12.22.4 Tensor Multilinear Operation Boundary

Tensor Multilinear Operation Boundary is the specific form of tensor operation boundary that limits an operation to arguments over which it acts separately linearly in each slot, marking the point beyond which an operation ceases to preserve linearity in every one of its inputs and therefore falls outside the class of multilinear tensor operations.

The Multilinearity Requirement

Linearity in Each Argument Separately

A multilinear operation is one that, when all but one of its arguments are held fixed, behaves as a linear function of the remaining argument, satisfying both additivity and compatibility with scalar multiplication in that argument.

The Boundary as the Limit of Joint but Separate Linearity

The multilinear operation boundary marks the requirement that linearity hold in each argument individually rather than jointly across arguments together, so that an operation failing to be linear in even one argument, while all others are fixed, lies outside this boundary regardless of any other favorable behavior it might exhibit.

Operations Falling Within the Boundary

Tensor Addition as a Linear, Degenerate Case

Tensor addition, viewed as a function of two arguments, is linear in each argument separately and therefore lies within the multilinear operation boundary, representing the simplest case of a multilinear operation with exactly one tensor slot per argument.

The Tensor Product as the Prototypical Multilinear Operation

The tensor product is linear in each of its operand tensors separately, since scaling or adding within one operand while holding the other fixed produces a corresponding scaling or additive change in the resulting product, making it the operation most closely associated with the defining property of the multilinear boundary.

Contraction as a Linear Operation

Contraction, applied to a single tensor argument, is linear in that argument, since summing over a repeated index commutes with both addition of tensors and scalar multiplication applied before the summation.

Operations Falling Outside the Boundary

Nonlinear Combinations of Components

An operation that combines components through a nonlinear formula, such as one involving a product of two components drawn from the same tensor argument, fails to be linear in that argument and therefore lies outside the multilinear operation boundary.

Normalization-Type Operations

An operation that divides a tensor's components by a quantity depending on those same components, such as a magnitude or norm, fails the additivity requirement of linearity and consequently falls outside the boundary, even though such operations may still be well defined as functions on tensors.

Function of the Boundary in Verification

Guiding Which Operations Require Multilinearity Checks

The multilinear operation boundary identifies which tensor operations are expected to satisfy linearity in each argument, allowing the verification procedure to include a check of additivity and scalar compatibility specifically for operations claimed to lie within this boundary.

Relation to Invariance Verification

Multilinearity in each argument is a structural precondition that supports the correct transformation behavior confirmed during invariance verification, since the standard tensor transformation law relies on the assumption that the operation producing a result is linear in the tensors it combines.

Relationship to Tensor Operation Notation

The multilinear operation boundary is reflected in tensor operation notation through the way indices from separate tensor arguments appear without being combined nonlinearly, so that an expression respecting standard indicial notation, with indices from different tensors joined only through multiplication and summation over repeated pairs, inherently signals an operation lying within the multilinear boundary.