4.11.3 Tensor Multilinear Output Calculation

Tensor Multilinear Output Calculation explores how tensors produce outputs through multilinear operations, essential in advanced algebraic structures and applications.

Tensor Multilinear Output Calculation is the procedural aspect of producing the numerical output of a tensor evaluation, focusing on how the sum of products prescribed by component evaluation is actually carried out step by step, in what order, and at what computational cost, rather than on the formula itself. Where tensor multilinear component evaluation establishes the formula relating a tensor's components and its arguments' coordinates to the resulting scalar, the output calculation is concerned with the concrete arithmetic procedure, and the choices of grouping and ordering, used to arrive at that scalar in practice.

From Formula to Procedure

The Nested Summation Structure

The component evaluation formula for a type (p, q) tensor T on an n-dimensional space expresses the output as a sum over p + q independent indices, each ranging from 1 to n:

S = \sum_{i_{1} = 1}^{n} \dots \sum_{j_{q} = 1}^{n} T_{j_{1} \dots j_{q}}^{i_{1} \dots i_{p}} α_{1}^{i_{1}} \dots v_{q}^{j_{q}}

As a computational procedure, this can be realized directly as p + q nested loops, one per index, with a running total accumulated inside the innermost loop by adding each product as it is formed.

Associativity and Order Independence

Ordinary addition of real or complex numbers is associative and commutative, so the value of the sum S does not depend on the order in which the n^{p+q} individual terms are added together, nor on how they are grouped; any nested loop ordering, or any other traversal of the index combinations, produces the identical output value, up to the effects of finite-precision rounding when the calculation is carried out numerically.

Reducing the Cost of the Calculation

Factoring the Sum by Grouping Indices

Rather than expanding the full p + q-fold sum directly, the calculation can be reorganized by summing over one index at a time and reusing the partial results for the remaining indices, since the multilinear structure allows the sum to be factored:

S = \sum_{i_{1} = 1}^{n} α_{1}^{i_{1}} (\sum_{i_{2} = 1}^{n} \dots)

This factoring, performing one contraction against a single index at a time, replaces the naive n^{p+q} term expansion with a sequence of p + q smaller calculations, each costing on the order of n times the size of the previous intermediate result, which substantially lowers the total number of arithmetic operations required to reach the same output.

Choosing an Efficient Contraction Order

When several indices are contracted in sequence, the order in which they are eliminated affects the size of the intermediate results produced along the way, even though it does not affect the final output value; choosing an order that keeps the intermediate arrays as small as possible is a standard consideration in the output calculation, particularly when the tensor and its arguments are large.

Accumulation and Numerical Behavior

Running Totals During Calculation

At each step of the calculation, a running total accumulates the products already formed, and this running total, together with the remaining indices yet to be summed, fully determines the eventual output; the calculation can therefore be paused and resumed without loss of information, since the partial sum captures everything needed to complete the computation.

Sensitivity to Rounding in Finite Precision

Although the true value of the sum is independent of the order of accumulation, when the calculation is performed with finite-precision arithmetic, different orders of summation can produce slightly different rounding behavior, so the mathematical order-independence of the exact sum does not fully carry over to floating-point computation, and calculations involving many terms of greatly differing magnitude may accumulate rounding error depending on the chosen order.

Special Calculation Patterns

Diagonal-Only Calculation

When only diagonal input tuples are of interest, meaning the same vector or covector is repeated across several slots of the same kind, the output calculation can often be simplified, since many of the distinct index combinations appearing in the general sum collapse into repeated evaluations of the same coordinate, reducing the effective number of independent terms.

Sparse Component Calculation

When most components of T are zero, the output calculation can skip every term whose tensor component vanishes, since such terms contribute nothing to the sum regardless of the argument coordinates multiplying them, which can dramatically reduce the number of multiplications actually carried out compared to the full dense sum.

Diagrammatic Summary

The diagram depicts the output calculation as a set of nested loops over the tensor's indices, each layer contributing products that accumulate into a single running total, which becomes the final scalar output once every index has been summed.