4.12.4 Tensor Partial Evaluation Linearity

Tensor Partial Evaluation Linearity is the property that the tensor multilinear partial evaluation operation depends linearly on the arguments fixed into each chosen slot, so that scaling or adding fixed arguments in any one slot produces a correspondingly scaled or added reduced arity result, while every other fixed slot and every open slot remains unaffected. It is the precise sense in which partial evaluation inherits, slot by slot, the same multilinear behavior that defines the original tensor as a multilinear map, ensuring that fixing arguments never introduces any nonlinear distortion into the resulting lower-arity object.

Statement of the Linearity Property

Linearity in a Single Fixed Slot

For a type (p, q) tensor T on a vector space V, with a subset S of slots fixed and one particular slot r ∈ S singled out, the reduced arity result depends linearly on the value x_r placed in that slot, holding every other fixed value and every open slot fixed:

where y is any other value of the appropriate kind for that slot and λ is any scalar from the base field. This is exactly the multilinearity of T in that slot, restricted to the situation where the remaining slots have already been fixed or left open as prescribed by S.

Linearity Holds Independently for Every Fixed Slot

The linearity described above holds separately for each individual slot in S; fixing several slots at once does not combine their linear behaviors into some more complicated joint nonlinearity, since the underlying multilinearity of T guarantees separate, slot-by-slot linear behavior no matter how many slots are simultaneously involved in a given partial evaluation.

Consequences for the Reduced Arity Result

The Reduced Result as a Linear Function of Each Fixed Argument

Because of this linearity, the reduced arity result obtained from partial evaluation can be regarded as a linear map in its own right, from the vector space or dual space associated with slot r into the space of multilinear maps on the remaining open slots, since fixing x_r produces a definite reduced arity result, and this assignment respects addition and scalar multiplication in x_r.

Multilinearity Across All Fixed Slots Simultaneously

Taking every fixed slot in S together, the assignment sending the full collection of fixed values {x_r}_{r ∈ S} to the resulting reduced arity result is itself multilinear in those values jointly, exactly mirroring the original multilinearity of T restricted to the slots in S, with the slots not in S treated as still-open, unevaluated arguments.

Linearity and the Component Description

Linear Combination of Component Contributions

In component form, relative to a basis, fixing a slot r with value x_r = ∑_i x_r^i e_i produces a reduced arity result whose components are a linear combination, weighted by the coordinates x_r^i, of the components of T obtained by fixing that slot to each basis vector in turn:

where T_{S, r=i} denotes the reduced arity result obtained by fixing slot r specifically to the i-th basis vector, holding the rest of the fixed slots as before. This decomposition makes the linearity in slot r an explicit, verifiable statement about how components combine.

Preservation Under Basis Change

Because the linearity property is a statement about T itself rather than about any particular basis, it survives unchanged under any change of basis; the specific coordinate expressions used to describe the linear combination change accordingly, but the underlying fact that the reduced arity result depends linearly on each fixed argument remains true in every basis.

Why Linearity Must Hold

Direct Consequence of the Definition of a Tensor

Partial evaluation linearity is not an additional assumption imposed on tensors but a direct logical consequence of the definition of a tensor as a multilinear map: since T is required to be linear in every one of its p + q slots by definition, fixing some slots while leaving others open cannot alter the linear behavior already guaranteed for the slots that remain fixed, because linearity in one slot is, by definition, independent of what values occupy the other slots.

Failure of Linearity Would Contradict Multilinearity

If a reduced arity result failed to depend linearly on some fixed argument, this would mean that T, evaluated with the other arguments held fixed, is not actually linear in that slot, directly contradicting the assumption that T is a multilinear map; partial evaluation linearity is therefore inseparable from the very definition of what it means for T to be a tensor in the first place.

Diagrammatic Summary

The diagram illustrates partial evaluation linearity by showing that scaling or adding the value fixed into a single slot produces the correspondingly scaled or added reduced arity result, exactly as required by multilinearity in that slot.