphasizes contracting or projecting a tensor along a designated subset of its modes, yielding a lower-order object while retaining the structure of the remaining modes. The concept is not universally standardized and appears primarily in niche literature and tutorials exploring structured tensor decompositions.

Definition: Let T ∈ V1 ⊗ V2 ⊗ ... ⊗ Vn be a tensor over a field F. Choose a subset S ⊆ {1,...,n} and for each i ∈ S select a covector φi ∈ Vi*. The tensorunder of T with respect to S and the covectors φi is the contraction obtained by applying φi to the i-th mode for all i ∈ S. The result lies in ⊗_{j∉S} Vj. If S is empty, T is unchanged; if |S| = n, the result is a scalar that is the full contraction of T with φ1, ..., φn.

Examples: A matrix T ∈ F^{m×n} contracted on the second mode with a covector φ ∈ F^n* yields a vector in F^m. More generally, a 3-way tensor contracted on two modes reduces to a vector or matrix depending on the remaining modes.

Relations: Tensorunder generalizes standard tensor contraction and encompasses partial contractions used in tensor networks and multilinear models. It provides a flexible formalism for describing operations that reduce dimensionality while preserving certain structural aspects of the data.

See also: tensor contraction, partial trace, mode-n product. History: The term usage is informal and appears in a small subset of texts and lecture notes produced in the 2010s–2020s.