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260 lines
9.7 KiB
Python
260 lines
9.7 KiB
Python
# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
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# Licensed under the BSD 3-clause license (see LICENSE.txt)
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import numpy as np
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from scipy import weave
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from ...util.misc import param_to_array
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from stationary import Stationary
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from GPy.util.caching import Cache_this
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class RBF(Stationary):
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"""
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Radial Basis Function kernel, aka squared-exponential, exponentiated quadratic or Gaussian kernel:
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.. math::
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k(r) = \sigma^2 \exp \\bigg(- \\frac{1}{2} r^2 \\bigg)
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"""
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def __init__(self, input_dim, variance=1., lengthscale=None, ARD=False, name='RBF'):
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super(RBF, self).__init__(input_dim, variance, lengthscale, ARD, name)
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self.weave_options = {}
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def K_of_r(self, r):
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return self.variance * np.exp(-0.5 * r**2)
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def dK_dr(self, r):
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return -r*self.K_of_r(r)
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#---------------------------------------#
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# PSI statistics #
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#---------------------------------------#
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def psi0(self, Z, variational_posterior):
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return self.Kdiag(variational_posterior.mean)
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def psi1(self, Z, variational_posterior):
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_, _, _, psi1 = self._psi1computations(Z, variational_posterior)
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return psi1
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def psi2(self, Z, variational_posterior):
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_, _, _, _, _, psi2 = self._psi2computations(Z, variational_posterior)
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return psi2
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def update_gradients_expectations(self, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, variational_posterior):
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l2 = self.lengthscale **2
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#contributions from psi0:
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self.variance.gradient = np.sum(dL_dpsi0)
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self.lengthscale.gradient = 0.
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#from psi1
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denom, _, dist_sq, psi1 = self._psi1computations(Z, variational_posterior)
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d_length = psi1[:,:,None] * ((dist_sq - 1.)/(self.lengthscale*denom) +1./self.lengthscale)
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dpsi1_dlength = d_length * dL_dpsi1[:, :, None]
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if not self.ARD:
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self.lengthscale.gradient += dpsi1_dlength.sum()
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else:
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self.lengthscale.gradient += dpsi1_dlength.sum(0).sum(0)
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self.variance.gradient += np.sum(dL_dpsi1 * psi1) / self.variance
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#from psi2
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S = variational_posterior.variance
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denom, _, Zdist_sq, _, mudist_sq, psi2 = self._psi2computations(Z, variational_posterior)
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d_length = 2.*psi2[:, :, :, None] * (Zdist_sq * denom + mudist_sq + S[:, None, None, :] / l2) / (self.lengthscale * denom)
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#TODO: combine denom and l2 as denom_l2??
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#TODO: tidy the above!
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#TODO: tensordot below?
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dpsi2_dlength = d_length * dL_dpsi2[:, :, :, None]
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if not self.ARD:
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self.lengthscale.gradient += dpsi2_dlength.sum()
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else:
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self.lengthscale.gradient += dpsi2_dlength.sum(0).sum(0).sum(0)
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self.variance.gradient += 2.*np.sum(dL_dpsi2 * psi2)/self.variance
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def gradients_Z_expectations(self, dL_dpsi1, dL_dpsi2, Z, variational_posterior):
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l2 = self.lengthscale **2
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#psi1
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denom, dist, dist_sq, psi1 = self._psi1computations(Z, variational_posterior)
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denominator = l2 * denom
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dpsi1_dZ = -psi1[:, :, None] * (dist / denominator)
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grad = np.sum(dL_dpsi1[:, :, None] * dpsi1_dZ, 0)
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#psi2
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denom, Zdist, Zdist_sq, mudist, mudist_sq, psi2 = self._psi2computations(Z, variational_posterior)
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term1 = Zdist / l2 # M, M, Q
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term2 = mudist / denom / l2 # N, M, M, Q
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dZ = psi2[:, :, :, None] * (term1[None, :, :, :] + term2) #N,M,M,Q
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grad += 2*(dL_dpsi2[:, :, :, None] * dZ).sum(0).sum(0)
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return grad
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def gradients_qX_expectations(self, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, variational_posterior):
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l2 = self.lengthscale **2
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#psi1
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denom, dist, dist_sq, psi1 = self._psi1computations(Z, variational_posterior)
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tmp = psi1[:, :, None] / l2 / denom
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grad_mu = np.sum(dL_dpsi1[:, :, None] * tmp * dist, 1)
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grad_S = np.sum(dL_dpsi1[:, :, None] * 0.5 * tmp * (dist_sq - 1), 1)
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#psi2
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denom, Zdist, Zdist_sq, mudist, mudist_sq, psi2 = self._psi2computations(Z, variational_posterior)
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tmp = psi2[:, :, :, None] / l2 / denom
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grad_mu += -2.*(dL_dpsi2[:, :, :, None] * tmp * mudist).sum(1).sum(1)
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grad_S += (dL_dpsi2[:, :, :, None] * tmp * (2.*mudist_sq - 1)).sum(1).sum(1)
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return grad_mu, grad_S
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#---------------------------------------#
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# Precomputations #
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#---------------------------------------#
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#TODO: this function is unused, but it will be useful in the stationary class
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def _dL_dlengthscales_via_K(self, dL_dK, X, X2):
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"""
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A helper function for update_gradients_* methods
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Computes the derivative of the objective L wrt the lengthscales via
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dL_dl = sum_{i,j}(dL_dK_{ij} dK_dl)
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assumes self._K_computations has just been called.
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This is only valid if self.ARD=True
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"""
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target = np.zeros(self.input_dim)
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dvardLdK = self._K_dvar * dL_dK
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var_len3 = self.variance / np.power(self.lengthscale, 3)
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if X2 is None:
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# save computation for the symmetrical case
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dvardLdK = dvardLdK + dvardLdK.T
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code = """
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int q,i,j;
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double tmp;
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for(q=0; q<input_dim; q++){
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tmp = 0;
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for(i=0; i<num_data; i++){
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for(j=0; j<i; j++){
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tmp += (X(i,q)-X(j,q))*(X(i,q)-X(j,q))*dvardLdK(i,j);
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}
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}
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target(q) += var_len3(q)*tmp;
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}
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"""
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num_data, num_inducing, input_dim = X.shape[0], X.shape[0], self.input_dim
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X, dvardLdK, var_len3 = param_to_array(X, dvardLdK, var_len3)
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weave.inline(code, arg_names=['num_data', 'num_inducing', 'input_dim', 'X', 'target', 'dvardLdK', 'var_len3'], type_converters=weave.converters.blitz, **self.weave_options)
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else:
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code = """
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int q,i,j;
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double tmp;
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for(q=0; q<input_dim; q++){
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tmp = 0;
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for(i=0; i<num_data; i++){
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for(j=0; j<num_inducing; j++){
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tmp += (X(i,q)-X2(j,q))*(X(i,q)-X2(j,q))*dvardLdK(i,j);
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}
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}
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target(q) += var_len3(q)*tmp;
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}
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"""
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num_data, num_inducing, input_dim = X.shape[0], X2.shape[0], self.input_dim
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X, X2, dvardLdK, var_len3 = param_to_array(X, X2, dvardLdK, var_len3)
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weave.inline(code, arg_names=['num_data', 'num_inducing', 'input_dim', 'X', 'X2', 'target', 'dvardLdK', 'var_len3'], type_converters=weave.converters.blitz, **self.weave_options)
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return target
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@Cache_this(limit=1)
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def _psi1computations(self, Z, vp):
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mu, S = vp.mean, vp.variance
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l2 = self.lengthscale **2
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denom = S[:, None, :] / l2 + 1. # N,1,Q
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dist = Z[None, :, :] - mu[:, None, :] # N,M,Q
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dist_sq = np.square(dist) / l2 / denom # N,M,Q
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exponent = -0.5 * np.sum(dist_sq + np.log(denom), -1)#N,M
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psi1 = self.variance * np.exp(exponent) # N,M
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return denom, dist, dist_sq, psi1
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#@cache_this(ignore_args=(1,))
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def _Z_distances(self, Z):
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Zhat = 0.5 * (Z[:, None, :] + Z[None, :, :]) # M,M,Q
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Zdist = 0.5 * (Z[:, None, :] - Z[None, :, :]) # M,M,Q
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return Zhat, Zdist
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@Cache_this(limit=1)
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def _psi2computations(self, Z, vp):
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mu, S = vp.mean, vp.variance
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N, Q = mu.shape
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M = Z.shape[0]
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#compute required distances
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Zhat, Zdist = self._Z_distances(Z)
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Zdist_sq = np.square(Zdist / self.lengthscale) # M,M,Q
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#allocate memory for the things we want to compute
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mudist = np.empty((N, M, M, Q))
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mudist_sq = np.empty((N, M, M, Q))
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exponent = np.zeros((N,M,M))
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psi2 = np.empty((N, M, M))
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l2 = self.lengthscale **2
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denom = (2.*S[:,None,None,:] / l2) + 1. # N,Q
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half_log_denom = 0.5 * np.log(denom[:,0,0,:])
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denom_l2 = denom[:,0,0,:]*l2
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variance_sq = float(np.square(self.variance))
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code = """
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double tmp, exponent_tmp;
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//#pragma omp parallel for private(tmp, exponent_tmp)
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for (int n=0; n<N; n++)
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{
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for (int m=0; m<M; m++)
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{
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for (int mm=0; mm<(m+1); mm++)
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{
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exponent_tmp = 0.0;
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for (int q=0; q<Q; q++)
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{
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//compute mudist
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tmp = mu(n,q) - Zhat(m,mm,q);
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mudist(n,m,mm,q) = tmp;
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mudist(n,mm,m,q) = tmp;
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//now mudist_sq
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tmp = tmp*tmp/denom_l2(n,q);
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mudist_sq(n,m,mm,q) = tmp;
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mudist_sq(n,mm,m,q) = tmp;
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//now exponent
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tmp = -Zdist_sq(m,mm,q) - tmp - half_log_denom(n,q);
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exponent_tmp += tmp;
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}
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//compute psi2 by exponontiating
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psi2(n,m,mm) = variance_sq * exp(exponent_tmp);
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psi2(n,mm,m) = psi2(n,m,mm);
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}
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}
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}
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"""
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support_code = """
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#include <omp.h>
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#include <math.h>
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"""
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mu = param_to_array(mu)
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weave.inline(code, support_code=support_code, libraries=['gomp'],
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arg_names=['N', 'M', 'Q', 'mu', 'Zhat', 'mudist_sq', 'mudist', 'denom_l2', 'Zdist_sq', 'half_log_denom', 'psi2', 'variance_sq'],
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type_converters=weave.converters.blitz, **self.weave_options)
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return denom, Zdist, Zdist_sq, mudist, mudist_sq, psi2
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def input_sensitivity(self):
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if self.ARD: return 1./self.lengthscale
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else: return (1./self.lengthscale).repeat(self.input_dim)
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