Merge branch 'params' of github.com:SheffieldML/GPy into params

GPy/core/parameterization/variational.py

@@ -9,7 +9,10 @@ from parameterized import Parameterized
 from param import Param
 from transformations import Logexp
 
-class VariationalPrior(object):
+class VariationalPrior(Parameterized):
+    def __init__(self, name=None, **kw):
+        super(VariationalPrior, self).__init__(name=name, **kw)
+        
     def KL_divergence(self, variational_posterior):
         raise NotImplementedError, "override this for variational inference of latent space"
 
@@ -30,6 +33,32 @@ class NormalPrior(VariationalPrior):
         variational_posterior.mean.gradient -= variational_posterior.mean
         variational_posterior.variance.gradient -= (1. - (1. / (variational_posterior.variance))) * 0.5
 
+class SpikeAndSlabPrior(VariationalPrior):
+    def __init__(self, variance = 1.0, pi = 0.5, name='SpikeAndSlabPrior', **kw):
+        super(VariationalPrior, self).__init__(name=name, **kw)
+        assert variance==1.0, "Not Implemented!"
+        self.pi = Param('pi', pi)
+        self.variance = Param('variance',variance)
+        self.add_parameters(self.pi, self.variance)
+        
+    def KL_divergence(self, variational_posterior):
+        mu = variational_posterior.mean
+        S = variational_posterior.variance
+        gamma = variational_posterior.binary_prob
+        var_mean = np.square(mu)
+        var_S = (S - np.log(S))
+        var_gamma = (gamma*np.log(gamma/self.pi)).sum()+((1-gamma)*np.log((1-gamma)/(1-self.pi))).sum()
+        return var_gamma+ 0.5 * (gamma* (var_mean + var_S -1)).sum()
+    
+    def update_gradients_KL(self, variational_posterior):
+        mu = variational_posterior.mean
+        S = variational_posterior.variance
+        gamma = variational_posterior.binary_prob
+
+        gamma.gradient -= np.log((1-self.pi)/self.pi*gamma/(1.-gamma))+(np.square(mu)+S-np.log(S)-1.)/2.
+        mu.gradient -= gamma*mu
+        S.gradient -= (1. - (1. / (S))) * gamma /2.
+
 
 class VariationalPosterior(Parameterized):
     def __init__(self, means=None, variances=None, name=None, **kw):
@@ -65,7 +94,7 @@ class NormalPosterior(VariationalPosterior):
         from ...plotting.matplot_dep import variational_plots
         return variational_plots.plot(self,*args)
 
-class SpikeAndSlab(VariationalPosterior):
+class SpikeAndSlabPosterior(VariationalPosterior):
     '''
     The SpikeAndSlab distribution for variational approximations.
     '''
@@ -73,7 +102,7 @@ class SpikeAndSlab(VariationalPosterior):
         """
         binary_prob : the probability of the distribution on the slab part.
         """
-        super(SpikeAndSlab, self).__init__(means, variances, name)
+        super(SpikeAndSlabPosterior, self).__init__(means, variances, name)
         self.gamma = Param("binary_prob",binary_prob,)
         self.add_parameter(self.gamma)
 

GPy/kern/__init__.py

@@ -9,3 +9,4 @@ from _src.mlp import MLP
 from _src.periodic import PeriodicExponential, PeriodicMatern32, PeriodicMatern52
 from _src.independent_outputs import IndependentOutputs, Hierarchical
 from _src.coregionalize import Coregionalize
+from _src.ssrbf import SSRBF

GPy/kern/_src/ssrbf.py

@@ -0,0 +1,252 @@
+# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
+# Licensed under the BSD 3-clause license (see LICENSE.txt)
+
+
+from kern import Kern
+import numpy as np
+from ...util.linalg import tdot
+from ...util.config import *
+from ...util.caching import cache_this
+from stationary import Stationary
+
+class SSRBF(Stationary):
+    """
+    Radial Basis Function kernel, aka squared-exponential, exponentiated quadratic or Gaussian kernel
+    for Spike-and-Slab GPLVM
+
+    .. math::
+
+       k(r) = \sigma^2 \exp \\bigg(- \\frac{1}{2} r^2 \\bigg) \ \ \ \ \  \\text{ where  } r^2 = \sum_{i=1}^d \\frac{ (x_i-x^\prime_i)^2}{\ell_i^2}
+
+    where \ell_i is the lengthscale, \sigma^2 the variance and d the dimensionality of the input.
+
+    :param input_dim: the number of input dimensions
+    :type input_dim: int
+    :param variance: the variance of the kernel
+    :type variance: float
+    :param lengthscale: the vector of lengthscale of the kernel
+    :type lengthscale: array or list of the appropriate size (or float if there is only one lengthscale parameter)
+    :param ARD: Auto Relevance Determination. If equal to "False", the kernel is isotropic (ie. one single lengthscale parameter \ell), otherwise there is one lengthscale parameter per dimension.
+    :type ARD: Boolean
+    :rtype: kernel object
+
+    .. Note: this object implements both the ARD and 'spherical' version of the function
+    """
+
+    def __init__(self, input_dim, variance=1., lengthscale=None, ARD=True, name='SSRBF'):
+        assert ARD==True, "Not Implemented!"
+        super(SSRBF, self).__init__(input_dim, variance, lengthscale, ARD, name)
+        
+    def K_of_r(self, r):
+        return self.variance * np.exp(-0.5 * r**2)
+
+    def dK_dr(self, r):
+        return -r*self.K_of_r(r)
+
+    def parameters_changed(self):
+        pass
+
+    def Kdiag(self, X):
+        ret = np.empty(X.shape[0])
+        ret[:] = self.variance
+        return ret
+        
+    #---------------------------------------#
+    #             PSI statistics            #
+    #---------------------------------------#
+    
+    def psi0(self, Z, posterior_variational):
+        ret = np.empty(posterior_variational.mean.shape[0])
+        ret[:] = self.variance
+        return ret
+
+    def psi1(self, Z, posterior_variational):
+        self._psi_computations(Z, posterior_variational.mean, posterior_variational.variance, posterior_variational.binary_prob)
+        return self._psi1
+
+    def psi2(self, Z, posterior_variational):
+        self._psi_computations(Z, posterior_variational.mean, posterior_variational.variance, posterior_variational.binary_prob)
+        return self._psi2
+
+    def dL_dpsi0_dmuSgamma(self, dL_dpsi0, Z, mu, S, gamma, target_mu, target_S, target_gamma):
+        pass
+
+
+    def dL_dpsi1_dmuSgamma(self, dL_dpsi1, Z, mu, S, gamma, target_mu, target_S, target_gamma):
+        self._psi_computations(Z, mu, S, gamma)
+        target_mu += (dL_dpsi1[:, :, None] * self._dpsi1_dmu).sum(axis=1)
+        target_S += (dL_dpsi1[:, :, None] * self._dpsi1_dS).sum(axis=1)
+        target_gamma += (dL_dpsi1[:,:,None] * self._dpsi1_dgamma).sum(axis=1)
+
+
+    def dL_dpsi2_dmuSgamma(self, dL_dpsi2, Z, mu, S, gamma, target_mu, target_S, target_gamma):
+        """Think N,num_inducing,num_inducing,input_dim """
+        self._psi_computations(Z, mu, S, gamma)
+        target_mu += (dL_dpsi2[:, :, :, None] * self._dpsi2_dmu).reshape(mu.shape[0],-1,mu.shape[1]).sum(axis=1)
+        target_S += (dL_dpsi2[:, :, :, None] * self._dpsi2_dS).reshape(S.shape[0],-1,S.shape[1]).sum(axis=1)
+        target_gamma += (dL_dpsi2[:,:,:, None] *self._dpsi2_dgamma).reshape(gamma.shape[0],-1,gamma.shape[1]).sum(axis=1)
+
+    def update_gradients_variational(self, dL_dKmm, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, posterior_variational):
+        self._psi_computations(Z, posterior_variational.mean, posterior_variational.variance, posterior_variational.binary_prob)
+
+        #contributions from psi0:
+        self.variance.gradient = np.sum(dL_dpsi0)
+
+        #from psi1
+        self.variance.gradient += np.sum(dL_dpsi1 * self._dpsi1_dvariance)
+        self.lengthscale.gradient = (dL_dpsi1[:,:,None]*self._dpsi1_dlengthscale).reshape(-1,self.input_dim).sum(axis=0) 
+    
+
+        #from psi2
+        self.variance.gradient += (dL_dpsi2 * self._dpsi2_dvariance).sum()
+        self.lengthscale.gradient += (dL_dpsi2[:,:,:,None] * self._dpsi2_dlengthscale).reshape(-1,self.input_dim).sum(axis=0)
+
+        #from Kmm
+        self._K_computations(Z, None)
+        dvardLdK = self._K_dvar * dL_dKmm
+        var_len3 = self.variance / (np.square(self.lengthscale)*self.lengthscale)
+
+        self.variance.gradient += np.sum(dvardLdK)
+        self.lengthscale.gradient += (np.square(Z[:,None,:]-Z[None,:,:])*dvardLdK[:,:,None]).reshape(-1,self.input_dim).sum(axis=0)*var_len3
+        
+        
+    def gradients_Z_variational(self, dL_dKmm, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, posterior_variational):
+        self._psi_computations(Z, posterior_variational.mean, posterior_variational.variance, posterior_variational.binary_prob)
+
+        #psi1
+        grad = (dL_dpsi1[:, :, None] * self._dpsi1_dZ).sum(axis=0)
+
+        #psi2
+        grad += (dL_dpsi2[:, :, :, None] * self._dpsi2_dZ).sum(axis=0).sum(axis=1)
+
+        grad += self.gradients_X(dL_dKmm, Z, None)
+
+        return grad
+
+    def gradients_q_variational(self, dL_dKmm, dL_dpsi0, dL_dpsi1, dL_dpsi2, Z, posterior_variational):
+        ndata = posterior_variational.mean.shape[0]
+        self._psi_computations(Z, posterior_variational.mean, posterior_variational.variance, posterior_variational.binary_prob)
+        #psi1
+        grad_mu = (dL_dpsi1[:, :, None] * self._dpsi1_dmu).sum(axis=1)
+        grad_S = (dL_dpsi1[:, :, None] * self._dpsi1_dS).sum(axis=1)
+        grad_gamma = (dL_dpsi1[:,:,None] * self._dpsi1_dgamma).sum(axis=1)
+        #psi2
+        grad_mu += (dL_dpsi2[:, :, :, None] * self._dpsi2_dmu).reshape(ndata,-1,self.input_dim).sum(axis=1)
+        grad_S += (dL_dpsi2[:, :, :, None] * self._dpsi2_dS).reshape(ndata,-1,self.input_dim).sum(axis=1)
+        grad_gamma += (dL_dpsi2[:,:,:, None] *self._dpsi2_dgamma).reshape(ndata,-1,self.input_dim).sum(axis=1)
+        
+        return grad_mu, grad_S, grad_gamma
+
+    def gradients_X(self, dL_dK, X, X2=None):
+        #if self._X is None or X.base is not self._X.base or X2 is not None:
+        if X2==None:
+            _K_dist = X[:,None,:] - X[None,:,:]
+            _K_dist2 = np.square(_K_dist/self.lengthscale).sum(axis=-1)
+            dK_dX = self.variance*np.exp(-0.5 * self._K_dist2[:,:,None]) * (-2.*_K_dist/np.square(self.lengthscale))
+            dL_dX = (dL_dK[:,:,None] * dK_dX).sum(axis=1)
+        else:
+            _K_dist = X[:,None,:] - X2[None,:,:]
+            _K_dist2 = np.square(_K_dist/self.lengthscale).sum(axis=-1)
+            dK_dX = self.variance*np.exp(-0.5 * self._K_dist2[:,:,None]) * (-_K_dist/np.square(self.lengthscale))
+            dL_dX = (dL_dK[:,:,None] * dK_dX).sum(axis=1)
+        return dL_dX
+        
+    #---------------------------------------#
+    #            Precomputations            #
+    #---------------------------------------#
+
+    @cache_this(1)
+    def _K_computations(self, X, X2):
+        """
+        K(X,X2) - X is NxQ 
+        Q -> input dimension (self.input_dim)
+        """
+        if X2 is None:
+            self._X2 = None
+                            
+            X = X / self.lengthscale
+            Xsquare = np.sum(np.square(X), axis=1)
+            self._K_dist2 = -2.*tdot(X) + (Xsquare[:, None] + Xsquare[None, :])
+        else:
+            self._X2 = X2.copy()
+            
+            X = X / self.lengthscale
+            X2 = X2 / self.lengthscale
+            self._K_dist2 = -2.*np.dot(X, X2.T) + (np.sum(np.square(X), axis=1)[:, None] + np.sum(np.square(X2), axis=1)[None, :])
+        self._K_dvar = np.exp(-0.5 * self._K_dist2)
+
+    @cache_this(1)
+    def _psi_computations(self, Z, mu, S, gamma):
+        """
+        Z - MxQ
+        mu - NxQ
+        S - NxQ
+        gamma - NxQ
+        """
+        # here are the "statistics" for psi1 and psi2
+        # Produced intermediate results:
+        # _psi1                NxM
+        # _dpsi1_dvariance     NxM
+        # _dpsi1_dlengthscale  NxMxQ
+        # _dpsi1_dZ            NxMxQ
+        # _dpsi1_dgamma        NxMxQ
+        # _dpsi1_dmu           NxMxQ
+        # _dpsi1_dS            NxMxQ
+        # _psi2                NxMxM
+        # _psi2_dvariance      NxMxM
+        # _psi2_dlengthscale   NxMxMxQ
+        # _psi2_dZ             NxMxMxQ
+        # _psi2_dgamma         NxMxMxQ
+        # _psi2_dmu            NxMxMxQ
+        # _psi2_dS             NxMxMxQ
+        
+        lengthscale2 = np.square(self.lengthscale)
+                    
+        _psi2_Zhat = 0.5 * (Z[:, None, :] + Z[None, :, :]) # M,M,Q
+        _psi2_Zdist = 0.5 * (Z[:, None, :] - Z[None, :, :]) # M,M,Q
+        _psi2_Zdist_sq = np.square(_psi2_Zdist / self.lengthscale) # M,M,Q
+        _psi2_Z_sq_sum = (np.square(Z[:,None,:])+np.square(Z[None,:,:]))/lengthscale2 # MxMxQ
+
+        # psi1
+        _psi1_denom = S[:, None, :] / lengthscale2 + 1.  # Nx1xQ
+        _psi1_denom_sqrt = np.sqrt(_psi1_denom) #Nx1xQ
+        _psi1_dist = Z[None, :, :] - mu[:, None, :]  # NxMxQ
+        _psi1_dist_sq = np.square(_psi1_dist) / (lengthscale2 * _psi1_denom) # NxMxQ
+        _psi1_common = gamma[:,None,:] / (lengthscale2*_psi1_denom*_psi1_denom_sqrt) #Nx1xQ
+        _psi1_exponent1 = np.log(gamma[:,None,:]) -0.5 * (_psi1_dist_sq + np.log(_psi1_denom)) # NxMxQ
+        _psi1_exponent2 = np.log(1.-gamma[:,None,:]) -0.5 * (np.square(Z[None,:,:])/lengthscale2) # NxMxQ
+        _psi1_exponent = np.log(np.exp(_psi1_exponent1) + np.exp(_psi1_exponent2)) #NxMxQ
+        _psi1_exp_sum = _psi1_exponent.sum(axis=-1) #NxM
+        _psi1_exp_dist_sq = np.exp(-0.5*_psi1_dist_sq) # NxMxQ
+        _psi1_exp_Z = np.exp(-0.5*np.square(Z[None,:,:])/lengthscale2) # 1xMxQ
+        _psi1_q = self.variance * np.exp(_psi1_exp_sum[:,:,None] - _psi1_exponent) # NxMxQ
+        self._psi1 = self.variance * np.exp(_psi1_exp_sum) # NxM
+        self._dpsi1_dvariance = self._psi1 / self.variance # NxM
+        self._dpsi1_dgamma = _psi1_q * (_psi1_exp_dist_sq/_psi1_denom_sqrt-_psi1_exp_Z) # NxMxQ
+        self._dpsi1_dmu = _psi1_q * (_psi1_exp_dist_sq * _psi1_dist * _psi1_common) # NxMxQ
+        self._dpsi1_dS = _psi1_q * (_psi1_exp_dist_sq * _psi1_common * 0.5 * (_psi1_dist_sq - 1.)) # NxMxQ
+        self._dpsi1_dZ = _psi1_q * (- _psi1_common * _psi1_dist * _psi1_exp_dist_sq - (1-gamma[:,None,:])/lengthscale2*Z[None,:,:]*_psi1_exp_Z) # NxMxQ
+        self._dpsi1_dlengthscale = 2.*self.lengthscale*_psi1_q * (0.5*_psi1_common*(S[:,None,:]/lengthscale2+_psi1_dist_sq)*_psi1_exp_dist_sq + 0.5*(1-gamma[:,None,:])*np.square(Z[None,:,:]/lengthscale2)*_psi1_exp_Z) # NxMxQ
+
+
+        # psi2
+        _psi2_denom = 2.*S[:, None, None, :] / lengthscale2 + 1. # Nx1x1xQ
+        _psi2_denom_sqrt = np.sqrt(_psi2_denom)
+        _psi2_mudist = mu[:,None,None,:]-_psi2_Zhat #N,M,M,Q
+        _psi2_mudist_sq = np.square(_psi2_mudist)/(lengthscale2*_psi2_denom)
+        _psi2_common = gamma[:,None,None,:]/(lengthscale2 * _psi2_denom * _psi2_denom_sqrt) # Nx1x1xQ
+        _psi2_exponent1 = -_psi2_Zdist_sq -_psi2_mudist_sq -0.5*np.log(_psi2_denom)+np.log(gamma[:,None,None,:]) #N,M,M,Q
+        _psi2_exponent2 = np.log(1.-gamma[:,None,None,:]) - 0.5*(_psi2_Z_sq_sum) # NxMxMxQ
+        _psi2_exponent = np.log(np.exp(_psi2_exponent1) + np.exp(_psi2_exponent2))
+        _psi2_exp_sum = _psi2_exponent.sum(axis=-1) #NxM
+        _psi2_q = np.square(self.variance) * np.exp(_psi2_exp_sum[:,:,:,None]-_psi2_exponent) # NxMxMxQ 
+        _psi2_exp_dist_sq = np.exp(-_psi2_Zdist_sq -_psi2_mudist_sq) # NxMxMxQ
+        _psi2_exp_Z = np.exp(-0.5*_psi2_Z_sq_sum) # MxMxQ
+        self._psi2 = np.square(self.variance) * np.exp(_psi2_exp_sum) # N,M,M
+        self._dpsi2_dvariance = 2. * self._psi2/self.variance # NxMxM
+        self._dpsi2_dgamma = _psi2_q * (_psi2_exp_dist_sq/_psi2_denom_sqrt - _psi2_exp_Z) # NxMxMxQ
+        self._dpsi2_dmu = _psi2_q * (-2.*_psi2_common*_psi2_mudist * _psi2_exp_dist_sq) # NxMxMxQ
+        self._dpsi2_dS = _psi2_q * (_psi2_common * (2.*_psi2_mudist_sq - 1.) * _psi2_exp_dist_sq) # NxMxMxQ
+        self._dpsi2_dZ = 2.*_psi2_q * (_psi2_common*(-_psi2_Zdist*_psi2_denom+_psi2_mudist)*_psi2_exp_dist_sq - (1-gamma[:,None,None,:])*Z[:,None,:]/lengthscale2*_psi2_exp_Z) # NxMxMxQ
+        self._dpsi2_dlengthscale = 2.*self.lengthscale* _psi2_q * (_psi2_common*(S[:,None,None,:]/lengthscale2+_psi2_Zdist_sq*_psi2_denom+_psi2_mudist_sq)*_psi2_exp_dist_sq+(1-gamma[:,None,None,:])*_psi2_Z_sq_sum*0.5/lengthscale2*_psi2_exp_Z) # NxMxMxQ
+        

GPy/models/__init__.py

@@ -15,3 +15,4 @@ from mrd import MRD
 from gradient_checker import GradientChecker
 from gp_multioutput_regression import GPMultioutputRegression
 from sparse_gp_multioutput_regression import SparseGPMultioutputRegression
+from ss_gplvm import SSGPLVM

GPy/models/ss_gplvm.py

@@ -0,0 +1,301 @@
+# Copyright (c) 2012, GPy authors (see AUTHORS.txt).
+# Licensed under the BSD 3-clause license (see LICENSE.txt)
+
+import numpy as np
+import itertools
+from matplotlib import pyplot
+
+from ..core.sparse_gp import SparseGP
+from .. import kern
+from ..likelihoods import Gaussian
+from ..inference.optimization import SCG
+from ..util import linalg
+from ..core.parameterization.variational import SpikeAndSlabPrior, SpikeAndSlabPosterior
+
+class SSGPLVM(SparseGP):
+    """
+    Spike-and-Slab Gaussian Process Latent Variable Model
+
+    :param Y: observed data (np.ndarray) or GPy.likelihood
+    :type Y: np.ndarray| GPy.likelihood instance
+    :param input_dim: latent dimensionality
+    :type input_dim: int
+    :param init: initialisation method for the latent space
+    :type init: 'PCA'|'random'
+
+    """
+    def __init__(self, Y, input_dim, X=None, X_variance=None, init='PCA', num_inducing=10,
+                 Z=None, kernel=None, inference_method=None, likelihood=None, name='Spike-and-Slab GPLVM', **kwargs):
+
+        if X == None: # The mean of variational approximation (mu)
+            from ..util.initialization import initialize_latent
+            X = initialize_latent(init, input_dim, Y)
+        self.init = init
+
+        if X_variance is None: # The variance of the variational approximation (S)
+            X_variance = np.random.uniform(0,.1,X.shape)
+            
+        gamma = np.empty_like(X) # The posterior probabilities of the binary variable in the variational approximation
+        gamma[:] = 0.5
+        
+        if Z is None:
+            Z = np.random.permutation(X.copy())[:num_inducing]
+        assert Z.shape[1] == X.shape[1]
+
+        if likelihood is None:
+            likelihood = Gaussian()
+
+        if kernel is None:
+            kernel = kern.SSRBF(input_dim)
+            
+        self.variational_prior = SpikeAndSlabPrior(pi=0.5) # the prior probability of the latent binary variable b
+        X = SpikeAndSlabPosterior(X, X_variance, gamma)
+
+        SparseGP.__init__(self, X, Y, Z, kernel, likelihood, inference_method, name, **kwargs)
+        self.add_parameter(self.X, index=0)
+
+    def parameters_changed(self):
+        super(SSGPLVM, self).parameters_changed()
+        self._log_marginal_likelihood -= self.variational_prior.KL_divergence(self.X)
+
+        self.X.mean.gradient, self.X.variance.gradient, self.X.binary_prob.gradient = self.kern.gradients_q_variational(posterior_variational=self.X, Z=self.Z, **self.grad_dict)
+
+        # update for the KL divergence
+        self.variational_prior.update_gradients_KL(self.X)
+
+    def plot_latent(self, plot_inducing=True, *args, **kwargs):
+        pass
+        #return plot_latent.plot_latent(self, plot_inducing=plot_inducing, *args, **kwargs)
+
+    def do_test_latents(self, Y):
+        """
+        Compute the latent representation for a set of new points Y
+
+        Notes:
+        This will only work with a univariate Gaussian likelihood (for now)
+        """
+        assert not self.likelihood.is_heteroscedastic
+        N_test = Y.shape[0]
+        input_dim = self.Z.shape[1]
+        means = np.zeros((N_test, input_dim))
+        covars = np.zeros((N_test, input_dim))
+
+        dpsi0 = -0.5 * self.output_dim * self.likelihood.precision
+        dpsi2 = self.dL_dpsi2[0][None, :, :] # TODO: this may change if we ignore het. likelihoods
+        V = self.likelihood.precision * Y
+
+        #compute CPsi1V
+        if self.Cpsi1V is None:
+            psi1V = np.dot(self.psi1.T, self.likelihood.V)
+            tmp, _ = linalg.dtrtrs(self._Lm, np.asfortranarray(psi1V), lower=1, trans=0)
+            tmp, _ = linalg.dpotrs(self.LB, tmp, lower=1)
+            self.Cpsi1V, _ = linalg.dtrtrs(self._Lm, tmp, lower=1, trans=1)
+
+        dpsi1 = np.dot(self.Cpsi1V, V.T)
+
+        start = np.zeros(self.input_dim * 2)
+
+        for n, dpsi1_n in enumerate(dpsi1.T[:, :, None]):
+            args = (self.kern, self.Z, dpsi0, dpsi1_n.T, dpsi2)
+            xopt, fopt, neval, status = SCG(f=latent_cost, gradf=latent_grad, x=start, optargs=args, display=False)
+
+            mu, log_S = xopt.reshape(2, 1, -1)
+            means[n] = mu[0].copy()
+            covars[n] = np.exp(log_S[0]).copy()
+
+        return means, covars
+
+    def dmu_dX(self, Xnew):
+        """
+        Calculate the gradient of the prediction at Xnew w.r.t Xnew.
+        """
+        dmu_dX = np.zeros_like(Xnew)
+        for i in range(self.Z.shape[0]):
+            dmu_dX += self.kern.dK_dX(self.Cpsi1Vf[i:i + 1, :], Xnew, self.Z[i:i + 1, :])
+        return dmu_dX
+
+    def dmu_dXnew(self, Xnew):
+        """
+        Individual gradient of prediction at Xnew w.r.t. each sample in Xnew
+        """
+        dK_dX = np.zeros((Xnew.shape[0], self.num_inducing))
+        ones = np.ones((1, 1))
+        for i in range(self.Z.shape[0]):
+            dK_dX[:, i] = self.kern.dK_dX(ones, Xnew, self.Z[i:i + 1, :]).sum(-1)
+        return np.dot(dK_dX, self.Cpsi1Vf)
+
+    def plot_steepest_gradient_map(self, fignum=None, ax=None, which_indices=None, labels=None, data_labels=None, data_marker='o', data_s=40, resolution=20, aspect='auto', updates=False, ** kwargs):
+        input_1, input_2 = significant_dims = most_significant_input_dimensions(self, which_indices)
+
+        X = np.zeros((resolution ** 2, self.input_dim))
+        indices = np.r_[:X.shape[0]]
+        if labels is None:
+            labels = range(self.output_dim)
+
+        def plot_function(x):
+            X[:, significant_dims] = x
+            dmu_dX = self.dmu_dXnew(X)
+            argmax = np.argmax(dmu_dX, 1)
+            return dmu_dX[indices, argmax], np.array(labels)[argmax]
+
+        if ax is None:
+            fig = pyplot.figure(num=fignum)
+            ax = fig.add_subplot(111)
+
+        if data_labels is None:
+            data_labels = np.ones(self.num_data)
+        ulabels = []
+        for lab in data_labels:
+            if not lab in ulabels:
+                ulabels.append(lab)
+        marker = itertools.cycle(list(data_marker))
+        from GPy.util import Tango
+        for i, ul in enumerate(ulabels):
+            if type(ul) is np.string_:
+                this_label = ul
+            elif type(ul) is np.int64:
+                this_label = 'class %i' % ul
+            else:
+                this_label = 'class %i' % i
+            m = marker.next()
+            index = np.nonzero(data_labels == ul)[0]
+            x = self.X[index, input_1]
+            y = self.X[index, input_2]
+            ax.scatter(x, y, marker=m, s=data_s, color=Tango.nextMedium(), label=this_label)
+
+        ax.set_xlabel('latent dimension %i' % input_1)
+        ax.set_ylabel('latent dimension %i' % input_2)
+
+        from matplotlib.cm import get_cmap
+        from GPy.util.latent_space_visualizations.controllers.imshow_controller import ImAnnotateController
+        if not 'cmap' in kwargs.keys():
+            kwargs.update(cmap=get_cmap('jet'))
+        controller = ImAnnotateController(ax,
+                                      plot_function,
+                                      tuple(self.X.min(0)[:, significant_dims]) + tuple(self.X.max(0)[:, significant_dims]),
+                                      resolution=resolution,
+                                      aspect=aspect,
+                                      **kwargs)
+        ax.legend()
+        ax.figure.tight_layout()
+        if updates:
+            pyplot.show()
+            clear = raw_input('Enter to continue')
+            if clear.lower() in 'yes' or clear == '':
+                controller.deactivate()
+        return controller.view
+
+    def plot_X_1d(self, fignum=None, ax=None, colors=None):
+        """
+        Plot latent space X in 1D:
+
+            - if fig is given, create input_dim subplots in fig and plot in these
+            - if ax is given plot input_dim 1D latent space plots of X into each `axis`
+            - if neither fig nor ax is given create a figure with fignum and plot in there
+
+        colors:
+            colors of different latent space dimensions input_dim
+
+        """
+        import pylab
+        if ax is None:
+            fig = pylab.figure(num=fignum, figsize=(8, min(12, (2 * self.X.shape[1]))))
+        if colors is None:
+            colors = pylab.gca()._get_lines.color_cycle
+            pylab.clf()
+        else:
+            colors = iter(colors)
+        plots = []
+        x = np.arange(self.X.shape[0])
+        for i in range(self.X.shape[1]):
+            if ax is None:
+                a = fig.add_subplot(self.X.shape[1], 1, i + 1)
+            elif isinstance(ax, (tuple, list)):
+                a = ax[i]
+            else:
+                raise ValueError("Need one ax per latent dimnesion input_dim")
+            a.plot(self.X, c='k', alpha=.3)
+            plots.extend(a.plot(x, self.X.T[i], c=colors.next(), label=r"$\mathbf{{X_{{{}}}}}$".format(i)))
+            a.fill_between(x,
+                            self.X.T[i] - 2 * np.sqrt(self.X_variance.T[i]),
+                            self.X.T[i] + 2 * np.sqrt(self.X_variance.T[i]),
+                            facecolor=plots[-1].get_color(),
+                            alpha=.3)
+            a.legend(borderaxespad=0.)
+            a.set_xlim(x.min(), x.max())
+            if i < self.X.shape[1] - 1:
+                a.set_xticklabels('')
+        pylab.draw()
+        if ax is None:
+            fig.tight_layout(h_pad=.01) # , rect=(0, 0, 1, .95))
+        return fig
+
+    def getstate(self):
+        """
+        Get the current state of the class,
+        here just all the indices, rest can get recomputed
+        """
+        return SparseGP._getstate(self) + [self.init]
+
+    def setstate(self, state):
+        self._const_jitter = None
+        self.init = state.pop()
+        SparseGP._setstate(self, state)
+
+
+def latent_cost_and_grad(mu_S, kern, Z, dL_dpsi0, dL_dpsi1, dL_dpsi2):
+    """
+    objective function for fitting the latent variables for test points
+    (negative log-likelihood: should be minimised!)
+    """
+    mu, log_S = mu_S.reshape(2, 1, -1)
+    S = np.exp(log_S)
+
+    psi0 = kern.psi0(Z, mu, S)
+    psi1 = kern.psi1(Z, mu, S)
+    psi2 = kern.psi2(Z, mu, S)
+
+    lik = dL_dpsi0 * psi0 + np.dot(dL_dpsi1.flatten(), psi1.flatten()) + np.dot(dL_dpsi2.flatten(), psi2.flatten()) - 0.5 * np.sum(np.square(mu) + S) + 0.5 * np.sum(log_S)
+
+    mu0, S0 = kern.dpsi0_dmuS(dL_dpsi0, Z, mu, S)
+    mu1, S1 = kern.dpsi1_dmuS(dL_dpsi1, Z, mu, S)
+    mu2, S2 = kern.dpsi2_dmuS(dL_dpsi2, Z, mu, S)
+
+    dmu = mu0 + mu1 + mu2 - mu
+    # dS = S0 + S1 + S2 -0.5 + .5/S
+    dlnS = S * (S0 + S1 + S2 - 0.5) + .5
+    return -lik, -np.hstack((dmu.flatten(), dlnS.flatten()))
+
+def latent_cost(mu_S, kern, Z, dL_dpsi0, dL_dpsi1, dL_dpsi2):
+    """
+    objective function for fitting the latent variables (negative log-likelihood: should be minimised!)
+    This is the same as latent_cost_and_grad but only for the objective
+    """
+    mu, log_S = mu_S.reshape(2, 1, -1)
+    S = np.exp(log_S)
+
+    psi0 = kern.psi0(Z, mu, S)
+    psi1 = kern.psi1(Z, mu, S)
+    psi2 = kern.psi2(Z, mu, S)
+
+    lik = dL_dpsi0 * psi0 + np.dot(dL_dpsi1.flatten(), psi1.flatten()) + np.dot(dL_dpsi2.flatten(), psi2.flatten()) - 0.5 * np.sum(np.square(mu) + S) + 0.5 * np.sum(log_S)
+    return -float(lik)
+
+def latent_grad(mu_S, kern, Z, dL_dpsi0, dL_dpsi1, dL_dpsi2):
+    """
+    This is the same as latent_cost_and_grad but only for the grad
+    """
+    mu, log_S = mu_S.reshape(2, 1, -1)
+    S = np.exp(log_S)
+
+    mu0, S0 = kern.dpsi0_dmuS(dL_dpsi0, Z, mu, S)
+    mu1, S1 = kern.dpsi1_dmuS(dL_dpsi1, Z, mu, S)
+    mu2, S2 = kern.dpsi2_dmuS(dL_dpsi2, Z, mu, S)
+
+    dmu = mu0 + mu1 + mu2 - mu
+    # dS = S0 + S1 + S2 -0.5 + .5/S
+    dlnS = S * (S0 + S1 + S2 - 0.5) + .5
+
+    return -np.hstack((dmu.flatten(), dlnS.flatten()))
+
+