13.7.4 Tensor Partial Contraction Intermediate Tensor

The Tensor Partial Contraction Intermediate Tensor results from contracting some indices of a tensor, preserving partial structure for further algebraic operations.

Tensor Partial Contraction Intermediate Tensor is the tensor object produced at a midpoint stage of a multi-step contraction process, one that has undergone at least one partial contraction but is not yet the final tensor intended by the overall computation. It denotes the transient result that carries forward from one contraction step to the next, distinguishing this in-progress object from both the original uncontracted tensor and the eventual final output of the complete sequence of operations.

Conceptual Basis

Role Within a Contraction Sequence

Complex tensor computations frequently require several contractions applied in succession, either to the same tensor or across a chain of tensors combined by products. The intermediate tensor is the object that exists between two such steps, having absorbed the effect of earlier contractions while still awaiting further reduction.

Transient Rather Than Final Status

An intermediate tensor is defined by its position within a computation rather than by any fixed structural property: the same tensor object could be treated as a final result in one context and as an intermediate stage in another, depending on whether further contractions are planned.

Necessity in Staged Computation

When a full computation involves contracting several index pairs distributed across multiple tensors, performing all contractions simultaneously is often impractical or ill-defined without first fixing an order, so intermediate tensors serve as the well-defined checkpoints through which the computation proceeds one contraction at a time.

Formal Description

Sequential Construction

Given an initial tensor $T_{0}$ and a sequence of partial contractions $C_{1}, C_{2}, \dots, C_{n}$ , each intermediate tensor is defined by:

T_{k} = C_{k} (T_{k - 1})

for $k = 1, \dots, n - 1$ , with $T_{n}$ designated as the final result rather than an intermediate.

Rank Tracking Across Steps

Each intermediate tensor carries a residual type reflecting the cumulative effect of all contractions applied up to that point, so that if the original tensor has type $(p, q)$ and $k$ contractions have been applied, the intermediate tensor at that stage has type $(p - k, q - k)$ .

Validity Requirements at Each Stage

Every intermediate tensor must itself be a well-formed tensor, meaning the indices it carries forward must have consistent variance and dimension, since a malformed intermediate tensor would render any subsequent contraction in the sequence undefined.

Properties

Path Dependence Versus Result Independence

While the specific intermediate tensors produced depend on the order in which contractions are performed, the final result of a sequence of independent contractions on disjoint index pairs does not depend on that order, meaning different valid sequences of intermediate tensors can lead to the same final tensor.

Storage and Reuse

An intermediate tensor may be computed once and reused if the same partial result is needed for more than one subsequent branch of a larger computation, avoiding redundant recomputation of the same contraction steps.

Basis Dependence

Because an intermediate tensor generally retains free indices, it remains basis-dependent in the same way any non-scalar tensor does, and only becomes basis-independent once the sequence reaches a final full contraction with no free indices remaining.

Practical Role

Symbolic Computation

In symbolic tensor algebra systems, intermediate tensors are explicitly represented and simplified at each stage, allowing index relabeling, symmetry exploitation, or algebraic simplification to be applied before the next contraction is carried out.

Numerical Computation

In numerical tensor computation, intermediate tensors correspond to actual arrays held in memory between successive contraction operations, and their size and rank directly affect the computational cost of the overall procedure.

Verification of Multi-Step Results

Examining intermediate tensors at each stage of a long contraction sequence provides a means of verifying correctness, since an error introduced at an early step will typically manifest as an inconsistency in the type or values of a later intermediate tensor.