14.16.2 Tensor Map Product Factor Evaluation

Tensor Map Product Factor Evaluation is the intermediate stage of evaluating a combined operator in which each individual factor map is applied only to its own corresponding vector within an elementary term, computed independently of what happens in every other factor, before the results from the different factors are joined together during assembly.

Isolating the Factor-Level Computation

One Map, One Vector, Independent of Other Factors

Within a single elementary term, factor evaluation consists of applying a single factor map to the single vector belonging to that same factor, a computation that requires no information about the other factors' maps or vectors.

Position Within the Overall Evaluation Procedure

Factor evaluation sits between the decomposition of the input tensor into elementary terms and the assembly of the transformed pieces into the final output tensor, forming the computational heart of the overall evaluation procedure for a combined operator.

Independence Across Factors

No Communication Required Between Factor Computations

Because each factor evaluation only reads its own factor's map and vector, the computations for different factors can be carried out in any order, or simultaneously, without either computation needing to wait for or depend on the outcome of the other.

Parallelizability of Factor Evaluation

This independence means that, in a setting with more than one factor, the individual factor evaluations are natural candidates for parallel computation, since there is no data dependency between the evaluation performed for one factor and the evaluation performed for another.

Diagram of Independent Factor Evaluation

Two Separate Computations Feeding Into Assembly

The diagram below shows the factor evaluation for two separate factors proceeding independently before their results are joined during assembly.

Factor Evaluation Relative to a Basis

Matrix-Vector Multiplication Within a Factor

Once a basis is fixed for a factor space, factor evaluation for that factor reduces to ordinary matrix-vector multiplication: the coordinate column of the input vector is multiplied by the matrix representing the factor map to produce the coordinate column of the output vector.

Cost Comparison With the Full Composite Matrix

Because each factor evaluation involves only a matrix of the size of that individual factor, the total cost of performing factor evaluation separately for each factor is typically far smaller than the cost of multiplying the input by the full Kronecker product matrix directly.

Reuse of Factor Evaluation Results

Sharing a Factor's Evaluation Across Several Terms

If the same vector appears as the component for a given factor across multiple elementary terms in a decomposition, the factor evaluation for that vector needs to be computed only once and can be reused for every term in which it appears, avoiding redundant computation.

Sharing Across Different Combined Operators

If several different combined operators share the same map for a particular factor, the factor evaluation of a given vector under that shared map can be computed once and reused across all of the different combined operators that rely on it.

Correctness of Factor Evaluation

Faithfulness to the Original Factor Map

Because factor evaluation applies the factor map exactly as it is defined on its own factor space, without any alteration introduced by the surrounding tensor product structure, the result of factor evaluation always coincides with the ordinary evaluation of that map on that vector, considered entirely on its own.

No Special Interpretation Required

There is no additional interpretation or adjustment needed when a factor map is evaluated within the context of a larger tensor product construction; the map behaves exactly as it would if it were being studied and evaluated in isolation on its own factor space.

Extension to Several Factors

Factor Evaluation Across Many Factors Simultaneously

When the tensor product involves three or more factor spaces, factor evaluation is carried out independently for every individual factor within an elementary term, with each factor's map applied only to its own corresponding vector, exactly as in the two-factor case.

Scaling of Independence With the Number of Factors

As the number of factors grows, the independence of factor evaluation across factors becomes an increasingly significant computational advantage, since the total work of evaluating all factors separately grows only in proportion to the sum of the individual factor sizes rather than their product.