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If ``x`` and ``y`` are vectors, ``\frac{\partial y}{\partial x}`` becomes a Jacobian. Importantly, because we are implementing reverse mode we actually left-multiply the Jacobian, i.e. `v'J`, rather than the more usual `J*v`. Transposing `v` to a row vector and back `(v'J)'` is equivalent to `J'v` so our gradient rules actually implement the *adjoint* of the Jacobian. This is relevant even for scalar code: the adjoint for `y = sin(x)` is `x̄ = sin(x)'*ȳ`; the conjugation is usually moot but gives the correct behaviour for complex code. "Pullbacks" are therefore sometimes called "vector-Jacobian products" (VJPs), and we refer to the reverse mode rules themselves as "adjoints".
If ``x`` and ``y`` are vectors, ``\frac{\partial y}{\partial x}`` becomes a Jacobian. Importantly, because we are implementing reverse mode we actually left-multiply the Jacobian, i.e. `v'J`, rather than the more usual `J*v`. Transposing `v` to a row vector and back `(v'J)'` is equivalent to `J'v` so our gradient rules actually implement the *adjoint* of the Jacobian. This is relevant even for scalar code: the adjoint for `y = sin(x)` is `x̄ = ȳ*cos(x)'`; the conjugation is usually moot but gives the correct behaviour for complex code, if `y(x)` is holomorphic. "Pullbacks" are therefore sometimes called "vector-Jacobian products" (VJPs), and we refer to the reverse mode rules themselves as "adjoints".