initial messing with svgp to diagonalize

GPy/inference/latent_function_inference/svgp.py

@@ -7,115 +7,96 @@ from scipy.linalg.blas import dgemm, dsymm, dtrmm
 
 class SVGP(LatentFunctionInference):
 
-    def inference(self, q_u_mean, q_u_chol, kern, X, Z, likelihood, Y, mean_function=None, Y_metadata=None, KL_scale=1.0, batch_scale=1.0):
+    def inference(self, q_v_mean, q_v_chol, kern, X, Z, likelihood, Y, mean_function=None, Y_metadata=None, KL_scale=1.0, batch_scale=1.0):
+
+        if mean_function is not None:
+            raise NotImplementedError
 
         num_data, _ = Y.shape
-        num_inducing, num_outputs = q_u_mean.shape
+        num_inducing, num_outputs = q_v_mean.shape
 
         #expand cholesky representation
-        L = choleskies.flat_to_triang(q_u_chol)
+        Lv = choleskies.flat_to_triang(q_v_chol)
 
+        #deal with posterior copvariance
+        Sv = np.zeros((num_outputs, num_inducing, num_inducing))
+        for i in range(num_outputs):
+            Sv[i] = Lv[i].dot(Lv[i].T)
+        logdetS = np.array([2.*np.sum(np.log(np.abs(np.diag(Lv[i,:,:])))) for i in range(Lv.shape[0])])
+        traceS = np.array([np.sum(np.square(np.diag(Lv[i,:,:]))) for i in range(Lv.shape[0])])
 
-        S = np.empty((num_outputs, num_inducing, num_inducing))
-        [np.dot(L[i,:,:], L[i,:,:].T, S[i,:,:]) for i in range(num_outputs)]
-        #Si,_ = linalg.dpotri(np.asfortranarray(L), lower=1)
-        Si = choleskies.multiple_dpotri(L)
-        logdetS = np.array([2.*np.sum(np.log(np.abs(np.diag(L[i,:,:])))) for i in range(L.shape[0])])
-
-        if np.any(np.isinf(Si)):
-            raise ValueError("Cholesky representation unstable")
-
-        #compute mean function stuff
-        if mean_function is not None:
-            prior_mean_u = mean_function.f(Z)
-            prior_mean_f = mean_function.f(X)
-        else:
-            prior_mean_u = np.zeros((num_inducing, num_outputs))
-            prior_mean_f = np.zeros((num_data, num_outputs))
 
         #compute kernel related stuff
         Kmm = kern.K(Z)
         Kmn = kern.K(Z, X)
         Knn_diag = kern.Kdiag(X)
-        Lm = linalg.jitchol(Kmm)
+        R = linalg.jitchol(Kmm)
-        logdetKmm = 2.*np.sum(np.log(np.diag(Lm)))
-        Kmmi, _ = linalg.dpotri(Lm)
 
         #compute the marginal means and variances of q(f)
-        A, _ = linalg.dpotrs(Lm, Kmn)
+        AT, _ = linalg.dtrtrs(R, Kmn)
-        mu = prior_mean_f + np.dot(A.T, q_u_mean - prior_mean_u)
+        A = AT.T
-        v = np.empty((num_data, num_outputs))
+        mu = np.dot(A, q_v_mean)
+        var = np.empty((num_data, num_outputs))
         for i in range(num_outputs):
-            tmp = dtrmm(1.0,L[i].T, A, lower=0, trans_a=0)
+            tmp = dtrmm(1.0,Lv[i].T, A.T, lower=0, trans_a=0)
-            v[:,i] = np.sum(np.square(tmp),0)
+            var[:,i] = np.sum(np.square(tmp),0)
-        v += (Knn_diag - np.sum(A*Kmn,0))[:,None]
+        var += (Knn_diag - np.sum(np.square(A),1))[:,None]
 
         #compute the KL term
-        Kmmim = np.dot(Kmmi, q_u_mean)
+        KL = -0.5*logdetS.sum() + 0.5*np.sum(np.square(q_v_mean)) + 0.5*traceS.sum()
-        KLs = -0.5*logdetS -0.5*num_inducing + 0.5*logdetKmm + 0.5*np.sum(Kmmi[None,:,:]*S,1).sum(1) + 0.5*np.sum(q_u_mean*Kmmim,0)
+        dL_dmv = q_v_mean*1
-        KL = KLs.sum()
+        dL_dL = np.zeros_like(Lv)
-        #gradient of the KL term (assuming zero mean function)
+        for k in range(num_outputs):
-        dKL_dm = Kmmim.copy()
+            Lii = np.diagonal(Lv[i])
-        dKL_dS = 0.5*(Kmmi[None,:,:] - Si)
+            diag = np.diagonal(dL_dL[i])
-        dKL_dKmm = 0.5*num_outputs*Kmmi - 0.5*Kmmi.dot(S.sum(0)).dot(Kmmi) - 0.5*Kmmim.dot(Kmmim.T)
+            diag = Lii - 1./Lii # write in place, need numpy 1.9+
-
-        if mean_function is not None:
-            #adjust KL term for mean function
-            Kmmi_mfZ = np.dot(Kmmi, prior_mean_u)
-            KL += -np.sum(q_u_mean*Kmmi_mfZ)
-            KL += 0.5*np.sum(Kmmi_mfZ*prior_mean_u)
-
-            #adjust gradient for mean fucntion
-            dKL_dm -= Kmmi_mfZ
-            dKL_dKmm += Kmmim.dot(Kmmi_mfZ.T)
-            dKL_dKmm -= 0.5*Kmmi_mfZ.dot(Kmmi_mfZ.T)
-
-            #compute gradients for mean_function
-            dKL_dmfZ = Kmmi_mfZ - Kmmim
 
         #quadrature for the likelihood
-        F, dF_dmu, dF_dv, dF_dthetaL = likelihood.variational_expectations(Y, mu, v, Y_metadata=Y_metadata)
+        F, dF_dmu, dF_dv, dF_dthetaL = likelihood.variational_expectations(Y, mu, var, Y_metadata=Y_metadata)
 
         #rescale the F term if working on a batch
         F, dF_dmu, dF_dv =  F*batch_scale, dF_dmu*batch_scale, dF_dv*batch_scale
         if dF_dthetaL is not None:
             dF_dthetaL =  dF_dthetaL.sum(1).sum(1)*batch_scale
 
-        #derivatives of expected likelihood, assuming zero mean function
+        #mv
-        Adv = A[None,:,:]*dF_dv.T[:,None,:] # As if dF_Dv is diagonal, D, M, N
+        dL_dmv += A.T.dot(dF_dmu)
-        Admu = A.dot(dF_dmu)
-        Adv = np.ascontiguousarray(Adv) # makes for faster operations later...(inc dsymm)
-        AdvA = np.dot(Adv.reshape(-1, num_data),A.T).reshape(num_outputs, num_inducing, num_inducing )
-        tmp = np.sum([np.dot(a,s) for a, s in zip(AdvA, S)],0).dot(Kmmi)
-        dF_dKmm = -Admu.dot(Kmmim.T) + AdvA.sum(0) - tmp - tmp.T
-        dF_dKmm = 0.5*(dF_dKmm + dF_dKmm.T) # necessary? GPy bug?
-        tmp = S.reshape(-1, num_inducing).dot(Kmmi).reshape(num_outputs, num_inducing , num_inducing )
-        tmp = 2.*(tmp - np.eye(num_inducing)[None, :,:])
 
-        dF_dKmn = Kmmim.dot(dF_dmu.T)
+        #Kfu
-        for a,b in zip(tmp, Adv):
+        RiTm, _ = linalg.dtrtrs(R, q_v_mean, lower=1, trans=1)
-            dF_dKmn += np.dot(a.T, b)
+        dL_dKmn = np.zeros((num_inducing, num_data))
+        for i in range(num_outputs):
+            tmp, _ = linalg.dtrtrs(R, np.eye(num_inducing)-Sv[i], trans=1, lower=1)
+            dL_dKmn += -2*np.dot(tmp, A.T*dF_dv[:,i])
+        dL_dKmn += np.dot(RiTm, dF_dmu.T)
 
-        dF_dm = Admu
+        #L
-        dF_dS = AdvA
+        for i in range(num_outputs):
+            dL_dL[i] += np.dot(Lv[i].T, A.T).dot(A*dF_dv[:,i][:,None])
 
-        #adjust gradient to account for mean function
+        #R
-        if mean_function is not None:
+        dL_dR = np.zeros((num_inducing, num_inducing))
-            dF_dmfX = dF_dmu.copy()
+        for i in range(num_outputs):
-            dF_dmfZ = -Admu
+            tmp = np.eye(num_inducing) - Sv[i]
-            dF_dKmn -= np.dot(Kmmi_mfZ, dF_dmu.T)
+            tmp = np.dot(tmp, A.T)
-            dF_dKmm += Admu.dot(Kmmi_mfZ.T)
+            tmp = np.dot(tmp, A*dF_dv[:,i][:,None])
+            tmp, _ = linalg.dtrtrs(R, tmp, trans=1, lower=1)
+            dL_dR += 2*tmp.T
+        dL_dR -= A.T.dot(dF_dmu).dot(RiTm.T)
+
+        #backprop dL_dR for dL_dKmm
+        dL_dKmm = choleskies.backprop_gradient(dL_dR, R)
 
 
         #sum (gradients of) expected likelihood and KL part
         log_marginal = F.sum() - KL
-        dL_dm, dL_dS, dL_dKmm, dL_dKmn = dF_dm - dKL_dm, dF_dS- dKL_dS, dF_dKmm- dKL_dKmm, dF_dKmn
 
-        dL_dchol = 2.*np.array([np.dot(a,b) for a, b in zip(dL_dS, L) ])
+        dL_dchol = choleskies.triang_to_flat(dL_dL)
-        dL_dchol = choleskies.triang_to_flat(dL_dchol)
 
-        grad_dict = {'dL_dKmm':dL_dKmm, 'dL_dKmn':dL_dKmn, 'dL_dKdiag': dF_dv.sum(1), 'dL_dm':dL_dm, 'dL_dchol':dL_dchol, 'dL_dthetaL':dF_dthetaL}
+        grad_dict = {'dL_dKmm':dL_dKmm, 'dL_dKmn':dL_dKmn, 'dL_dKdiag': dF_dv.sum(1), 'dL_dm':dL_dmv, 'dL_dchol':dL_dchol, 'dL_dthetaL':dF_dthetaL}
         if mean_function is not None:
             grad_dict['dL_dmfZ'] = dF_dmfZ - dKL_dmfZ
             grad_dict['dL_dmfX'] = dF_dmfX
-        return Posterior(mean=q_u_mean, cov=S.T, K=Kmm, prior_mean=prior_mean_u), log_marginal, grad_dict
+
+        q_u_mean = np.dot(R, q_v_mean)
+        return Posterior(mean=q_u_mean, cov=Sv.T, K=Kmm, prior_mean=0.), log_marginal, grad_dict