4.12 Tensor Multilinear Partial Evaluation Operation

Tensor Multilinear Partial Evaluation Operation is the operation of supplying values for a chosen subset of the arguments of a multilinear map while leaving the remaining arguments unspecified, producing a new multilinear map of reduced arity that depends only on the arguments not yet supplied. It generalizes the idea of currying a function of several variables to the tensorial setting, allowing a tensor to be treated as a family of lower-rank multilinear maps parameterized by whichever arguments have already been fixed.

Defining Partial Evaluation

Choosing a Subset of Slots

For a type (p, q) tensor T on a vector space V, partial evaluation designates a subset S of the p + q slots, of size m, and supplies a specific vector or covector, matching the kind required by each slot, into every slot in S. The remaining p + q - m slots are left open, and the result is a new multilinear map depending only on those open slots:

where S̄ denotes the complement of S among the p + q slots, and each factor in the product is V or V* according to whether the corresponding slot is covariant or contravariant.

General Form of the Reduced Map

Concretely, if the fixed arguments are α^1, ..., α^m supplied into m chosen slots, the partially evaluated map sends the remaining arguments to

with the placeholders ⋯ in the argument list marking the open slots that still receive whatever arguments the reduced map is subsequently applied to.

Rank of the Partially Evaluated Map

Counting Remaining Contravariant and Covariant Slots

If k of the m fixed slots were contravariant and the remaining m - k were covariant, the partially evaluated map T_S behaves as a type (p - k, q - (m - k)) tensor in its remaining arguments, since exactly k contravariant slots and m - k covariant slots have been removed from the original type (p, q).

Special Case of Full Evaluation

When S includes all p + q slots, so that m = p + q, the partially evaluated map has no remaining arguments at all, and T_S collapses to a single scalar, coinciding exactly with the tensor multilinear evaluation operation applied to the full input tuple.

Special Case of No Evaluation

When S is empty, no arguments have been fixed, and T_S is simply T itself, unchanged, so the partial evaluation operation includes the identity case as one of its extremes, alongside full evaluation as the other.

Iterated and Order-Independent Partial Evaluation

Building Up Partial Evaluation from Slot Substitution

Partial evaluation over a subset S can always be obtained by performing tensor multilinear slot substitution once for each slot in S, in any order, since multilinearity guarantees that filling several slots one at a time, regardless of the sequence chosen, yields the same reduced map as filling them all together as prescribed by S.

Composability of Partial Evaluations

If a subset S_1 is partially evaluated first, and then a further subset S_2, disjoint from S_1, is partially evaluated on the resulting reduced map, the outcome is identical to performing a single partial evaluation over the combined subset S_1 ∪ S_2 directly; partial evaluation therefore composes associatively, and the eventual reduced map depends only on which slots have been filled and with what, never on the sequence of steps taken to fill them.

Partial Evaluation and the Tensor Product Structure

Reduced Maps as Tensors in Their Own Right

Because the partially evaluated map T_S is itself multilinear in its remaining arguments, it corresponds to its own element of a tensor product space, built from copies of V and V* matching the open slots; partial evaluation therefore produces not merely a function but a genuine tensor of lower rank, obtainable as a contraction of T against the fixed arguments.

Dependence on the Fixed Arguments

The specific tensor obtained by partial evaluation depends linearly on each of the fixed arguments individually, so varying one fixed argument while holding the choice of subset S and the other fixed arguments constant produces a linear family of reduced tensors, mirroring the multilinearity of the original map T.

Uses of Partial Evaluation

Producing Families of Lower-Rank Tensors

Partial evaluation is the mechanism by which a single high-rank tensor generates an entire family of lower-rank tensors, one for each choice of fixed arguments, which is useful whenever a tensor needs to be applied repeatedly with some arguments held constant across many separate evaluations of the remaining ones.

Relation to Associated Linear and Multilinear Operators

Fixing all but one argument of a type (1, 1) tensor through partial evaluation recovers the associated linear operator on V, and fixing all but two arguments of a higher-rank tensor recovers an associated bilinear map, showing that many of the standard operators built from tensors arise as specific instances of the partial evaluation operation.

Diagrammatic Summary

The diagram shows some slots of the tensor T filled with fixed arguments while others remain open, illustrating how partial evaluation produces a reduced multilinear map T_S depending only on the still-open slots.