13.17.5 Tensor Contraction Sequence Result Preservation

Tensor Contraction Sequence Result Preservation is the guarantee that a well-formed multi-step contraction sequence, regardless of which valid grouping or admissible reordering of its disjoint steps is chosen, always produces the same final residual tensor, so that the identity of the result is preserved independently of the particular computational path taken to reach it.

Definition

Given a contraction sequence acting on tensors $$T_1,\dots ,T_m$$ with a fixed set of index pairs to be contracted, result preservation asserts that for any two admissible execution strategies $${\pi }_1$$ and $${\pi }_2$$ realizing the same set of contractions:

Sources of Preservation

Associativity

The associativity context guarantees that different groupings of the same contraction steps yield equal intermediate and final products, since finite sums of products can be regrouped freely without changing their total.

Commutativity of Disjoint Steps

When two contraction steps act on entirely disjoint index pairs, swapping the order in which they are carried out does not change the final residual tensor, since each step's summation is independent of the other's outcome.

Combined Guarantee

Result preservation is the composite consequence of both associativity and commutativity of disjoint steps applied together: any admissible reordering or regrouping consistent with the sequence's dependency structure yields an identical final tensor.

Scope of the Guarantee

What Is Preserved

Only the identity of which specific index pairs are contracted is fixed by the sequence's specification; result preservation guarantees the final tensor is unaffected by variation in execution order or grouping strategy, as long as that specification itself is held fixed.

What Is Not Guaranteed

Result preservation makes no claim about numerical round-off behavior in finite-precision floating point computation, nor does it extend to sequences in which the set of contracted index pairs itself differs between the two compared executions; changing which indices are paired produces, in general, a genuinely different tensor, outside the scope of this preservation guarantee.

Diagram Illustration

Practical Significance

Result preservation is the theoretical basis that justifies choosing among alternative execution orders purely for computational efficiency, as addressed by the sequence cost effect, without any concern that a cheaper ordering might silently alter the mathematical answer. It also underlies the correctness of parallelizing independent contraction steps, since disjoint steps may be executed concurrently and later combined without affecting the outcome.