Linearize And Transpose

linearize and linear_transpose are separate graph transforms.

Linearize

linearize takes a resolved primitive computation graph, selected output keys, selected input keys, a DiffPassId, mutable AD context, and input aliases. It returns a LinearizedGraph.

The returned graph has tangent inputs for the selected primal inputs and tangent outputs for the selected primal outputs. Use try_linearize when primitive rules can fail.

DiffPassId is a u64 that tags one linearization pass; it is woven into the tangent input keys (via ADKey::tangent_of) so repeated transforms do not collide. Pass a fresh value per linearize call (see Repeated Transforms).

The input-aliases map (HashMap<InputKey, ValueKey>) is empty in the common case. A non-empty entry says “treat this graph input as identical to that existing value,” so the input inherits the aliased value’s tangent instead of being an independent leaf — used for checkpointing and shared sub-graphs.

Linear Transpose

linear_transpose takes a LinearizedGraph and returns another LinearizedGraph whose inputs are cotangent seeds and whose outputs are cotangents for the original active inputs.

Use try_linear_transpose when transpose rules can fail. Use try_linear_transpose_with_builder when a downstream eager runtime wants to execute the transposed linear map directly through a concrete builder.

Cotangent Accumulation

When a single value feeds more than one consumer in the primal graph, the transposed graph receives a separate cotangent contribution from each consumer. linear_transpose sums these by emitting add nodes — obtained from Primitive::add() — so the transposed map yields one accumulated cotangent per active input.

For example, if x feeds both a Mul and an Add, transposition produces a cotangent for x from each, and an add node combines them before the value is reported as an input cotangent. Eager runtimes perform the same accumulation at execution time through BackwardExecutor::add_operands.

Repeated Transforms

Each linearize call needs a unique DiffPassId. If a transform result should be transformed again, resolve the graph collection that includes the previous result before calling linearize again.