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unfinished work on ratinoal quadratic kern

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This commit is contained in:
James Hensman 2014-02-24 09:41:13 +00:00
parent 712be15f6d
commit ff23a59d2d
5 changed files with 134 additions and 192 deletions
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12
GPy/kern/__init__.py
View file

@ -1,12 +1,11 @@
from _src.rbf import RBF
from _src.white import White
from _src.kern import Kern
from _src.linear import Linear
from _src.bias import Bias
from _src.static import Bias, White
from _src.brownian import Brownian
from _src.stationary import Exponential, Matern32, Matern52, ExpQuad
from _src.stationary import Exponential, Matern32, Matern52, ExpQuad, RatQuad
from _src.mlp import MLP
#import coregionalize
#import exponential
#import eq_ode1
#import finite_dimensional
#import fixed
@ -14,10 +13,6 @@ from _src.stationary import Exponential, Matern32, Matern52, ExpQuad
#import hetero
#import hierarchical
#import independent_outputs
#import linear
#import Matern32
#import Matern52
#import mlp
#import ODE_1
#import periodic_exponential
#import periodic_Matern32
@ -31,4 +26,3 @@ from _src.stationary import Exponential, Matern32, Matern52, ExpQuad
#import rbf_inv
#import spline
#import symmetric
#import white

138
GPy/kern/_src/mlp.py
View file

@ -1,11 +1,13 @@
# Copyright (c) 2013, GPy authors (see AUTHORS.txt).
# Licensed under the BSD 3-clause license (see LICENSE.txt)
from kernpart import Kernpart
from kern import Kern
from ...core.parameterization import Param
from ...core.parameterization.transformations import Logexp
import numpy as np
four_over_tau = 2./np.pi
class MLP(Kernpart):
class MLP(Kern):
"""
Multi layer perceptron kernel (also known as arc sine kernel or neural network kernel)
@ -13,10 +15,10 @@ class MLP(Kernpart):
.. math::
k(x,y) = \\sigma^{2}\\frac{2}{\\pi } \\text{asin} \\left ( \\frac{ \\sigma_w^2 x^\\top y+\\sigma_b^2}{\\sqrt{\\sigma_w^2x^\\top x + \\sigma_b^2 + 1}\\sqrt{\\sigma_w^2 y^\\top y \\sigma_b^2 +1}} \\right )
:param input_dim: the number of input dimensions
:type input_dim: int
:type input_dim: int
:param variance: the variance :math:`\sigma^2`
:type variance: float
:param weight_variance: the vector of the variances of the prior over input weights in the neural network :math:`\sigma^2_w`
@ -29,85 +31,58 @@ class MLP(Kernpart):
"""
def __init__(self, input_dim, variance=1., weight_variance=None, bias_variance=100., ARD=False):
self.input_dim = input_dim
self.ARD = ARD
if not ARD:
self.num_params=3
if weight_variance is not None:
weight_variance = np.asarray(weight_variance)
assert weight_variance.size == 1, "Only one weight variance needed for non-ARD kernel"
else:
weight_variance = 100.*np.ones(1)
else:
self.num_params = self.input_dim + 2
if weight_variance is not None:
weight_variance = np.asarray(weight_variance)
assert weight_variance.size == self.input_dim, "bad number of weight variances"
else:
weight_variance = np.ones(self.input_dim)
raise NotImplementedError
def __init__(self, input_dim, variance=1., weight_variance=1., bias_variance=100., name='mlp'):
super(Linear, self).__init__(input_dim, name)
self.variance = Param('variance', variance, Logexp)
self.weight_variance = Param('weight_variance', weight_variance, Logexp)
self.bias_variance = Param('bias_variance', bias_variance, Logexp)
self.add_parameters(self.variance, self.weight_variance, self.bias_variance)
self.name='mlp'
self._set_params(np.hstack((variance, weight_variance.flatten(), bias_variance)))
def _get_params(self):
return np.hstack((self.variance, self.weight_variance.flatten(), self.bias_variance))
def _set_params(self, x):
assert x.size == (self.num_params)
self.variance = x[0]
self.weight_variance = x[1:-1]
self.weight_std = np.sqrt(self.weight_variance)
self.bias_variance = x[-1]
def _get_param_names(self):
if self.num_params == 3:
return ['variance', 'weight_variance', 'bias_variance']
else:
return ['variance'] + ['weight_variance_%i' % i for i in range(self.lengthscale.size)] + ['bias_variance']
def K(self, X, X2, target):
"""Return covariance between X and X2."""
def K(self, X, X2=None):
self._K_computations(X, X2)
target += self.variance*self._K_dvar
return self.variance*self._K_dvar
def Kdiag(self, X, target):
def Kdiag(self, X):
"""Compute the diagonal of the covariance matrix for X."""
self._K_diag_computations(X)
target+= self.variance*self._K_diag_dvar
return self.variance*self._K_diag_dvar
def _param_grad_helper(self, dL_dK, X, X2, target):
def update_gradients_full(self, dL_dK, X, X2=None):
"""Derivative of the covariance with respect to the parameters."""
self._K_computations(X, X2)
denom3 = self._K_denom*self._K_denom*self._K_denom
self.variance.gradient = np.sum(self._K_dvar*dL_dK)
denom3 = self._K_denom**3
base = four_over_tau*self.variance/np.sqrt(1-self._K_asin_arg*self._K_asin_arg)
base_cov_grad = base*dL_dK
if X2 is None:
vec = np.diag(self._K_inner_prod)
target[1] += ((self._K_inner_prod/self._K_denom
self.weight_variance.gradient = ((self._K_inner_prod/self._K_denom
-.5*self._K_numer/denom3
*(np.outer((self.weight_variance*vec+self.bias_variance+1.), vec)
*(np.outer((self.weight_variance*vec+self.bias_variance+1.), vec)
+np.outer(vec,(self.weight_variance*vec+self.bias_variance+1.))))*base_cov_grad).sum()
target[2] += ((1./self._K_denom
-.5*self._K_numer/denom3
self.bias_variance.gradient = ((1./self._K_denom
-.5*self._K_numer/denom3
*((vec[None, :]+vec[:, None])*self.weight_variance
+2.*self.bias_variance + 2.))*base_cov_grad).sum()
else:
vec1 = (X*X).sum(1)
vec2 = (X2*X2).sum(1)
target[1] += ((self._K_inner_prod/self._K_denom
self.weight_variance.gradient = ((self._K_inner_prod/self._K_denom
-.5*self._K_numer/denom3
*(np.outer((self.weight_variance*vec1+self.bias_variance+1.), vec2) + np.outer(vec1, self.weight_variance*vec2 + self.bias_variance+1.)))*base_cov_grad).sum()
target[2] += ((1./self._K_denom
-.5*self._K_numer/denom3
self.bias_variance.gradient = ((1./self._K_denom
-.5*self._K_numer/denom3
*((vec1[:, None]+vec2[None, :])*self.weight_variance
+ 2*self.bias_variance + 2.))*base_cov_grad).sum()
target[0] += np.sum(self._K_dvar*dL_dK)
def gradients_X(self, dL_dK, X, X2, target):
def update_gradients_diag(self, X):
raise NotImplementedError, "TODO"
def gradients_X(self, dL_dK, X, X2):
"""Derivative of the covariance matrix with respect to X"""
self._K_computations(X, X2)
arg = self._K_asin_arg
@ -116,47 +91,38 @@ class MLP(Kernpart):
denom3 = denom*denom*denom
if X2 is not None:
vec2 = (X2*X2).sum(1)*self.weight_variance+self.bias_variance + 1.
target += four_over_tau*self.weight_variance*self.variance*((X2[None, :, :]/denom[:, :, None] - vec2[None, :, None]*X[:, None, :]*(numer/denom3)[:, :, None])*(dL_dK/np.sqrt(1-arg*arg))[:, :, None]).sum(1)
return four_over_tau*self.weight_variance*self.variance*((X2[None, :, :]/denom[:, :, None] - vec2[None, :, None]*X[:, None, :]*(numer/denom3)[:, :, None])*(dL_dK/np.sqrt(1-arg*arg))[:, :, None]).sum(1)
else:
vec = (X*X).sum(1)*self.weight_variance+self.bias_variance + 1.
target += 2*four_over_tau*self.weight_variance*self.variance*((X[None, :, :]/denom[:, :, None] - vec[None, :, None]*X[:, None, :]*(numer/denom3)[:, :, None])*(dL_dK/np.sqrt(1-arg*arg))[:, :, None]).sum(1)
return 2*four_over_tau*self.weight_variance*self.variance*((X[None, :, :]/denom[:, :, None] - vec[None, :, None]*X[:, None, :]*(numer/denom3)[:, :, None])*(dL_dK/np.sqrt(1-arg*arg))[:, :, None]).sum(1)
def dKdiag_dX(self, dL_dKdiag, X, target):
"""Gradient of diagonal of covariance with respect to X"""
self._K_diag_computations(X)
arg = self._K_diag_asin_arg
denom = self._K_diag_denom
numer = self._K_diag_numer
target += four_over_tau*2.*self.weight_variance*self.variance*X*(1/denom*(1 - arg)*dL_dKdiag/(np.sqrt(1-arg*arg)))[:, None]
return four_over_tau*2.*self.weight_variance*self.variance*X*(1./denom*(1. - arg)*dL_dKdiag/(np.sqrt(1-arg*arg)))[:, None]
def _K_computations(self, X, X2):
"""Pre-computations for the covariance matrix (used for computing the covariance and its gradients."""
if self.ARD:
pass
if X2 is None:
self._K_inner_prod = np.dot(X,X.T)
vec = np.diag(self._K_numer) + 1.
self._K_denom = np.sqrt(np.outer(vec,vec))
else:
if X2 is None:
self._K_inner_prod = np.dot(X,X.T)
self._K_numer = self._K_inner_prod*self.weight_variance+self.bias_variance
vec = np.diag(self._K_numer) + 1.
self._K_denom = np.sqrt(np.outer(vec,vec))
self._K_asin_arg = self._K_numer/self._K_denom
self._K_dvar = four_over_tau*np.arcsin(self._K_asin_arg)
else:
self._K_inner_prod = np.dot(X,X2.T)
self._K_numer = self._K_inner_prod*self.weight_variance + self.bias_variance
vec1 = (X*X).sum(1)*self.weight_variance + self.bias_variance + 1.
vec2 = (X2*X2).sum(1)*self.weight_variance + self.bias_variance + 1.
self._K_denom = np.sqrt(np.outer(vec1,vec2))
self._K_asin_arg = self._K_numer/self._K_denom
self._K_dvar = four_over_tau*np.arcsin(self._K_asin_arg)
self._K_inner_prod = np.dot(X,X2.T)
vec1 = (X*X).sum(1)*self.weight_variance + self.bias_variance + 1.
vec2 = (X2*X2).sum(1)*self.weight_variance + self.bias_variance + 1.
self._K_denom = np.sqrt(np.outer(vec1,vec2))
self._K_numer = self._K_inner_prod*self.weight_variance + self.bias_variance
self._K_asin_arg = self._K_numer/self._K_denom
self._K_dvar = four_over_tau*np.arcsin(self._K_asin_arg)
def _K_diag_computations(self, X):
"""Pre-computations concerning the diagonal terms (used for computation of diagonal and its gradients)."""
if self.ARD:
pass
else:
self._K_diag_numer = (X*X).sum(1)*self.weight_variance + self.bias_variance
self._K_diag_denom = self._K_diag_numer+1.
self._K_diag_asin_arg = self._K_diag_numer/self._K_diag_denom
self._K_diag_dvar = four_over_tau*np.arcsin(self._K_diag_asin_arg)
self._K_diag_numer = (X*X).sum(1)*self.weight_variance + self.bias_variance
self._K_diag_denom = self._K_diag_numer+1.
self._K_diag_asin_arg = self._K_diag_numer/self._K_diag_denom
self._K_diag_dvar = four_over_tau*np.arcsin(self._K_diag_asin_arg)

83
GPy/kern/_src/bias.py → GPy/kern/_src/static.py
View file

@ -7,8 +7,63 @@ from ...core.parameterization import Param
from ...core.parameterization.transformations import Logexp
import numpy as np
class Bias(Kern):
def __init__(self,input_dim,variance=1.,name=None):
class Static(Kern):
def gradients_X(self, dL_dK, X, X2, target):
return np.zeros(X.shape)
def gradients_X_diag(self, dL_dKdiag, X, target):
return np.zeros(X.shape)
def gradients_Z_variational(self, dL_dKmm, dL_dpsi0, dL_dpsi1, dL_dpsi2, mu, S, Z):
return np.zeros(Z.shape)
def gradients_muS_variational(self, dL_dKmm, dL_dpsi0, dL_dpsi1, dL_dpsi2, mu, S, Z):
return np.zeros(mu.shape), np.zeros(S.shape)
def psi0(self, Z, mu, S):
return self.Kdiag(mu)
def psi1(self, Z, mu, S, target):
return self.K(mu, Z)
def psi2(Z, mu, S):
K = self.K(mu, Z)
return K[:,:,None]*K[:,None,:] # NB. more efficient implementations on inherriting classes
class White(Static):
def __init__(self, input_dim, variance=1., name='white'):
super(White, self).__init__(input_dim, name)
self.input_dim = input_dim
self.variance = Param('variance', variance, Logexp())
self.add_parameters(self.variance)
def K(self, X, X2=None):
if X2 is None:
return np.eye(X.shape[0])*self.variance
else:
return np.zeros((X.shape[0], X2.shape[0]))
def Kdiag(self, X):
ret = np.ones(X.shape[0])
ret[:] = self.variance
return ret
def psi2(self, Z, mu, S, target):
return np.zeros((mu.shape[0], Z.shape[0], Z.shape[0]), dtype=np.float64)
def update_gradients_full(self, dL_dK, X):
self.variance.gradient = np.trace(dL_dK)
def update_gradients_diag(self, dL_dKdiag, X):
self.variance.gradient = dL_dKdiag.sum()
def update_gradients_variational(self, dL_dKmm, dL_dpsi0, dL_dpsi1, dL_dpsi2, mu, S, Z):
self.variance.gradient = np.trace(dL_dKmm) + dL_dpsi0.sum()
class Bias(Static):
def __init__(self, input_dim, variance=1., name=None):
super(Bias, self).__init__(input_dim, name)
self.variance = Param("variance", variance, Logexp())
self.add_parameter(self.variance)
@ -19,7 +74,7 @@ class Bias(Kern):
ret[:] = self.variance
return ret
def Kdiag(self,X):
def Kdiag(self, X):
ret = np.empty((X.shape[0],), dtype=np.float64)
ret[:] = self.variance
return ret
@ -30,23 +85,6 @@ class Bias(Kern):
def update_gradients_diag(self, dL_dKdiag, X):
self.variance.gradient = dL_dK.sum()
def gradients_X(self, dL_dK,X, X2, target):
return np.zeros(X.shape)
def gradients_X_diag(self,dL_dKdiag,X,target):
return np.zeros(X.shape)
#---------------------------------------#
# PSI statistics #
#---------------------------------------#
def psi0(self, Z, mu, S):
return self.Kdiag(mu)
def psi1(self, Z, mu, S, target):
return self.K(mu, S)
def psi2(self, Z, mu, S, target):
ret = np.empty((mu.shape[0], Z.shape[0], Z.shape[0]), dtype=np.float64)
ret[:] = self.variance**2
@ -55,8 +93,3 @@ class Bias(Kern):
def update_gradients_variational(self, dL_dKmm, dL_dpsi0, dL_dpsi1, dL_dpsi2, mu, S, Z):
self.variance.gradient = dL_dKmm.sum() + dL_dpsi0.sum() + dL_dpsi1.sum() + 2.*self.variance*dL_dpsi2.sum()
def gradients_Z_variational(self, dL_dKmm, dL_dpsi0, dL_dpsi1, dL_dpsi2, mu, S, Z):
return np.zeros(Z.shape)
def gradients_muS_variational(self, dL_dKmm, dL_dpsi0, dL_dpsi1, dL_dpsi2, mu, S, Z):
return np.zeros(mu.shape), np.zeros(S.shape)

23
GPy/kern/_src/stationary.py
View file

@ -193,8 +193,6 @@ class Matern52(Stationary):
return(1./self.variance* (G_coef*G + orig + orig2))
class ExpQuad(Stationary):
def __init__(self, input_dim, variance=1., lengthscale=None, ARD=False, name='ExpQuad'):
super(ExpQuad, self).__init__(input_dim, variance, lengthscale, ARD, name)
@ -207,5 +205,26 @@ class ExpQuad(Stationary):
dist = self._scaled_dist(X, X2)
return -dist*self.K(X, X2)
class RatQuad(Stationary):
def __init__(self, input_dim, variance=1., lengthscale=None, power=2., ARD=False, name='ExpQuad'):
super(RatQuad, self).__init__(input_dim, variance, lengthscale, ARD, name)
self.power = Param('power', power, Logexp)
self.add_parameters(self.power)
def K(self, X, X2=None):
r = self._scaled_dist(X, X2)
return self.variance*(1. + r**2/2.)**(-self.power)
def dK_dr(self, X, X2):
r = self._scaled_dist(X, X2)
return -self.variance*self.power*r*(1. + r**2/2)**(-self.power - 1.)
def update_gradients_full(self, dL_dK, X, X2=None):
super(RatQuad, self).update_gradients_full(dL_dK, X, X2)
r = self._scaled_dist(X, X2)
r2 = r**2
dpow = -2.**self.power*(r2 + 2.)**(-self.power)*np.log(0.5*(r2+2.))
self.power.gradient = np.sum(dL_dK*dpow)

70
GPy/kern/_src/white.py
View file

@ -6,73 +6,3 @@ import numpy as np
from ...core.parameterization import Param
from ...core.parameterization.transformations import Logexp
class White(Kern):
"""
White noise kernel.
:param input_dim: the number of input dimensions
:type input_dim: int
:param variance:
:type variance: float
"""
def __init__(self,input_dim,variance=1., name='white'):
super(White, self).__init__(input_dim, name)
self.input_dim = input_dim
self.variance = Param('variance', variance, Logexp())
self.add_parameters(self.variance)
def K(self, X, X2=None):
if X2 is None:
return np.eye(X.shape[0])*self.variance
else:
return np.zeros((X.shape[0], X2.shape[0]))
def Kdiag(self,X):
ret = np.ones(X.shape[0])
ret[:] = self.variance
return ret
def update_gradients_full(self, dL_dK, X):
self.variance.gradient = np.trace(dL_dK)
def update_gradients_sparse(self, dL_dKmm, dL_dKnm, dL_dKdiag, X, Z):
self.variance.gradient = np.trace(dL_dKmm) + np.sum(dL_dKdiag)
def update_gradients_variational(self, dL_dKmm, dL_dpsi0, dL_dpsi1, dL_dpsi2, mu, S, Z):
raise NotImplementedError
def gradients_X(self,dL_dK,X,X2):
return np.zeros_like(X)
def psi0(self,Z,mu,S,target):
pass # target += self.variance
def dpsi0_dtheta(self,dL_dpsi0,Z,mu,S,target):
pass # target += dL_dpsi0.sum()
def dpsi0_dmuS(self,dL_dpsi0,Z,mu,S,target_mu,target_S):
pass
def psi1(self,Z,mu,S,target):
pass
def dpsi1_dtheta(self,dL_dpsi1,Z,mu,S,target):
pass
def dpsi1_dZ(self,dL_dpsi1,Z,mu,S,target):
pass
def dpsi1_dmuS(self,dL_dpsi1,Z,mu,S,target_mu,target_S):
pass
def psi2(self,Z,mu,S,target):
pass
def dpsi2_dZ(self,dL_dpsi2,Z,mu,S,target):
pass
def dpsi2_dtheta(self,dL_dpsi2,Z,mu,S,target):
pass
def dpsi2_dmuS(self,dL_dpsi2,Z,mu,S,target_mu,target_S):
pass