def _initialize_posterior_distribution(self, RecognitionParams):
# Now actually compute the precisions (from their square roots)
self.Lambda = T.batched_dot(self.LambdaChol, self.LambdaChol.dimshuffle(0,2,1))
# dynamics matrix & initialize the innovations precision, xDim x xDim
self.A = theano.shared(value=RecognitionParams['A'].astype(theano.config.floatX) ,name='A' )
self.QinvChol = theano.shared(value=RecognitionParams['QinvChol'].astype(theano.config.floatX) ,name='QinvChol' )
self.Q0invChol = theano.shared(value=RecognitionParams['Q0invChol'].astype(theano.config.floatX),name='Q0invChol')
self.Qinv = T.dot(self.QinvChol,self.QinvChol.T)
self.Q0inv = T.dot(self.Q0invChol,self.Q0invChol.T)
################## put together the total precision matrix ######################
AQinvA = T.dot(T.dot(self.A.T, self.Qinv), self.A)
# for now we (suboptimally) replicate a bunch of times
AQinvrep = Tsla.kron(T.ones([self.Tt-1,1,1]),-T.dot(self.A.T, self.Qinv)) # off-diagonal blocks (upper triangle)
AQinvArep = Tsla.kron(T.ones([self.Tt-2,1,1]), AQinvA+self.Qinv)
AQinvArepPlusQ = T.concatenate([T.shape_padleft(self.Q0inv + AQinvA), AQinvArep, T.shape_padleft(self.Qinv)])
# This is our inverse covariance matrix: diagonal (AA) and off-diagonal (BB) blocks.
self.AA = self.Lambda + AQinvArepPlusQ
self.BB = AQinvrep
# symbolic recipe for computing the the diagonal (V) and
# off-diagonal (VV) blocks of the posterior covariance
self.V, self.VV, self.S = compute_sym_blk_tridiag(self.AA, self.BB)
# now compute the posterior mean
LambdaMu = T.batched_dot(self.Lambda, self.Mu) # scale by precision (no need for transpose; lambda is symmetric)
#self.old_postX = compute_sym_blk_tridiag_inv_b(self.S,self.V,LambdaMu) # apply inverse
# compute cholesky decomposition
self.the_chol = blk_tridag_chol(self.AA, self.BB)
# intermediary (mult by R^T) -
ib = blk_chol_inv(self.the_chol[0], self.the_chol[1], LambdaMu)
# final result (mult by R)-
self.postX = blk_chol_inv(self.the_chol[0], self.the_chol[1], ib, lower=False, transpose=True)
# The determinant of the covariance is the square of the determinant of the cholesky factor.
# Determinant of the Cholesky factor is the product of the diagonal elements of the block-diagonal.
def comp_log_det(L):
return T.log(T.diag(L)).sum()
self.ln_determinant = -2*theano.scan(fn=comp_log_det, sequences=self.the_chol[0])[0].sum()
def test_numpy_2d(self):
for shp0 in [(2, 3)]:
x = tensor.tensor(dtype="floatX", broadcastable=(False,) * len(shp0))
a = numpy.asarray(self.rng.rand(*shp0)).astype(config.floatX)
for shp1 in [(6, 7)]:
if len(shp0) + len(shp1) == 2:
continue
y = tensor.tensor(dtype="floatX", broadcastable=(False,) * len(shp1))
f = function([x, y], kron(x, y))
b = self.rng.rand(*shp1).astype(config.floatX)
out = f(a, b)
assert numpy.allclose(out, numpy.kron(a, b))
def test_numpy_2d(self):
for shp0 in [(2, 3)]:
x = tensor.tensor(dtype="floatX", broadcastable=(False,) * len(shp0))
a = np.asarray(self.rng.rand(*shp0)).astype(config.floatX)
for shp1 in [(6, 7)]:
if len(shp0) + len(shp1) == 2:
continue
y = tensor.tensor(dtype="floatX", broadcastable=(False,) * len(shp1))
f = function([x, y], kron(x, y))
b = self.rng.rand(*shp1).astype(config.floatX)
out = f(a, b)
assert np.allclose(out, np.kron(a, b))
def test_perform(self):
if not imported_scipy:
raise SkipTest('kron tests need the scipy package to be installed')
for shp0 in [(2,), (2, 3), (2, 3, 4), (2, 3, 4, 5)]:
for shp1 in [(6,), (6, 7), (6, 7, 8), (6, 7, 8, 9)]:
if len(shp0) + len(shp1) == 2:
continue
x = tensor.tensor(dtype='floatX',
broadcastable=(False,) * len(shp0))
y = tensor.tensor(dtype='floatX',
broadcastable=(False,) * len(shp1))
f = function([x, y], kron(x, y))
a = numpy.asarray(self.rng.rand(*shp0)).astype(config.floatX)
b = self.rng.rand(*shp1).astype(config.floatX)
out = f(a, b)
assert numpy.allclose(out, scipy.linalg.kron(a, b))
def test_perform(self):
if not imported_scipy:
raise SkipTest('kron tests need the scipy package to be installed')
for shp0 in [(2, ), (2, 3), (2, 3, 4), (2, 3, 4, 5)]:
for shp1 in [(6, ), (6, 7), (6, 7, 8), (6, 7, 8, 9)]:
if len(shp0) + len(shp1) == 2:
continue
x = tensor.tensor(dtype='floatX',
broadcastable=(False, ) * len(shp0))
y = tensor.tensor(dtype='floatX',
broadcastable=(False, ) * len(shp1))
f = function([x, y], kron(x, y))
a = numpy.asarray(self.rng.rand(*shp0)).astype(config.floatX)
b = self.rng.rand(*shp1).astype(config.floatX)
out = f(a, b)
assert numpy.allclose(out, scipy.linalg.kron(a, b))
def test_perform(self):
scipy = pytest.importorskip("scipy")
for shp0 in [(2,), (2, 3), (2, 3, 4), (2, 3, 4, 5)]:
x = tensor.tensor(dtype="floatX", broadcastable=(False,) * len(shp0))
a = np.asarray(self.rng.rand(*shp0)).astype(config.floatX)
for shp1 in [(6,), (6, 7), (6, 7, 8), (6, 7, 8, 9)]:
if len(shp0) + len(shp1) == 2:
continue
y = tensor.tensor(dtype="floatX", broadcastable=(False,) * len(shp1))
f = function([x, y], kron(x, y))
b = self.rng.rand(*shp1).astype(config.floatX)
out = f(a, b)
# Newer versions of scipy want 4 dimensions at least,
# so we have to add a dimension to a and flatten the result.
if len(shp0) + len(shp1) == 3:
scipy_val = scipy.linalg.kron(a[np.newaxis, :], b).flatten()
else:
scipy_val = scipy.linalg.kron(a, b)
utt.assert_allclose(out, scipy_val)
def test_perform(self):
if not imported_scipy:
raise SkipTest("kron tests need the scipy package to be installed")
for shp0 in [(2,), (2, 3), (2, 3, 4), (2, 3, 4, 5)]:
x = tensor.tensor(dtype="floatX", broadcastable=(False,) * len(shp0))
a = numpy.asarray(self.rng.rand(*shp0)).astype(config.floatX)
for shp1 in [(6,), (6, 7), (6, 7, 8), (6, 7, 8, 9)]:
if len(shp0) + len(shp1) == 2:
continue
y = tensor.tensor(dtype="floatX", broadcastable=(False,) * len(shp1))
f = function([x, y], kron(x, y))
b = self.rng.rand(*shp1).astype(config.floatX)
out = f(a, b)
# Newer versions of scipy want 4 dimensions at least,
# so we have to add a dimension to a and flatten the result.
if len(shp0) + len(shp1) == 3:
scipy_val = scipy.linalg.kron(a[numpy.newaxis, :], b).flatten()
else:
scipy_val = scipy.linalg.kron(a, b)
utt.assert_allclose(out, scipy_val)
def test_perform(self):
if not imported_scipy:
raise SkipTest('kron tests need the scipy package to be installed')
for shp0 in [(2,), (2, 3), (2, 3, 4), (2, 3, 4, 5)]:
x = tensor.tensor(dtype='floatX',
broadcastable=(False,) * len(shp0))
a = numpy.asarray(self.rng.rand(*shp0)).astype(config.floatX)
for shp1 in [(6,), (6, 7), (6, 7, 8), (6, 7, 8, 9)]:
if len(shp0) + len(shp1) == 2:
continue
y = tensor.tensor(dtype='floatX',
broadcastable=(False,) * len(shp1))
f = function([x, y], kron(x, y))
b = self.rng.rand(*shp1).astype(config.floatX)
out = f(a, b)
# Newer versions of scipy want 4 dimensions at least,
# so we have to add a dimension to a and flatten the result.
if len(shp0) + len(shp1) == 3:
scipy_val = scipy.linalg.kron(
a[numpy.newaxis, :], b).flatten()
else:
scipy_val = scipy.linalg.kron(a, b)
utt.assert_allclose(out, scipy_val)
def __init__(self,
numberOfInducingPoints, # Number of inducing ponts in sparse GP
batchSize, # Size of mini batch
dimX, # Dimensionality of the latent co-ordinates
dimZ, # Dimensionality of the latent variables
data, # [NxP] matrix of observations
kernelType='ARD',
encoderType_qX='FreeForm2', # 'MLP', 'Kernel'.
encoderType_rX='FreeForm2', # 'MLP', 'Kernel'
Xu_optimise=False,
numberOfEncoderHiddenUnits=10
):
self.numTestSamples = 5000
# set the data
data = np.asarray(data, dtype=precision)
self.N = data.shape[0] # Number of observations
self.P = data.shape[1] # Dimension of each observation
self.M = numberOfInducingPoints
self.B = batchSize
self.R = dimX
self.Q = dimZ
self.H = numberOfEncoderHiddenUnits
self.encoderType_qX = encoderType_qX
self.encoderType_rX = encoderType_rX
self.Xu_optimise = Xu_optimise
self.y = th.shared(data)
self.y.name = 'y'
if kernelType == 'RBF':
self.numberOfKernelParameters = 2
elif kernelType == 'RBFnn':
self.numberOfKernelParameters = 1
elif kernelType == 'ARD':
self.numberOfKernelParameters = self.R + 1
else:
raise RuntimeError('Unrecognised kernel type')
self.lowerBound = -np.inf # Lower bound
self.numberofBatchesPerEpoch = int(np.ceil(np.float32(self.N) / self.B))
numPad = self.numberofBatchesPerEpoch * self.B - self.N
self.batchStream = srng.permutation(n=self.N)
self.padStream = srng.choice(size=(numPad,), a=self.N,
replace=False, p=None, ndim=None, dtype='int32')
self.batchStream.name = 'batchStream'
self.padStream.name = 'padStream'
self.iterator = th.shared(0)
self.iterator.name = 'iterator'
self.allBatches = T.reshape(T.concatenate((self.batchStream, self.padStream)), [self.numberofBatchesPerEpoch, self.B])
self.currentBatch = T.flatten(self.allBatches[self.iterator, :])
self.allBatches.name = 'allBatches'
self.currentBatch.name = 'currentBatch'
self.y_miniBatch = self.y[self.currentBatch, :]
self.y_miniBatch.name = 'y_miniBatch'
self.jitterDefault = np.float64(0.0001)
self.jitterGrowthFactor = np.float64(1.1)
self.jitter = th.shared(np.asarray(self.jitterDefault, dtype='float64'), name='jitter')
kfactory = kernelFactory(kernelType)
# kernel parameters
self.log_theta = sharedZeroMatrix(1, self.numberOfKernelParameters, 'log_theta', broadcastable=(True,False)) # parameters of Kuu, Kuf, Kff
self.log_omega = sharedZeroMatrix(1, self.numberOfKernelParameters, 'log_omega', broadcastable=(True,False)) # parameters of Kuu, Kuf, Kff
self.log_gamma = sharedZeroMatrix(1, self.numberOfKernelParameters, 'log_gamma', broadcastable=(True,False)) # parameters of Kuu, Kuf, Kff
# Random variables
self.xi = srng.normal(size=(self.B, self.R), avg=0.0, std=1.0, ndim=None)
self.alpha = srng.normal(size=(self.M, self.Q), avg=0.0, std=1.0, ndim=None)
self.beta = srng.normal(size=(self.B, self.Q), avg=0.0, std=1.0, ndim=None)
self.xi.name = 'xi'
self.alpha.name = 'alpha'
self.beta.name = 'beta'
self.sample_xi = th.function([], self.xi)
self.sample_alpha = th.function([], self.alpha)
self.sample_beta = th.function([], self.beta)
self.sample_batchStream = th.function([], self.batchStream)
self.sample_padStream = th.function([], self.padStream)
self.getCurrentBatch = th.function([], self.currentBatch, no_default_updates=True)
# Compute parameters of q(X)
if self.encoderType_qX == 'FreeForm1' or self.encoderType_qX == 'FreeForm2':
# Have a normal variational distribution over location of latent co-ordinates
self.phi_full = sharedZeroMatrix(self.N, self.R, 'phi_full')
self.phi = self.phi_full[self.currentBatch, :]
self.phi.name = 'phi'
if encoderType_qX == 'FreeForm1':
self.Phi_full_sqrt = sharedZeroMatrix(self.N, self.N, 'Phi_full_sqrt')
Phi_batch_sqrt = self.Phi_full_sqrt[self.currentBatch][:, self.currentBatch]
Phi_batch_sqrt.name = 'Phi_batch_sqrt'
self.Phi = dot(Phi_batch_sqrt, Phi_batch_sqrt.T, 'Phi')
self.cPhi, _, self.logDetPhi = cholInvLogDet(self.Phi, self.B, 0)
self.qX_vars = [self.Phi_full_sqrt, self.phi_full]
else:
self.Phi_full_logdiag = sharedZeroArray(self.N, 'Phi_full_logdiag')
Phi_batch_logdiag = self.Phi_full_logdiag[self.currentBatch]
Phi_batch_logdiag.name = 'Phi_batch_logdiag'
self.Phi, self.cPhi, _, self.logDetPhi \
= diagCholInvLogDet_fromLogDiag(Phi_batch_logdiag, 'Phi')
self.qX_vars = [self.Phi_full_logdiag, self.phi_full]
elif self.encoderType_qX == 'MLP':
# Auto encode
self.W1_qX = sharedZeroMatrix(self.H, self.P, 'W1_qX')
self.W2_qX = sharedZeroMatrix(self.R, self.H, 'W2_qX')
self.W3_qX = sharedZeroMatrix(1, self.H, 'W3_qX')
self.b1_qX = sharedZeroVector(self.H, 'b1_qX', broadcastable=(False, True))
self.b2_qX = sharedZeroVector(self.R, 'b2_qX', broadcastable=(False, True))
self.b3_qX = sharedZeroVector(1, 'b3_qX', broadcastable=(False, True))
# [HxB] = softplus( [HxP] . [BxP]^T + repmat([Hx1],[1,B]) )
h_qX = softplus(plus(dot(self.W1_qX, self.y_miniBatch.T), self.b1_qX), 'h_qX' )
# [RxB] = sigmoid( [RxH] . [HxB] + repmat([Rx1],[1,B]) )
mu_qX = plus(dot(self.W2_qX, h_qX), self.b2_qX, 'mu_qX')
# [1xB] = 0.5 * ( [1xH] . [HxB] + repmat([1x1],[1,B]) )
log_sigma_qX = mul( 0.5, plus(dot(self.W3_qX, h_qX), self.b3_qX), 'log_sigma_qX')
self.phi = mu_qX.T # [BxR]
self.Phi, self.cPhi, self.iPhi,self.logDetPhi \
= diagCholInvLogDet_fromLogDiag(log_sigma_qX, 'Phi')
self.qX_vars = [self.W1_qX, self.W2_qX, self.W3_qX, self.b1_qX, self.b2_qX, self.b3_qX]
elif self.encoderType_qX == 'Kernel':
# Draw the latent coordinates from a GP with data co-ordinates
self.Phi = kfactory.kernel(self.y_miniBatch, None, self.log_gamma, 'Phi')
self.phi = sharedZeroMatrix(self.B, self.R, 'phi')
(self.cPhi, self.iPhi, self.logDetPhi) = cholInvLogDet(self.Phi, self.B, self.jitter)
self.qX_vars = [self.log_gamma]
else:
raise RuntimeError('Unrecognised encoding for q(X): ' + self.encoderType_qX)
# Variational distribution q(u)
self.kappa = sharedZeroMatrix(self.M, self.Q, 'kappa')
self.Kappa_sqrt = sharedZeroMatrix(self.M, self.M, 'Kappa_sqrt')
self.Kappa = dot(self.Kappa_sqrt, self.Kappa_sqrt.T, 'Kappa')
(self.cKappa, self.iKappa, self.logDetKappa) \
= cholInvLogDet(self.Kappa, self.M, 0)
self.qu_vars = [self.Kappa_sqrt, self.kappa]
# Calculate latent co-ordinates Xf
# [BxR] = [BxR] + [BxB] . [BxR]
self.Xz = plus( self.phi, dot(self.cPhi, self.xi), 'Xf' )
# Inducing points co-ordinates
self.Xu = sharedZeroMatrix(self.M, self.R, 'Xu')
# Kernels
self.Kzz = kfactory.kernel(self.Xz, None, self.log_theta, 'Kff')
self.Kuu = kfactory.kernel(self.Xu, None, self.log_theta, 'Kuu')
self.Kzu = kfactory.kernel(self.Xz, self.Xu, self.log_theta, 'Kfu')
self.cKuu, self.iKuu, self.logDetKuu = cholInvLogDet(self.Kuu, self.M, self.jitter)
# Variational distribution
# A has dims [BxM] = [BxM] . [MxM]
self.A = dot(self.Kzu, self.iKuu, 'A')
# L is the covariance of conditional distribution q(z|u,Xf)
self.C = minus( self.Kzz, dot(self.A, self.Kzu.T), 'C')
self.cC, self.iC, self.logDetC = cholInvLogDet(self.C, self.B, self.jitter)
# Sample u_q from q(u_q) = N(u_q; kappa_q, Kappa ) [MxQ]
self.u = plus(self.kappa, (dot(self.cKappa, self.alpha)), 'u')
# compute mean of z [QxB]
# [BxQ] = [BxM] * [MxQ]
self.mu = dot(self.A, self.u, 'mu')
# Sample f from q(f|u,X) = N( mu_q, C )
# [BxQ] =
self.z = plus(self.mu, (dot(self.cC, self.beta)), 'z')
self.qz_vars = [self.log_theta]
self.iUpsilon = plus(self.iKappa, dot(self.A.T, dot(self.iC, self.A) ), 'iUpsilon')
_, self.Upsilon, self.negLogDetUpsilon = cholInvLogDet(self.iUpsilon, self.M, self.jitter)
if self.encoderType_rX == 'MLP':
self.W1_rX = sharedZeroMatrix(self.H, self.Q+self.P, 'W1_rX')
self.W2_rX = sharedZeroMatrix(self.R, self.H, 'W2_rX')
self.W3_rX = sharedZeroMatrix(self.R, self.H, 'W3_rX')
self.b1_rX = sharedZeroVector(self.H, 'b1_rX', broadcastable=(False, True))
self.b2_rX = sharedZeroVector(self.R, 'b2_rX', broadcastable=(False, True))
self.b3_rX = sharedZeroVector(self.R, 'b3_rX', broadcastable=(False, True))
# [HxB] = softplus( [Hx(Q+P)] . [(Q+P)xB] + repmat([Hx1], [1,B]) )
h_rX = softplus(plus(dot(self.W1_rX, T.concatenate((self.z.T, self.y_miniBatch.T))), self.b1_rX), 'h_rX')
# [RxB] = softplus( [RxH] . [HxB] + repmat([Rx1], [1,B]) )
mu_rX = plus(dot(self.W2_rX, h_rX), self.b2_rX, 'mu_rX')
# [RxB] = 0.5*( [RxH] . [HxB] + repmat([Rx1], [1,B]) )
log_sigma_rX = mul( 0.5, plus(dot(self.W3_rX, h_rX), self.b3_rX), 'log_sigma_rX')
self.tau = mu_rX.T
# Diagonal optimisation of Tau
self.Tau_isDiagonal = True
self.Tau = T.reshape(log_sigma_rX, [self.B * self.R, 1])
self.logDetTau = T.sum(log_sigma_rX)
self.Tau.name = 'Tau'
self.logDetTau.name = 'logDetTau'
self.rX_vars = [self.W1_rX, self.W2_rX, self.W3_rX, self.b1_rX, self.b2_rX, self.b3_rX]
elif self.encoderType_rX == 'Kernel':
self.tau = sharedZeroMatrix(self.B, self.R, 'tau')
# Tau_r [BxB] = kernel( [[BxQ]^T,[BxP]^T].T )
Tau_r = kfactory.kernel(T.concatenate((self.z.T, self.y_miniBatch.T)).T, None, self.log_omega, 'Tau_r')
(cTau_r, iTau_r, logDetTau_r) = cholInvLogDet(Tau_r, self.B, self.jitter)
# self.Tau = slinalg.kron(T.eye(self.R), Tau_r)
self.cTau = slinalg.kron(cTau_r, T.eye(self.R))
self.iTau = slinalg.kron(iTau_r, T.eye(self.R))
self.logDetTau = logDetTau_r * self.R
self.tau.name = 'tau'
# self.Tau.name = 'Tau'
self.cTau.name = 'cTau'
self.iTau.name = 'iTau'
self.logDetTau.name = 'logDetTau'
self.Tau_isDiagonal = False
self.rX_vars = [self.log_omega]
else:
raise RuntimeError('Unrecognised encoding for r(X|z)')
# Gradient variables - should be all the th.shared variables
# We always want to optimise these variables
if self.Xu_optimise:
self.gradientVariables = [self.Xu]
else:
self.gradientVariables = []
self.gradientVariables.extend(self.qu_vars)
self.gradientVariables.extend(self.qz_vars)
self.gradientVariables.extend(self.qX_vars)
self.gradientVariables.extend(self.rX_vars)
self.lowerBounds = []
self.condKappa = myCond()(self.Kappa)
self.condKappa.name = 'condKappa'
self.Kappa_conditionNumber = th.function([], self.condKappa, no_default_updates=True)
self.condKuu = myCond()(self.Kuu)
self.condKuu.name = 'condKuu'
self.Kuu_conditionNumber = th.function([], self.condKuu, no_default_updates=True)
self.condC = myCond()(self.C)
self.condC.name = 'condC'
self.C_conditionNumber = th.function([], self.condC, no_default_updates=True)
self.condUpsilon = myCond()(self.Upsilon)
self.condUpsilon.name = 'condUpsilon'
self.Upsilon_conditionNumber = th.function([], self.condUpsilon, no_default_updates=True)
self.Xz_get_value = th.function([], self.Xz, no_default_updates=True)