class

DetBackward

DetBackward

source edit

Python wrappers

pylucid.linalg.detfunction

Autograd node for the determinant $\det(A)$ .

Implements Jacobi's formula by reusing differentiable primitives (inv_op, mT_op, mul_op, broadcast_to_op) so the backward sub-graph is itself differentiable — second-order gradients (Hessians) work without additional code.

Both inputs and outputs are saved: $A$ is required to form $(A^{-1})^\top$ in the backward and $\det(A)$ is the scalar multiplicative factor.

Math

d = \det(A), \qquad g = \frac{\partial L}{\partial d},

\frac{\partial L}{\partial A} = \mathrm{broadcast}(d \cdot g,\; \text{shape}(A)) \;\odot\; (A^{-1})^\top.

Shape

A: (..., N, N).
grad_out: (...,) (matches forward output shape).
Output gradient: (..., N, N).

References

Petersen & Pedersen, The Matrix Cookbook (2012), §9.2.1. Magnus & Neudecker, Matrix Differential Calculus (3rd ed.), §8.4.

See Also

InvBackward : Used internally to form $(A^{-1})^\top$ .

Attributes

schema_v1OpSchema

"det", one input, AmpPolicy::KeepInput to prevent AMP downcasting before the inverse call.

saved_inputs_[0]Storage

Storage of the input matrix

A

saved_output_Storage

Storage of the scalar (per-batch) determinant

\det(A)

Instance methods

apply

→std::vector<Storage>

int apply(int grad_out)

source edit

Compute the gradient {∂L/∂A} from the upstream scalar gradient.

Parameters

grad_outStorage

Upstream gradient

\partial L/\partial \det

with shape matching out_shape_.

Returns

std::vector<Storage>

Single-entry vector {∂L/∂A} aligned with the one differentiable input slot.

class

DetBackward

DetBackward

source edit

Python wrappers

pylucid.linalg.detfunction

Autograd node for the determinant $\det(A)$ .

Both inputs and outputs are saved: $A$ is required to form $(A^{-1})^\top$ in the backward and $\det(A)$ is the scalar multiplicative factor.

Math

d = \det(A), \qquad g = \frac{\partial L}{\partial d},

\frac{\partial L}{\partial A} = \mathrm{broadcast}(d \cdot g,\; \text{shape}(A)) \;\odot\; (A^{-1})^\top.

Shape

A: (..., N, N).
grad_out: (...,) (matches forward output shape).
Output gradient: (..., N, N).

References

Petersen & Pedersen, The Matrix Cookbook (2012), §9.2.1. Magnus & Neudecker, Matrix Differential Calculus (3rd ed.), §8.4.

See Also

InvBackward : Used internally to form $(A^{-1})^\top$ .

Attributes

schema_v1OpSchema

"det", one input, AmpPolicy::KeepInput to prevent AMP downcasting before the inverse call.

saved_inputs_[0]Storage

Storage of the input matrix

A

saved_output_Storage

Storage of the scalar (per-batch) determinant

\det(A)

Instance methods

apply

→std::vector<Storage>

int apply(int grad_out)

source edit

Compute the gradient {∂L/∂A} from the upstream scalar gradient.

Parameters

grad_outStorage

Upstream gradient

\partial L/\partial \det

with shape matching out_shape_.

Returns

std::vector<Storage>

Single-entry vector {∂L/∂A} aligned with the one differentiable input slot.