def beta_fisher(x1, x2, params={'alpha': .5, 'beta': .5}):
score1 = beta_score(x1, params)
score2 = beta_score(x2, params)
temp = polygamma(1, params['alpha'] + params['beta'])
fisher_info_mat = np.array([[polygamma(1, params['alpha']) - temp, -temp],
[-temp,
polygamma(1, params['beta']) - temp]])
return np.dot(np.dot(score1, inv(fisher_info_mat)), score2.T)[0, 0]
def gamma_diffusion(x1, x2, params={'t': .09}):
try:
n = len(x1)
except:
n = 1
geo = polygamma(1, x1) - polygamma(1, x2)
squ_geo = np.sum(geo * geo)
return np.exp((-.25 / params['t']) * squ_geo)
def metric_det(param_a, param_b):
"""Compute the determinant of the metric.
Parameters
----------
param_a : array-like, shape=[n_samples,]
param_b : array-like, shape=[n_samples,]
Returns
-------
metric_det : array-like, shape=[n_samples,]
"""
metric_det = polygamma(1, param_a) * polygamma(1, param_b) - \
polygamma(1, param_a + param_b) * (polygamma(1, param_a) +
polygamma(1, param_b))
return metric_det
def gamma_fisher(x1, x2, params={'alpha': 1., 'theta': 1.}):
score1 = gamma_score(x1, params)
score2 = gamma_score(x2, params)
temp = 1. / params['theta']
fisher_info_mat = np.array([[polygamma(1, params['alpha']), temp],
[temp, params['alpha'] / params['theta']**2]])
return np.dot(np.dot(score1, inv(fisher_info_mat)), score2.T)[0, 0]
def inner_product_matrix(self, base_point=None):
"""Compute inner product matrix at the tangent space at base point.
Parameters
----------
base_point : array-like, shape=[n_samples, 2]
Returns
-------
base_point : array-like, shape=[n_samples, 2, 2]
"""
if base_point is not None:
raise ValueError('The metric depends on the base point.')
base_point = gs.to_ndarray(base_point, to_ndim=2)
matrices = []
for point in base_point:
param_a, param_b = point
g0 = gs.array(
[polygamma(1, param_a) - polygamma(1, param_a + param_b),
- polygamma(1, param_a + param_b)])
g1 = gs.array(
[- polygamma(1, param_a + param_b), polygamma(1, param_b)
- polygamma(1, param_a + param_b)])
matrices.append(gs.stack([g0, g1]))
return gs.stack(matrices)
def grep_gradient(alpha, m, x, K, alphaz):
gradient = np.zeros((alpha.shape[0], 2))
lmbda = npr.gamma(alpha, 1.)
lmbda[lmbda < 1e-5] = 1e-5
Tinv_val = fun_Tinv(lmbda, alpha)
h_val = fun_H(Tinv_val, alpha)
u_val = fun_U(Tinv_val, alpha)
zw = m * lmbda / alpha
zw[zw < 1e-5] = 1e-5
logp_der = grad_logp(zw, K, x, alphaz)
logp_val = logp(zw, K, x, alphaz)
logq_der = grad_logQ_Z(zw, alpha)
gradient[:, 0] = logp_der * (h_val - lmbda / alpha) * m / alpha
gradient[:, 1] = logp_der * lmbda / alpha
gradient[:, 0] += logp_val * (
np.log(lmbda) + (alpha / lmbda - 1.) * h_val - sp.digamma(alpha) +
sp.polygamma(2, alpha) / 2. / sp.polygamma(1, alpha))
gradient += grad_entropy(alpha, m)
return gradient
def grad_entropy(alpha, m):
gradient = np.zeros((alpha.shape[0], 2))
gradient[:, 0] = 1. - 1. / alpha + (1. - alpha) * sp.polygamma(1, alpha)
gradient[:, 1] = 1. / m
return gradient
def coefficients(param_a, param_b):
metric_det = self.metric_det(param_a, param_b)
c1 = (polygamma(2, param_a) * polygamma(1, param_b) -
polygamma(2, param_a) * polygamma(1, param_a + param_b) -
polygamma(1, param_b) * polygamma(2, param_a + param_b)) / (
2 * metric_det)
c2 = - polygamma(1, param_b) * polygamma(2, param_a + param_b) / (
2 * metric_det)
c3 = (polygamma(2, param_b) * polygamma(1, param_a + param_b) -
polygamma(1, param_b) * polygamma(2, param_a + param_b)) / (
2 * metric_det)
return c1, c2, c3
def fun_Tinv(z, alpha):
return (np.log(z) - sp.digamma(alpha)) / np.sqrt(sp.polygamma(1, alpha))
def fun_U(eps, alpha):
poly1 = sp.polygamma(1, alpha)
poly2 = sp.polygamma(2, alpha)
return 0.5 * poly2 / poly1 + 0.5 * poly2 * eps / np.sqrt(poly1) + poly1
def fun_H(eps, alpha):
poly1 = sp.polygamma(1, alpha)
poly2 = sp.polygamma(2, alpha)
T_val = fun_T(eps, alpha)
return T_val * (0.5 * eps * poly2 / np.sqrt(poly1) + poly1)
def fun_T(eps, alpha):
return np.exp(eps * np.sqrt(sp.polygamma(1, alpha)) + sp.digamma(alpha))
from __future__ import absolute_import
import autograd.numpy as np
import autograd.scipy.special as sp
### Gamma functions ###
sp.polygamma.defjvp(lambda g, ans, gvs, vs, n, x: g * sp.polygamma(n + 1, x),
argnum=1)
sp.psi.defjvp(lambda g, ans, gvs, vs, x: g * sp.polygamma(1, x))
sp.digamma.defjvp(lambda g, ans, gvs, vs, x: g * sp.polygamma(1, x))
sp.gamma.defjvp(lambda g, ans, gvs, vs, x: g * ans * sp.psi(x))
sp.gammaln.defjvp(lambda g, ans, gvs, vs, x: g * sp.psi(x))
sp.rgamma.defjvp(lambda g, ans, gvs, vs, x: g * sp.psi(x) / -sp.gamma(x))
sp.multigammaln.defjvp(lambda g, ans, gvs, vs, a, d: g * np.sum(
sp.digamma(np.expand_dims(a, -1) - np.arange(d) / 2.), -1))
### Bessel functions ###
sp.j0.defjvp(lambda g, ans, gvs, vs, x: -g * sp.j1(x))
sp.y0.defjvp(lambda g, ans, gvs, vs, x: -g * sp.y1(x))
sp.j1.defjvp(lambda g, ans, gvs, vs, x: g * (sp.j0(x) - sp.jn(2, x)) / 2.0)
sp.y1.defjvp(lambda g, ans, gvs, vs, x: g * (sp.y0(x) - sp.yn(2, x)) / 2.0)
sp.jn.defjvp(lambda g, ans, gvs, vs, n, x: g *
(sp.jn(n - 1, x) - sp.jn(n + 1, x)) / 2.0,
argnum=1)
sp.yn.defjvp(lambda g, ans, gvs, vs, n, x: g *
(sp.yn(n - 1, x) - sp.yn(n + 1, x)) / 2.0,
argnum=1)
### Error Function ###
sp.erf.defjvp(
lambda g, ans, gvs, vs, x: 2. * g * sp.inv_root_pi * np.exp(-x**2))
def beta_score(x, params):
v1 = np.log(x) - polygamma(0, params['alpha']) + polygamma(
0, params['alpha'] + params['beta'])
v2 = np.log(1 - x) - polygamma(0, params['beta']) + polygamma(
0, params['alpha'] + params['beta'])
return np.array([[v1, v2]])
def gamma_kld(a1, a2, theta1=10., theta2=10.):
return np.sum((a1 - a2) * polygamma(0, a1) - np.log(gammaFn(a1)) +
np.log(gammaFn(a2)) + a2 *
(np.log(theta2) - np.log(theta1)) + a1 *
((theta1 - theta2) / theta2))
def gamma_score(x, params):
v1 = np.log(x) - np.log(params['theta']) - polygamma(0, params['alpha'])
v2 = x / params['theta']**2 - params['alpha'] / params['theta']
return np.array([[v1, v2]])