Added pep.py -- Sparse Gaussian processes using Power Expectation Propagation

This allows interpolation between FITC (EP or alpha = 1), and Titsias's variational (VarDTC, VFE when alpha = 0).

Reference: A Unifying Framework for Sparse Gaussian Process Approximation using Power Expectation Propagation https://arxiv.org/abs/1605.07066

GPy/inference/latent_function_inference/pep.py

@@ -0,0 +1,93 @@
+from .posterior import Posterior
+from ...util.linalg import jitchol, tdot, dtrtrs, dtrtri, pdinv
+from ...util import diag
+import numpy as np
+from . import LatentFunctionInference
+log_2_pi = np.log(2*np.pi)
+
+class PEP(LatentFunctionInference):
+    '''
+    Sparse Gaussian processes using Power-Expectation Propagation
+    for regression: alpha \approx 0 gives VarDTC and alpha = 1 gives FITC
+    
+    Reference: A Unifying Framework for Sparse Gaussian Process Approximation using 
+    Power Expectation Propagation, https://arxiv.org/abs/1605.07066
+    
+    '''
+    const_jitter = 1e-6
+    
+    def __init__(self, alpha):
+        super(PEP, self).__init__()
+        self.alpha = alpha
+
+    def inference(self, kern, X, Z, likelihood, Y, mean_function=None, Y_metadata=None):
+        assert mean_function is None, "inference with a mean function not implemented"
+
+        num_inducing, _ = Z.shape
+        num_data, output_dim = Y.shape
+
+        #make sure the noise is not hetero
+        sigma_n = likelihood.gaussian_variance(Y_metadata)
+        if sigma_n.size >1:
+            raise NotImplementedError("no hetero noise with this implementation of PEP")
+
+        Kmm = kern.K(Z)
+        Knn = kern.Kdiag(X)
+        Knm = kern.K(X, Z)
+        U = Knm
+
+        #factor Kmm
+        diag.add(Kmm, self.const_jitter)
+        Kmmi, L, Li, _ = pdinv(Kmm)
+
+        #compute beta_star, the effective noise precision
+        LiUT = np.dot(Li, U.T)
+        sigma_star = sigma_n + self.alpha * (Knn - np.sum(np.square(LiUT),0))
+        beta_star = 1./sigma_star
+
+        # Compute and factor A
+        A = tdot(LiUT*np.sqrt(beta_star)) + np.eye(num_inducing)
+        LA = jitchol(A)
+
+        # back substitute to get b, P, v
+        URiy = np.dot(U.T*beta_star,Y)
+        tmp, _ = dtrtrs(L, URiy, lower=1)
+        b, _ = dtrtrs(LA, tmp, lower=1)
+        tmp, _ = dtrtrs(LA, b, lower=1, trans=1)
+        v, _ = dtrtrs(L, tmp, lower=1, trans=1)
+        tmp, _ = dtrtrs(LA, Li, lower=1, trans=0)
+        P = tdot(tmp.T)
+
+        alpha_const_term = (1.0-self.alpha) / self.alpha
+
+        #compute log marginal
+        log_marginal = -0.5*num_data*output_dim*np.log(2*np.pi) + \
+                       -np.sum(np.log(np.diag(LA)))*output_dim + \
+                       0.5*output_dim*(1+alpha_const_term)*np.sum(np.log(beta_star)) + \
+                       -0.5*np.sum(np.square(Y.T*np.sqrt(beta_star))) + \
+                       0.5*np.sum(np.square(b)) + 0.5*alpha_const_term*num_data*np.log(sigma_n)
+        #compute dL_dR
+        Uv = np.dot(U, v)
+        dL_dR = 0.5*(np.sum(U*np.dot(U,P), 1) - (1.0+alpha_const_term)/beta_star + np.sum(np.square(Y), 1) - 2.*np.sum(Uv*Y, 1) \
+            + np.sum(np.square(Uv), 1))*beta_star**2 
+
+        # Compute dL_dKmm
+        vvT_P = tdot(v.reshape(-1,1)) + P
+        dL_dK = 0.5*(Kmmi - vvT_P)
+        KiU = np.dot(Kmmi, U.T)
+        dL_dK += self.alpha * np.dot(KiU*dL_dR, KiU.T)
+
+        # Compute dL_dU
+        vY = np.dot(v.reshape(-1,1),Y.T)
+        dL_dU = vY - np.dot(vvT_P, U.T)
+        dL_dU *= beta_star
+        dL_dU -= self.alpha * 2.*KiU*dL_dR
+
+        dL_dthetaL = likelihood.exact_inference_gradients(dL_dR)
+        dL_dthetaL += 0.5*alpha_const_term*num_data / sigma_n
+        grad_dict = {'dL_dKmm': dL_dK, 'dL_dKdiag':dL_dR * self.alpha, 'dL_dKnm':dL_dU.T, 'dL_dthetaL':dL_dthetaL}
+
+        #construct a posterior object
+        post = Posterior(woodbury_inv=Kmmi-P, woodbury_vector=v, K=Kmm, mean=None, cov=None, K_chol=L)
+
+        return post, log_marginal, grad_dict