13.17.2 Tensor Contraction Sequence Intermediate Tensor

The Tensor Contraction Sequence Intermediate Tensor arises in algebraic operations, facilitating the reduction of tensor rank through systematic contraction steps.

Tensor Contraction Sequence Intermediate Tensor is the tensor object that exists at any stage strictly between the beginning and the end of a multi-step contraction sequence procedure, produced by applying some but not all of the sequence's steps, and serving as both the output of the preceding step and the input to the following step.

Definition

Within a sequence $T_{0} \to T_{1} \to \dots \to T_{k}$ , every tensor $T_{i}$ with $0 < i < k$ is an intermediate tensor of the sequence: it is neither the original input $T_{0}$ nor the final residual $T_{k}$ , but a fully valid tensor in its own right at the moment it exists.

Role of Intermediate Tensors

Dual Function

Each intermediate tensor $T_{i}$ serves two roles simultaneously: it is the result of contracting $T_{i - 1}$ according to step $C_{i}$ , and it is simultaneously the tensor to which step $C_{i + 1}$ will next be applied.

Type Progression

Each intermediate tensor has a determined type based on how many steps precede it. If the original tensor has type $(p_{0}, q_{0})$ , then:

type (T_{i}) = (p_{0} - i, q_{0} - i)

Independent Validity

An intermediate tensor is a genuine, well-formed tensor, obeying the standard transformation law for its own type, independent of whether further contraction steps are later applied to it. Its status as merely "intermediate" reflects only its position within a particular sequence, not any structural incompleteness.

Existence and Availability

Availability Constraint

An intermediate tensor $T_{i}$ only exists, and only has meaning, after all steps $C_{1}, \dots, C_{i}$ have actually been carried out in order. It cannot be referenced or computed before its preceding steps are complete.

Discardability

Once step $C_{i + 1}$ has been applied to produce $T_{i + 1}$ , the intermediate tensor $T_{i}$ is no longer needed to continue the sequence, though it may still be retained separately if its value is of independent interest.

Diagram of Intermediate Stages

Significance in Analysis

Intermediate tensors are essential when examining or validating a contraction sequence step by step, since properties such as symmetry, type consistency, or the correctness of index bookkeeping can be checked at each intermediate stage individually, rather than only at the final residual tensor, making errors introduced by any single step localizable to that step's specific intermediate output.