helperfunctions.py

import numpy as np
from scipy.spatial.distance import cdist


def sigmoid(x):
    """
    Logistic Sigmoid function $\sigma(x) = (1 + exp(-x))^(-1)$
    """
    return 1. / (1 + np.exp(-np.clip(x, -100, 100)))


def lambda_fun(xi):
    """
    Lambda function for the Jaakkola bound (see e.g. Bishop, ch 10)
    """
    res = (sigmoid(xi) - 0.5) / (2. * xi)
    res[xi == 0.0] = 0.125
    return res


class RBF(object):
    def __init__(self, lengthscale=1.0, variance=1.0):
        """
        RBF Kernel with k(x,z) = var * np.exp(-0.5 * ||x-z||^2/length^2)
        """
        self.length = lengthscale
        self.var = variance

    def compute(self, X1, X2=None):
        if X2 is None:
            X2 = X1
        dist = cdist(X1, X2)
        return self.var * np.exp(-0.5 * np.square(dist) / self.length**2)

    def compute_diag(self, X1):
        return np.identity(X1.shape[0]) * self.var