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estimator_double.py
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import numpy as np
from scipy.special import binom, digamma, gamma
import warnings
warnings.filterwarnings("error")
from utility import compute_normalized_volumes
class NNDoubleFunctionalEstimator:
def __init__(self, ks=None, ls=None, alphas=None):
"""
Parameters
----------
ks: np.array
array of integer k values
ls: np.array
array of integer l values
alphas: np.array
array of alpha values for the (logarithmic) alpha divergences
"""
assert len(ks) == len(ls)
self.ks = np.array(ks)
self.ls = np.array(ls)
self.alphas = alphas
self.functional_names = ['KL divergence'] + \
['{}-divergence'.format(alpha) for alpha in alphas] + \
['Logarithmic {}-divergence'.format(alpha) for alpha in alphas]
self.num_functionals = len(self.functional_names)
def phi(self, u, v):
assert (u > 0).all() and (v > 0).all()
phis = np.stack([phi_kl_divergence(u, v, self.ks, self.ls)] +
[phi_alpha_divergence(u, v, self.ks, self.ls, alpha) for alpha in self.alphas] +
[phi_logarithmic_alpha_divergence(u, v, self.ks, self.ls, alpha) for alpha in self.alphas],
0) # (num_functionals, m, len(ks))
return phis
def estimate(self, x, y):
"""
Arguments
---------
x: np.array
data points (m, dim)
y: np.array
data points (n, dim)
"""
u = compute_normalized_volumes(x, ks=self.ks)
v = compute_normalized_volumes(x, ks=self.ls, y=y)
# Compute estimates for each k in ks by taking the mean over samples
estimates = self.phi(u, v).mean(1) # (num_functionals, len(ks))
return estimates
def phi_kl_divergence(u, v, ks, ls):
return -np.log(u) + np.log(v) + digamma(ks) - digamma(ls)
def phi_alpha_divergence(u, v, ks, ls, alpha):
return np.exp(np.log(gamma(ks)) - np.log(gamma(np.maximum(ks, np.ceil(alpha - 1 + 1e-5)) - alpha + 1)) +
np.log(gamma(ls)) - np.log(gamma(np.maximum(ls, np.ceil(1 - alpha + 1e-5)) + alpha - 1))) * \
(u / v) ** (1 - alpha)
def phi_logarithmic_alpha_divergence(u, v, ks, ls, alpha):
return np.exp(np.log(gamma(ks)) - np.log(gamma(np.maximum(ks, np.ceil(alpha - 1 + 1e-5)) - alpha + 1)) +
np.log(gamma(ls)) - np.log(gamma(np.maximum(ls, np.ceil(1 - alpha + 1e-5)) + alpha - 1))) * \
(v / u) ** (alpha - 1) * (digamma(np.maximum(ks, np.ceil(alpha - 1 + 1e-5)) - alpha + 1) -
digamma(np.maximum(ls, np.ceil(1 - alpha + 1e-5)) + alpha - 1) +
np.log(v) - np.log(u))
def phi_asymptotic_nn_classification_error(u, v, ks, ls):
"""
Parameters
----------
u: np.array
(m, len(ks))
v: np.array
(m, len(ls))
ks: np.array
ls: np.array
Returns
-------
phi: (1, m)
"""
assert u.shape[1] == len(ks)
assert v.shape[1] == len(ls)
assert len(ks) == len(ls)
def phi_kl(w, k, l):
"""
Parameters
----------
w: (m, )
shorthand notation for u/v
k: int
l: int
Returns
-------
phi: (m, )
"""
phi = - (w >= 1).astype(float) * (1 - 1 / w) ** (k + l - 2)
for i in range(l):
phi += binom(k + l - 2, i) * (-1 / w) ** i
phi *= (-w) ** (l - 1) / binom(k + l - 2, k - 1)
phi = 1 - phi
return phi
w = u / v
phi = np.stack([
phi_kl(w[:, idx_k], ks[idx_k], ls[idx_k])
for idx_k in range(len(ks))], 1) # (m, len(ks))
return phi
def phi_jensen_shannon_divergence(u, v, ks, ls):
"""
Parameters
----------
u: np.array
(m, len(ks))
v: np.array
(m, len(ls))
ks: np.array
ls: np.array
Returns
-------
phi: (1, m)
"""
assert u.shape[1] == len(ks)
assert v.shape[1] == len(ls)
assert len(ks) == len(ls)
def phi_kl(u, v, k, l):
"""
Parameters
----------
u: (m, )
v: (m, )
k: int
l: int
Returns
-------
phi: (m, )
"""
def b1(w, k, l):
# bkl when w is less than 1
s = 0
for j in range(l - 1):
s += binom(k + l - 2, j) * (-w) ** (l - 1 - j) / (l - 1 - j)
s /= binom(k + l - 2, k - 1)
return s
def b2(w, k, l):
# bkl when w is greater than 1
s = 0
for i in range(k - 1):
s += binom(k + l - 2, i) * (-1 / w) ** (k - 1 - i) / (k - 1 - i)
for i in range(k + l - 1):
if i != k - 1:
s -= binom(k + l - 2, i) * (-1.) ** (k - 1 - i) / (k - 1 - i)
s /= binom(k + l - 2, k - 1)
s -= np.log(w)
return s
def b(w, k, l):
return b1(w, k, l) * (w < 1) + b2(w, k, l) * (w >= 1)
l = max(l, 2)
w = u / v
phi = np.log(2) + (l - 1) / k * w * (np.log(2) + digamma(l - 1 + 1e-5) - digamma(k + 1) + np.log(w)) + \
(b(w, k, l) + (l - 1) / k * w * b(w, k + 1, l - 1))
phi /= 2
return phi
phi = np.stack([
phi_kl(u[:, idx_k], v[:, idx_k], ks[idx_k], ls[idx_k])
for idx_k in range(len(ks))], 1) # (m, len(ks))
return phi