Function sparsemap_gradient 

pub fn sparsemap_gradient<F>(
    result: &SparsemapResult<F>,
    upstream_grad: &[F],
) -> Vec<F>
where
    F: Float + FromPrimitive + Debug + Clone,

Compute gradient of a loss through SparseMAP via the active-set theorem.

For SparseMAP, the Jacobian of the optimal solution μ*(θ) w.r.t. θ is:

dμ*/dθ = Π_S  (projection onto tangent space of active support S)

Concretely, for the simplex case, only active coordinates (support) can receive gradient. The backward pass is:

dL/dθ = Π_S (upstream_grad)
      = upstream_grad[support] - mean(upstream_grad[support]) · 1_S

This is the projection of upstream_grad onto the tangent space of the simplex face defined by the active support.

Arguments

• result – the forward-pass SparsemapResult.
• upstream_grad – gradient of the scalar loss w.r.t. solution (∂L/∂μ).

Returns

Gradient ∂L/∂θ of the same length as the score input.