def __init__(self):
self.likelihood = lambda x, y: utils.logistic_sigmoid(x, y)
self.loglikelihood = lambda x, y: np.log(self.likelihood(x, y))
self.max_iter = 100
self.posterior_mode = None
self.covariance_function = utils.squared_exponential
self.covariance_explicit = utils.squared_exponential_explicit
self.covariance_params = [0., 0., 0.]
self.is_fitted = False
def hessian(A, x, y):
""" Computes the hessian of the logistic loss function at the current
estimate x.
H(x) = (1/n)(<A',D,A>)
D is the diagonal matrix D_ii=(sigmoid(z_i)(1 - sigmoid(z_i)) : R^(n,n)
z_i = y_i(<A_i,x>) : R^1
:param x: Regression weight vector : R^p
:param A: Design/Sensing matrix : R^(n,p)
:param y: Observation label vector : {-1,1}^n
:return: Hessian H(x) : R^(p,p)
"""
z = y * A.dot(x) # decision value for each observation
# transform z:R^n into diag_matrix:R^(n,n)
D = np.diag(logistic_sigmoid(z))
return A.T.dot(D).dot(A) / y.size # normalized hessian
def gradient(A, x, y):
""" Computes the gradient of the logistic loss function at the current
estimate x.
grad_x = (1/n) * sum_i{(y_i * A_i)(sigmoid(z_i) - 1)} : R^p
z_i = y_i(<A_i,x>) : R^1
:param x: Regression weight vector : R^p
:param A: Design/Sensing matrix : R^(n,p)
:param y: Observation label vector : {-1,1}^n
:return: Normalized gradient grad_x : R^p
"""
z = y * A.dot(x) # decision value for each observation
phi = (logistic_sigmoid(z) - 1) * y
grad_x = A.T.dot(phi)
# Gradient normalized by the num obs
return grad_x / y.size
