14.16.3 Tensor Map Product Output Tensor Assembly

Tensor Map Product Output Tensor Assembly combines tensor mappings to construct new tensors through structured algebraic operations.

Tensor Map Product Output Tensor Assembly is the final stage of evaluating a combined operator, in which the individual images produced by applying each factor map to its corresponding component are joined back together into elementary tensors, and those elementary tensors are then summed to form the single output tensor in the codomain tensor product space.

The Assembly Step in Context

Position Within the Evaluation Procedure

Output tensor assembly is the last of the operations carried out when evaluating a combined operator: after the input tensor has been decomposed into elementary terms and each factor map has been applied to its corresponding vector within each term, the resulting pieces must be reassembled into the final answer.

T_{1} (u_{k}) \otimes T_{2} (v_{k})

Two Layers of Assembly

Assembly occurs at two layers: first, the two transformed component vectors within a single term are joined into a single elementary tensor using the tensor product symbol, and second, all such elementary tensors arising from every term of the original decomposition are added together.

Assembling a Single Elementary Term

Joining Transformed Components

Given the image of the first component under its factor map and the image of the second component under its factor map, these two vectors are joined together using the tensor product operation to produce a single elementary tensor in the codomain tensor product space.

w = T_{1} (u) \otimes T_{2} (v)

No Further Simplification Required at This Stage

The joined elementary tensor is already a valid element of the codomain tensor product space and requires no additional processing beyond this joining step, since the tensor product operation itself is what defines membership in that space.

Assembling the Full Output Across Several Terms

Summing the Elementary Term Images

Once every term of the original decomposition has produced its own elementary tensor image, the output tensor assembly step adds all of these elementary tensors together to form the single final output tensor.

y = \sum_{k}^{} T_{1} (u_{k}) \otimes T_{2} (v_{k})

Order of Summation Does Not Matter

Because addition of tensors in the codomain space is commutative and associative, the order in which the individual elementary term images are added together during assembly has no effect on the final output tensor.

Diagram of the Assembly Process

From Transformed Pieces to a Single Output

The diagram below shows several transformed elementary pieces being joined and then summed into a single output tensor.

Assembly at the Coordinate Level

Coordinate Assembly as Ordinary Vector Addition

Once bases are fixed and elementary tensors are represented as coordinate columns via the Kronecker product of the individual coordinate vectors, assembling the output tensor from several term images reduces to ordinary vector addition of the corresponding coordinate columns.

Equivalence With Matrix Multiplication

Carrying out the full decomposition, evaluation, and assembly procedure at the coordinate level reproduces exactly the result of multiplying the coordinate column of the original input tensor by the Kronecker product matrix representing the combined operator, confirming the consistency of the two computational viewpoints.

Practical Notes on Assembly

Assembly Cost Depends on the Number of Terms

The computational cost of the assembly step grows with the number of elementary terms in the chosen decomposition of the input tensor, so choosing a decomposition with fewer terms, when possible, reduces the work required at this final stage.

Partial Assembly for Efficiency

In some computational strategies, elementary term images are grouped and partially summed as they are produced, rather than waiting until every term has been evaluated, which can reduce the amount of intermediate storage needed during the overall evaluation procedure.

Extension to Several Factors

Assembly With Many Factor Components

When the tensor product involves three or more factor spaces, each elementary term's transformed components, one per factor, are joined together using the tensor product operation across all factors simultaneously, before the resulting elementary tensors from every term are summed exactly as in the two-factor case.

Consistency Regardless of the Number of Factors

The commutativity and associativity of tensor addition that permits any order of summation during assembly hold regardless of how many factor spaces are involved, so the assembly procedure generalizes directly without any additional considerations for a larger number of factors.