svm.py

class SVMTrainer(object):
    def __init__(self, kernel, c):
        self._kernel = kernel
        self._c = c

    def train(self, X, y):
        """Given the training features X with labels y, returns a SVM
        predictor representing the trained SVM.
        """
        lagrange_multipliers = self._compute_multipliers(X, y)
        return self._construct_predictor(X, y, lagrange_multipliers)

    def _gram_matrix(self, X):
        n_samples, n_features = X.shape
        K = np.zeros((n_samples, n_samples))
        # TODO(tulloch) - vectorize
        for i, x_i in enumerate(X):
            for j, x_j in enumerate(X):
                K[i, j] = self._kernel(x_i, x_j)
        return K

    def _construct_predictor(self, X, y, lagrange_multipliers):
        support_vector_indices = \
            lagrange_multipliers > MIN_SUPPORT_VECTOR_MULTIPLIER

        support_multipliers = lagrange_multipliers[support_vector_indices]
        support_vectors = X[support_vector_indices]
        support_vector_labels = y[support_vector_indices]

        # http://www.cs.cmu.edu/~guestrin/Class/10701-S07/Slides/kernels.pdf
        # bias = y_k - \sum z_i y_i  K(x_k, x_i)
        # Thus we can just predict an example with bias of zero, and
        # compute error.
        bias = np.mean(
            [y_k - SVMPredictor(
                kernel=self._kernel,
                bias=0.0,
                weights=support_multipliers,
                support_vectors=support_vectors,
                support_vector_labels=support_vector_labels).predict(x_k)
             for (y_k, x_k) in zip(support_vector_labels, support_vectors)])

        return SVMPredictor(
            kernel=self._kernel,
            bias=bias,
            weights=support_multipliers,
            support_vectors=support_vectors,
            support_vector_labels=support_vector_labels)

    def _compute_multipliers(self, X, y):
        n_samples, n_features = X.shape

        K = self._gram_matrix(X)
        # Solves
        # min 1/2 x^T P x + q^T x
        # s.t.
        #  Gx \coneleq h
        #  Ax = b

        P = cvxopt.matrix(np.outer(y, y) * K)
        q = cvxopt.matrix(-1 * np.ones(n_samples))

        # -a_i \leq 0
        # TODO(tulloch) - modify G, h so that we have a soft-margin classifier
        G_std = cvxopt.matrix(np.diag(np.ones(n_samples) * -1))
        h_std = cvxopt.matrix(np.zeros(n_samples))

        # a_i \leq c
        G_slack = cvxopt.matrix(np.diag(np.ones(n_samples)))
        h_slack = cvxopt.matrix(np.ones(n_samples) * self._c)

        G = cvxopt.matrix(np.vstack((G_std, G_slack)))
        h = cvxopt.matrix(np.vstack((h_std, h_slack)))

        A = cvxopt.matrix(y, (1, n_samples))
        b = cvxopt.matrix(0.0)

        solution = cvxopt.solvers.qp(P, q, G, h, A, b)

        # Lagrange multipliers
        return np.ravel(solution['x'])