Semi-supervised SVM objective with unlabeled hinge

Score the transductive SVM objective

A transductive SVM (Joachims, 1999) adds a penalty on unlabeled points that pushes them away from the decision boundary. For a linear score f(x) = w^T x + b, the objective is

J = 0.5 ||w||^2 + C * sum_l max(0, 1 - y_l f(x_l)) + C_u * sum_u max(0, 1 - |f(x_u)|).

The unlabeled term is zero when a point lies at least margin 1 from the boundary, whichever side it is on, so it rewards separation and not a particular label.

Implement s3vm_objective(w, b, X_l, y_l, X_u, C, C_u) and return J as a float.

Constraints

• w is 1-D. X_l is (n_l, d), y_l holds values in {-1, +1}, and X_u is (n_u, d).
• C and C_u must be nonnegative.
• Labels other than -1 and +1 raise ValueError.
• Shape mismatches raise ValueError.
• X_u may be empty, in which case its term is zero.
• Do not modify the inputs.

Hints