def define_process(self):
# Prior
self.prior_freedom = self.freedom()
self.prior_mean = self.location_space
self.prior_covariance = self.kernel_f_space * self.prior_freedom
self.prior_variance = tnl.extract_diag(self.prior_covariance)
self.prior_std = tt.sqrt(self.prior_variance)
self.prior_noise = tt.sqrt(tnl.extract_diag(self.kernel_space * self.prior_freedom))
self.prior_median = self.prior_mean
sigma = 2
self.prior_quantile_up = self.prior_mean + sigma * self.prior_std
self.prior_quantile_down = self.prior_mean - sigma * self.prior_std
self.prior_noise_up = self.prior_mean + sigma * self.prior_noise
self.prior_noise_down = self.prior_mean - sigma * self.prior_noise
self.prior_sampler = self.prior_mean + self.random_scalar * cholesky_robust(self.prior_covariance).dot(self.random_th)
# Posterior
self.posterior_freedom = self.prior_freedom + self.inputs.shape[1]
beta = (self.mapping_outputs - self.location_inputs).T.dot(tsl.solve(self.kernel_inputs, self.mapping_outputs - self.location_inputs))
coeff = (self.prior_freedom + beta - 2)/(self.posterior_freedom - 2)
self.posterior_mean = self.location_space + self.kernel_f_space_inputs.dot( tsl.solve(self.kernel_inputs, self.mapping_outputs - self.location_inputs))
self.posterior_covariance = coeff * (self.kernel_f.cov(self.space_th) - self.kernel_f_space_inputs.dot(
tsl.solve(self.kernel_inputs, self.kernel_f_space_inputs.T)))
self.posterior_variance = tnl.extract_diag(self.posterior_covariance)
self.posterior_std = tt.sqrt(self.posterior_variance)
self.posterior_noise = coeff * tt.sqrt(tnl.extract_diag(self.kernel.cov(self.space_th) - self.kernel_f_space_inputs.dot(
tsl.solve(self.kernel_inputs, self.kernel_f_space_inputs.T))))
self.posterior_median = self.posterior_mean
self.posterior_quantile_up = self.posterior_mean + sigma * self.posterior_std
self.posterior_quantile_down = self.posterior_mean - sigma * self.posterior_std
self.posterior_noise_up = self.posterior_mean + sigma * self.posterior_noise
self.posterior_noise_down = self.posterior_mean - sigma * self.posterior_noise
self.posterior_sampler = self.posterior_mean + self.random_scalar * cholesky_robust(self.posterior_covariance).dot(self.random_th)
def __init__(self, tau2_0=0.1, sigma2_0=0.1, l_0=0.1, eta=0.1, debug=1):
"""
:type sigma_0: float
:param sigma_0: starting value for variance.
:type l_0: float
:param l_0: starting value for length scale.
:type eta: float
:param eta: learning rate
:type debug: int
:param debug: verbosity
"""
print "GP Initing..." if debug > 0 else 0
##################################################
#### Prepare the -loglik gradient descent
##Init the shared vars
X = T.dmatrix('X')
f = T.dmatrix('f')
self.tau2 = theano.shared(tau2_0)
self.l = theano.shared(l_0)
self.sigma2 = theano.shared(sigma2_0)
#Make the covar matrix
K = self.covFunc(X, X, self.l)
#Get a numerically safe decomp
L = LA.cholesky(K + self.tau2 * T.identity_like(K))
#Calculate the weights for each of the training data; predictions are a weighted sum.
alpha = LA.solve(T.transpose(L), LA.solve(L, f))
##Calculate - log marginal likelihood
nloglik = -T.reshape(
-0.5 * T.dot(T.transpose(f), alpha) - T.sum(T.log(T.diag(L))), [])
#Get grad
grads = [
T.grad(nloglik, self.tau2),
T.grad(nloglik, self.l),
T.grad(nloglik, self.sigma2)
]
#Updates, make sure to keep the params positive
updates = [
(var, T.max([var - eta * grad, 0.1]))
for var, grad in zip([self.tau2, self.l, self.sigma2], grads)
]
self._gd = theano.function(inputs=[X, f], updates=updates)
print "Done" if debug > 0 else 0
def inv_as_solve(node):
if not imported_scipy:
return False
if isinstance(node.op, (Dot, Dot22)):
l, r = node.inputs
if l.owner and l.owner.op == matrix_inverse:
return [solve(l.owner.inputs[0], r)]
if r.owner and r.owner.op == matrix_inverse:
if is_symmetric(r.owner.inputs[0]):
return [solve(r.owner.inputs[0], l.T).T]
else:
return [solve(r.owner.inputs[0].T, l.T).T]
def subprocess_gp(self, subkernel, cov=False, noise=False):
k_ni = subkernel.cov(self.space, self.inputs)
mu = self.mean(self.space) + k_ni.dot(sL.solve(self.cov_inputs, self.inv_outputs - self.mean_inputs))
if noise:
k_cov = self.kernel.cov(self.space) - k_ni.dot(sL.solve(self.cov_inputs, k_ni.T))
else:
k_cov = self.kernel_f.cov(self.space) - k_ni.dot(sL.solve(self.cov_inputs, k_ni.T))
var = nL.extract_diag(debug(k_cov, 'k_cov'))
if cov:
return mu, var, k_cov
else:
return mu, var
def inv_as_solve(node):
if not imported_scipy:
return False
if isinstance(node.op, (Dot, Dot22)):
l, r = node.inputs
if l.owner and l.owner.op == matrix_inverse:
return [solve(l.owner.inputs[0], r)]
if r.owner and r.owner.op == matrix_inverse:
if is_symmetric(r.owner.inputs[0]):
return [solve(r.owner.inputs[0], l.T).T]
else:
return [solve(r.owner.inputs[0].T, l.T).T]
def th_define_process(self):
#print('stochastic_define_process')
# Basic Tensors
self.mapping_outputs = tt_to_num(self.f_mapping.inv(self.th_outputs))
self.mapping_latent = tt_to_num(self.f_mapping(self.th_outputs))
#self.mapping_scalar = tt_to_num(self.f_mapping.inv(self.th_scalar))
self.prior_location_space = self.f_location(self.th_space)
self.prior_location_inputs = self.f_location(self.th_inputs)
self.prior_kernel_space = tt_to_cov(self.f_kernel_noise.cov(self.th_space))
self.prior_kernel_inputs = tt_to_cov(self.f_kernel_noise.cov(self.th_inputs))
self.prior_cholesky_space = cholesky_robust(self.prior_kernel_space)
self.prior_kernel_f_space = self.f_kernel.cov(self.th_space)
self.prior_kernel_f_inputs = self.f_kernel.cov(self.th_inputs)
self.prior_cholesky_f_space = cholesky_robust(self.prior_kernel_f_space)
self.cross_kernel_space_inputs = tt_to_num(self.f_kernel_noise.cov(self.th_space, self.th_inputs))
self.cross_kernel_f_space_inputs = tt_to_num(self.f_kernel.cov(self.th_space, self.th_inputs))
self.posterior_location_space = self.prior_location_space + self.cross_kernel_space_inputs.dot(
tsl.solve(self.prior_kernel_inputs, self.mapping_outputs - self.prior_location_inputs))
self.posterior_location_f_space = self.prior_location_space + self.cross_kernel_f_space_inputs.dot(
tsl.solve(self.prior_kernel_inputs, self.mapping_outputs - self.prior_location_inputs))
self.posterior_kernel_space = self.prior_kernel_space - self.cross_kernel_space_inputs.dot(
tsl.solve(self.prior_kernel_inputs, self.cross_kernel_space_inputs.T))
self.posterior_cholesky_space = cholesky_robust(self.posterior_kernel_space)
self.posterior_kernel_f_space = self.prior_kernel_f_space - self.cross_kernel_f_space_inputs.dot(
tsl.solve(self.prior_kernel_inputs, self.cross_kernel_f_space_inputs.T))
self.posterior_cholesky_f_space = cholesky_robust(self.posterior_kernel_f_space)
self.prior_kernel_diag_space = tt_to_bounded(tnl.extract_diag(self.prior_kernel_space), zero32)
self.prior_kernel_diag_f_space = tt_to_bounded(tnl.extract_diag(self.prior_kernel_f_space), zero32)
self.posterior_kernel_diag_space = tt_to_bounded(tnl.extract_diag(self.posterior_kernel_space), zero32)
self.posterior_kernel_diag_f_space = tt_to_bounded(tnl.extract_diag(self.posterior_kernel_f_space), zero32)
self.prior_kernel_sd_space = tt.sqrt(self.prior_kernel_diag_space)
self.prior_kernel_sd_f_space = tt.sqrt(self.prior_kernel_diag_f_space)
self.posterior_kernel_sd_space = tt.sqrt(self.posterior_kernel_diag_space)
self.posterior_kernel_sd_f_space = tt.sqrt(self.posterior_kernel_diag_f_space)
self.prior_cholesky_diag_space = tnl.alloc_diag(self.prior_kernel_sd_space)
self.prior_cholesky_diag_f_space = tnl.alloc_diag(self.prior_kernel_sd_f_space)
self.posterior_cholesky_diag_space = tnl.alloc_diag(self.posterior_kernel_sd_space)
self.posterior_cholesky_diag_f_space = tnl.alloc_diag(self.posterior_kernel_sd_f_space)
def init_train_updates(self):
network_output = self.variables.network_output
prediction_func = self.variables.train_prediction_func
last_error = self.variables.last_error
error_func = self.variables.error_func
mu = self.variables.mu
new_mu = ifelse(
T.lt(last_error, error_func),
mu * self.mu_update_factor,
mu / self.mu_update_factor,
)
se_for_each_sample = ((network_output - prediction_func)**2).ravel()
params = parameter_values(self.connection)
param_vector = T.concatenate([param.flatten() for param in params])
J = compute_jacobian(se_for_each_sample, params)
n_params = J.shape[1]
updated_params = param_vector - slinalg.solve(
J.T.dot(J) + new_mu * T.eye(n_params), J.T.dot(se_for_each_sample))
updates = [(mu, new_mu)]
parameter_updates = setup_parameter_updates(params, updated_params)
updates.extend(parameter_updates)
return updates
def __init__(self, f1, vs1, f2, vs2):
self.f1 = f1
self.f2 = f2
self.vs1 = vs1
self.vs2 = vs2
self.sz1 = [shape(v)[0] for v in self.vs1]
self.sz2 = [shape(v)[0] for v in self.vs2]
for i in range(1, len(self.sz1)):
self.sz1[i] += self.sz1[i-1]
self.sz1 = [(0 if i==0 else self.sz1[i-1], self.sz1[i]) for i in range(len(self.sz1))]
for i in range(1, len(self.sz2)):
self.sz2[i] += self.sz2[i-1]
self.sz2 = [(0 if i==0 else self.sz2[i-1], self.sz2[i]) for i in range(len(self.sz2))]
self.df1 = grad(self.f1, vs1)
self.new_vs1 = [tt.vector() for v in self.vs1]
self.func1 = th.function(self.new_vs1, [-self.f1, -self.df1], givens=zip(self.vs1, self.new_vs1))
def f1_and_df1(x0):
return self.func1(*[x0[a:b] for a, b in self.sz1])
self.f1_and_df1 = f1_and_df1
J = jacobian(grad(f1, vs2), vs1)
H = hessian(f1, vs1)
g = grad(f2, vs1)
self.df2 = -tt.dot(J, ts.solve(H, g))+grad(f2, vs2)
self.func2 = th.function([], [-self.f2, -self.df2])
def f2_and_df2(x0):
for v, (a, b) in zip(self.vs2, self.sz2):
v.set_value(x0[a:b])
self.maximize1()
return self.func2()
self.f2_and_df2 = f2_and_df2
def test_solve_dtype(self):
pytest.importorskip("scipy")
dtypes = [
"uint8",
"uint16",
"uint32",
"uint64",
"int8",
"int16",
"int32",
"int64",
"float16",
"float32",
"float64",
]
A_val = np.eye(2)
b_val = np.ones((2, 1))
# try all dtype combinations
for A_dtype, b_dtype in itertools.product(dtypes, dtypes):
A = tensor.matrix(dtype=A_dtype)
b = tensor.matrix(dtype=b_dtype)
x = solve(A, b)
fn = function([A, b], x)
x_result = fn(A_val.astype(A_dtype), b_val.astype(b_dtype))
assert x.dtype == x_result.dtype
def __init__(self, f1, vs1, f2, vs2):
self.f1 = f1
self.f2 = f2
self.vs1 = vs1
self.vs2 = vs2
self.sz1 = [shape(v)[0] for v in self.vs1]
self.sz2 = [shape(v)[0] for v in self.vs2]
for i in range(1, len(self.sz1)):
self.sz1[i] += self.sz1[i-1]
self.sz1 = [(0 if i==0 else self.sz1[i-1], self.sz1[i]) for i in range(len(self.sz1))]
for i in range(1, len(self.sz2)):
self.sz2[i] += self.sz2[i-1]
self.sz2 = [(0 if i==0 else self.sz2[i-1], self.sz2[i]) for i in range(len(self.sz2))]
self.df1 = grad(self.f1, vs1)
self.new_vs1 = [tt.vector() for v in self.vs1]
self.func1 = th.function(self.new_vs1, [-self.f1, -self.df1], givens=zip(self.vs1, self.new_vs1))
def f1_and_df1(x0):
return self.func1(*[x0[a:b] for a, b in self.sz1])
self.f1_and_df1 = f1_and_df1
J = jacobian(grad(f1, vs2), vs1)
H = hessian(f1, vs1)
g = grad(f2, vs1)
self.df2 = -tt.dot(J, ts.solve(H, g))+grad(f2, vs2)
self.func2 = th.function([], [-self.f2, -self.df2])
def f2_and_df2(x0):
for v, (a, b) in zip(self.vs2, self.sz2):
v.set_value(x0[a:b])
self.maximize1()
return self.func2()
self.f2_and_df2 = f2_and_df2
def init_train_updates(self):
network_output = self.variables.network_output
prediction_func = self.variables.train_prediction_func
last_error = self.variables.last_error
error_func = self.variables.error_func
mu = self.variables.mu
new_mu = ifelse(
T.lt(last_error, error_func),
mu * self.mu_update_factor,
mu / self.mu_update_factor,
)
se_for_each_sample = (
(network_output - prediction_func) ** 2
).ravel()
params = parameter_values(self.connection)
param_vector = parameters2vector(self)
J = compute_jacobian(se_for_each_sample, params)
n_params = J.shape[1]
updated_params = param_vector - slinalg.solve(
J.T.dot(J) + new_mu * T.eye(n_params),
J.T.dot(se_for_each_sample)
)
updates = [(mu, new_mu)]
parameter_updates = setup_parameter_updates(params, updated_params)
updates.extend(parameter_updates)
return updates
def lp_to_phi0_fs(lp, Tobs):
Tobs2 = Tobs * Tobs
Tobs3 = Tobs2 * Tobs
phis = tts.solve(poly_basis_to_legendre_basis, lp)
phi0 = phis[0]
f0 = phis[1] / (2.0 * pi * Tobs)
fdot = phis[2] / (pi * Tobs2)
fddot = phis[3] / (pi / 3.0 * Tobs3)
return (phi0, f0, fdot, fddot)
def second_moments(i, j, M2, beta, R, logk_c, logk_r, z_, Sx, *args):
# This comes from Deisenroth's thesis ( Eqs 2.51- 2.54 )
Rij = R[i, j]
n2 = logk_c[i] + logk_r[j]
n2 += utils.maha(z_[i], -z_[j], 0.5 * solve(Rij, Sx))
Q = tt.exp(n2) / tt.sqrt(det(Rij))
# Eq 2.55
m2 = matrix_dot(beta[i], Q, beta[j])
m2 = theano.ifelse.ifelse(tt.eq(i, j), m2 + 1e-6, m2)
M2 = tt.set_subtensor(M2[i, j], m2)
return M2
def test_local_lift_solve():
A = tensor.fmatrix()
b = tensor.fmatrix()
o = slinalg.solve(A, b)
f_cpu = theano.function([A, b], o)
f_gpu = theano.function([A, b], o, mode=mode_with_gpu)
assert not any(isinstance(n.op, slinalg.Solve)
for n in f_gpu.maker.fgraph.apply_nodes)
assert any(isinstance(n.op, GpuCusolverSolve)
for n in f_gpu.maker.fgraph.apply_nodes)
A_val = numpy.random.uniform(-0.4, 0.4, (5, 5)).astype("float32")
b_val = numpy.random.uniform(-0.4, 0.4, (5, 3)).astype("float32")
utt.assert_allclose(f_cpu(A_val, b_val), f_gpu(A_val, b_val))
def __init__(self, f1, vs1, f2, vs2):
# creates a isolated var of the vars
self.f1 = f1
self.f2 = f2
self.vs1 = vs1
self.vs2 = vs2
#-----
#
self.sz1 = [
shape(v)[0] for v in self.vs1
] # converts from tensor variable to normal array (uses also some math magic)
self.sz2 = [shape(v)[0] for v in self.vs2]
for i in range(1, len(
self.sz1)): # adds together all future sz1 with old sz1
self.sz1[i] += self.sz1[i - 1]
self.sz1 = [
(0 if i == 0 else self.sz1[i - 1], self.sz1[i])
for i in range(len(self.sz1))
] #makes the array into a 2d-array with (prevoius var, var)
for i in range(1, len(self.sz2)): # same thing az sz1
self.sz2[i] += self.sz2[i - 1]
self.sz2 = [(0 if i == 0 else self.sz2[i - 1], self.sz2[i])
for i in range(len(self.sz2))] # samme thing as sz1
self.df1 = grad(self.f1, vs1) # IMPORTANT: VERY SLOW
self.new_vs1 = [tt.vector() for v in self.vs1
] # back from normal array to tensorVector
self.func1 = th.function(
self.new_vs1, [-self.f1, -self.df1],
givens=zip(self.vs1,
self.new_vs1)) # IMPORTANT: VERY VERY VERY SLOW
def f1_and_df1(x0):
return self.func1(*[x0[a:b] for a, b in self.sz1])
self.f1_and_df1 = f1_and_df1
J = jacobian(grad(f1, vs2), vs1) # IMPORTANT: VERY VERY VERY VERY SLOW
H = hessian(f1, vs1) # IMPORTANT: VERY VERY VERY VERY SLOW
g = grad(f2, vs1) # IMPORTANT: SLOW
self.df2 = -tt.dot(J, ts.solve(H, g)) + grad(f2,
vs2) # IMPORTANT: SLOW
self.func2 = th.function(
[], [-self.f2, -self.df2]) # IMPORTANT: EXREMELY SLOW
def f2_and_df2(x0):
for v, (a, b) in zip(self.vs2, self.sz2):
v.set_value(x0[a:b])
self.maximize1()
return self.func2()
self.f2_and_df2 = f2_and_df2
def grad(self, inputs, g_outputs):
# let A = I - rho W, and dRho(A) be the derivatrive of A wrt Rho
# dRho(log(|(AtA)|)) = dRho(log(|(AtA)|)))
# = dRho(log(|At|) + log(|A|))
# = dRho(log(|A| + log(|A|)))
# = 2 dRho(log(|A|))
# = 2 |A|^{-1} dRho(|A|) = 2|A|^{-1} tr(Adj(A)dRho(A))
# = 2 |A|^{-1} |A| tr(A^{-1}(-W)) = 2 * tr(A^{-1}W)
[gz] = g_outputs
[rho] = inputs
A = self.I - rho * self.W
trAiW = slinalg.solve(A, self.W).diagonal().sum()
#trAiW = (nlinalg.matrix_inverse(A).dot(self.W)).diagonal().sum()
return [trAiW]
def test_local_lift_solve():
if not cusolver_available:
raise SkipTest('No cuSolver')
A = tensor.fmatrix()
b = tensor.fmatrix()
o = slinalg.solve(A, b)
f_cpu = theano.function([A, b], o, mode_without_gpu)
f_gpu = theano.function([A, b], o, mode=mode_with_gpu)
assert not any(isinstance(n.op, slinalg.Solve)
for n in f_gpu.maker.fgraph.apply_nodes)
assert any(isinstance(n.op, GpuCusolverSolve) and n.op.inplace
for n in f_gpu.maker.fgraph.apply_nodes)
A_val = np.random.uniform(-0.4, 0.4, (5, 5)).astype("float32")
b_val = np.random.uniform(-0.4, 0.4, (5, 3)).astype("float32")
utt.assert_allclose(f_cpu(A_val, b_val), f_gpu(A_val, b_val))
def test_gpu_solve_not_inplace():
if not cusolver_available:
raise SkipTest('No cuSolver')
A = tensor.fmatrix()
b = tensor.fmatrix()
s = slinalg.solve(A, b)
o = tensor.dot(A, s)
f_cpu = theano.function([A, b], o, mode_without_gpu)
f_gpu = theano.function([A, b], o, mode=mode_with_gpu)
count_not_inplace = len([n.op for n in f_gpu.maker.fgraph.apply_nodes
if isinstance(n.op, GpuCusolverSolve) and not n.op.inplace])
assert count_not_inplace == 1, count_not_inplace
A_val = np.random.uniform(-0.4, 0.4, (5, 5)).astype("float32")
b_val = np.random.uniform(-0.4, 0.4, (5, 3)).astype("float32")
utt.assert_allclose(f_cpu(A_val, b_val), f_gpu(A_val, b_val))
def test_local_lift_solve():
if not cusolver_available:
raise SkipTest('No cuSolver')
A = tensor.fmatrix()
b = tensor.fmatrix()
o = slinalg.solve(A, b)
f_cpu = theano.function([A, b], o, mode_without_gpu)
f_gpu = theano.function([A, b], o, mode=mode_with_gpu)
assert not any(isinstance(n.op, slinalg.Solve)
for n in f_gpu.maker.fgraph.apply_nodes)
assert any(isinstance(n.op, GpuCusolverSolve) and n.op.inplace
for n in f_gpu.maker.fgraph.apply_nodes)
A_val = np.random.uniform(-0.4, 0.4, (5, 5)).astype("float32")
b_val = np.random.uniform(-0.4, 0.4, (5, 3)).astype("float32")
utt.assert_allclose(f_cpu(A_val, b_val), f_gpu(A_val, b_val))
def test_local_lift_solve():
A = tensor.fmatrix()
b = tensor.fmatrix()
o = slinalg.solve(A, b)
f_cpu = theano.function([A, b], o)
f_gpu = theano.function([A, b], o, mode=mode_with_gpu)
assert not any(
isinstance(n.op, slinalg.Solve)
for n in f_gpu.maker.fgraph.apply_nodes)
assert any(
isinstance(n.op, GpuCusolverSolve)
for n in f_gpu.maker.fgraph.apply_nodes)
A_val = numpy.random.uniform(-0.4, 0.4, (5, 5)).astype("float32")
b_val = numpy.random.uniform(-0.4, 0.4, (5, 3)).astype("float32")
utt.assert_allclose(f_cpu(A_val, b_val), f_gpu(A_val, b_val))
def test_gpu_solve_not_inplace():
if not cusolver_available:
raise SkipTest('No cuSolver')
A = tensor.fmatrix()
b = tensor.fmatrix()
s = slinalg.solve(A, b)
o = tensor.dot(A, s)
f_cpu = theano.function([A, b], o, mode_without_gpu)
f_gpu = theano.function([A, b], o, mode=mode_with_gpu)
count_not_inplace = len([n.op for n in f_gpu.maker.fgraph.apply_nodes
if isinstance(n.op, GpuCusolverSolve) and not n.op.inplace])
assert count_not_inplace == 1, count_not_inplace
A_val = np.random.uniform(-0.4, 0.4, (5, 5)).astype("float32")
b_val = np.random.uniform(-0.4, 0.4, (5, 3)).astype("float32")
utt.assert_allclose(f_cpu(A_val, b_val), f_gpu(A_val, b_val))
def init_train_updates(self):
n_parameters = count_parameters(self.connection)
parameters = parameter_values(self.connection)
param_vector = T.concatenate([param.flatten() for param in parameters])
penalty_const = asfloat(self.penalty_const)
hessian_matrix, full_gradient = find_hessian_and_gradient(
self.variables.error_func, parameters)
updated_parameters = param_vector - slinalg.solve(
hessian_matrix + penalty_const * T.eye(n_parameters),
full_gradient)
updates = setup_parameter_updates(parameters, updated_parameters)
return updates
def th_cross_mean(self, prior=False, noise=False, cross_kernel=None):
"""
Using two kernels calculate the media of one process given the other.
:param prior: if the process considers a prior of not
:param noise: if the process considers noise
:param cross_kernel: it's the covariance between two process
:return: returns a tensor with the location of a process given another process.
"""
if prior:
return self.prior_location_space
if cross_kernel is None:
cross_kernel = self.f_kernel
return self.prior_location_space + cross_kernel.cov(
self.th_space_, self.th_inputs_).dot(
tsl.solve(self.prior_kernel_inputs,
self.mapping_outputs - self.prior_location_inputs))
def init_train_updates(self):
n_parameters = count_parameters(self.connection)
parameters = parameter_values(self.connection)
param_vector = parameters2vector(self)
penalty_const = asfloat(self.penalty_const)
hessian_matrix, full_gradient = find_hessian_and_gradient(
self.variables.error_func, parameters
)
updated_parameters = param_vector - slinalg.solve(
hessian_matrix + penalty_const * T.eye(n_parameters),
full_gradient
)
updates = setup_parameter_updates(parameters, updated_parameters)
return updates
def logp(self, value):
"""
the sparse cached log determinant assumes I - rho W = A, and computes
the log determinant of A wrt rho with cached W.
To get this right with the SMA, we need to use -rho in the logdet.
"""
delta = value - self.mu
ld = self.spld(-self.rho)
out = -self.W.n / 2.0 * tt.log(np.pi * self.scale)
out -= ld
kern = slinalg.solve(self.AAt, delta)
kern = tt.mul(delta, kern)
kern = kern.sum()
kern *= self.scale**-2
kern /= 2.0
return out - kern
def test_solve_dtype(self):
if not imported_scipy:
raise SkipTest("Scipy needed for the Solve op.")
dtypes = ['uint8', 'uint16', 'uint32', 'uint64',
'int8', 'int16', 'int32', 'int64',
'float16', 'float32', 'float64']
A_val = numpy.eye(2)
b_val = numpy.ones((2, 1))
# try all dtype combinations
for A_dtype, b_dtype in itertools.product(dtypes, dtypes):
A = tensor.matrix(dtype=A_dtype)
b = tensor.matrix(dtype=b_dtype)
x = solve(A, b)
fn = function([A, b], x)
x_result = fn(A_val.astype(A_dtype), b_val.astype(b_dtype))
assert x.dtype == x_result.dtype
def test_solve_dtype(self):
if not imported_scipy:
raise SkipTest("Scipy needed for the Solve op.")
dtypes = [
'uint8', 'uint16', 'uint32', 'uint64', 'int8', 'int16', 'int32',
'int64', 'float16', 'float32', 'float64'
]
A_val = np.eye(2)
b_val = np.ones((2, 1))
# try all dtype combinations
for A_dtype, b_dtype in itertools.product(dtypes, dtypes):
A = tensor.matrix(dtype=A_dtype)
b = tensor.matrix(dtype=b_dtype)
x = solve(A, b)
fn = function([A, b], x)
x_result = fn(A_val.astype(A_dtype), b_val.astype(b_dtype))
assert x.dtype == x_result.dtype
def predict(self, mx, Sx, *args, **kwargs):
if self.N < self.n_inducing:
# stick with the full GP
return GP_UI.predict(self, mx, Sx)
idims = self.D
odims = self.E
# centralize inputs
zeta = self.X_sp - mx
# initialize some variables
sf2 = self.hyp[:, idims]**2
eyeE = tt.tile(tt.eye(idims), (odims, 1, 1))
lscales = self.hyp[:, :idims]
iL = eyeE/lscales.dimshuffle(0, 1, 'x')
# predictive mean
inp = iL.dot(zeta.T).transpose(0, 2, 1)
iLdotSx = iL.dot(Sx)
B = (iLdotSx[:, :, None, :]*iL[:, None, :, :]).sum(-1) + tt.eye(idims)
t = tt.stack([solve(B[i].T, inp[i].T).T for i in range(odims)])
c = sf2/tt.sqrt(tt.stack([det(B[i]) for i in range(odims)]))
l_ = tt.exp(-0.5*tt.sum(inp*t, 2))
lb = l_*self.beta_sp
M = tt.sum(lb, 1)*c
# input output covariance
tiL = tt.stack([t[i].dot(iL[i]) for i in range(odims)])
V = tt.stack([tiL[i].T.dot(lb[i]) for i in range(odims)]).T*c
# predictive covariance
logk = (tt.log(sf2))[:, None] - 0.5*tt.sum(inp*inp, 2)
logk_r = logk.dimshuffle(0, 'x', 1)
logk_c = logk.dimshuffle(0, 1, 'x')
Lambda = tt.square(iL)
LL = (Lambda.dimshuffle(0, 'x', 1, 2) + Lambda).transpose(0, 1, 3, 2)
R = tt.dot(LL, Sx.T).transpose(0, 1, 3, 2) + tt.eye(idims)
z_ = Lambda.dot(zeta.T).transpose(0, 2, 1)
M2 = tt.zeros((odims, odims))
# initialize indices
triu_indices = np.triu_indices(odims)
indices = [tt.as_index_variable(idx) for idx in triu_indices]
def second_moments(i, j, M2, beta, iK, sf2, R, logk_c, logk_r, z_, Sx):
# This comes from Deisenroth's thesis ( Eqs 2.51- 2.54 )
Rij = R[i, j]
n2 = logk_c[i] + logk_r[j]
n2 += utils.maha(z_[i], -z_[j], 0.5*solve(Rij, Sx))
Q = tt.exp(n2)/tt.sqrt(det(Rij))
# Eq 2.55
m2 = matrix_dot(beta[i], Q, beta[j])
m2 = theano.ifelse.ifelse(
tt.eq(i, j), m2 - tt.sum(iK[i]*Q) + sf2[i], m2)
M2 = tt.set_subtensor(M2[i, j], m2)
M2 = theano.ifelse.ifelse(
tt.eq(i, j), M2 + 1e-6, tt.set_subtensor(M2[j, i], m2))
return M2
nseq = [self.beta_sp, (self.iKmm - self.iBmm), sf2,
R, logk_c, logk_r, z_, Sx]
M2_, updts = theano.scan(
fn=second_moments, sequences=indices, outputs_info=[M2],
non_sequences=nseq, allow_gc=False)
M2 = M2_[-1]
S = M2 - tt.outer(M, M)
return M, S, V
def grad(self, inputs, g_outputs):
gz, = g_outputs
x, = inputs
return [slinalg.solve(x.T, gz)]
def predict_symbolic(self, mx, Sx, unroll_scan=False):
idims = self.D
odims = self.E
# centralize inputs
zeta = self.X - mx
# initialize some variables
sf2 = self.hyp[:, idims]**2
eyeE = tt.tile(tt.eye(idims), (odims, 1, 1))
lscales = self.hyp[:, :idims]
iL = eyeE / lscales.dimshuffle(0, 1, 'x')
# predictive mean
inp = iL.dot(zeta.T).transpose(0, 2, 1)
iLdotSx = iL.dot(Sx)
# TODO vectorize this
B = (iLdotSx[:, :, None, :] *
iL[:, None, :, :]).sum(-1) + tt.eye(idims)
t = tt.stack([solve(B[i].T, inp[i].T).T for i in range(odims)])
c = sf2 / tt.sqrt(tt.stack([det(B[i]) for i in range(odims)]))
l = tt.exp(-0.5 * tt.sum(inp * t, 2))
lb = l * self.beta # E x N dot E x N
M = tt.sum(lb, 1) * c
# input output covariance
tiL = (t[:, :, None, :] * iL[:, None, :, :]).sum(-1)
# tiL = tt.stack([t[i].dot(iL[i]) for i in range(odims)])
V = tt.stack([tiL[i].T.dot(lb[i]) for i in range(odims)]).T * c
# predictive covariance
logk = (tt.log(sf2))[:, None] - 0.5 * tt.sum(inp * inp, 2)
logk_r = logk.dimshuffle(0, 'x', 1)
logk_c = logk.dimshuffle(0, 1, 'x')
Lambda = tt.square(iL)
LL = (Lambda.dimshuffle(0, 'x', 1, 2) + Lambda).transpose(0, 1, 3, 2)
R = tt.dot(LL, Sx).transpose(0, 1, 3, 2) + tt.eye(idims)
z_ = Lambda.dot(zeta.T).transpose(0, 2, 1)
M2 = tt.zeros((odims, odims))
# initialize indices
triu_indices = np.triu_indices(odims)
indices = [tt.as_index_variable(idx) for idx in triu_indices]
def second_moments(i, j, M2, beta, iK, sf2, R, logk_c, logk_r, z_, Sx,
*args):
# This comes from Deisenroth's thesis ( Eqs 2.51- 2.54 )
Rij = R[i, j]
n2 = logk_c[i] + logk_r[j]
n2 += utils.maha(z_[i], -z_[j], 0.5 * solve(Rij, Sx))
Q = tt.exp(n2) / tt.sqrt(det(Rij))
# Eq 2.55
m2 = matrix_dot(beta[i], Q, beta[j])
m2 = theano.ifelse.ifelse(tt.eq(i, j),
m2 - tt.sum(iK[i] * Q) + sf2[i], m2)
M2 = tt.set_subtensor(M2[i, j], m2)
return M2
nseq = [self.beta, self.iK, sf2, R, logk_c, logk_r, z_, Sx, self.L]
if unroll_scan:
from lasagne.utils import unroll_scan
[M2_] = unroll_scan(second_moments, indices, [M2], nseq,
len(triu_indices[0]))
updts = {}
else:
M2_, updts = theano.scan(fn=second_moments,
sequences=indices,
outputs_info=[M2],
non_sequences=nseq,
allow_gc=False,
strict=True,
name="%s>M2_scan" % (self.name))
M2 = M2_[-1]
M2 = M2 + tt.triu(M2, k=1).T
S = M2 - tt.outer(M, M)
return M, S, V
def predict_symbolic(self, mx, Sx=None, unroll_scan=False):
idims = self.D
odims = self.E
# initialize some variables
sf2 = self.hyp[:, idims]**2
eyeE = tt.tile(tt.eye(idims), (odims, 1, 1))
lscales = self.hyp[:, :idims]
iL = eyeE / lscales.dimshuffle(0, 1, 'x')
if Sx is None:
# first check if we received a vector [D] or a matrix [nxD]
if mx.ndim == 1:
mx = mx[None, :]
# centralize inputs
zeta = self.X[:, None, :] - mx[None, :, :]
# predictive mean ( we don't need to do the rest )
inp = (iL[:, None, :, None, :] * zeta[:, None, :, :]).sum(2)
l = tt.exp(-0.5 * tt.sum(inp**2, -1))
lb = l * self.beta[:, :, None] # E x N
M = tt.sum(lb, 1).T * sf2
# apply saturating function to the output if available
if self.sat_func is not None:
# saturate the output
M = self.sat_func(M)
return M
# centralize inputs
zeta = self.X - mx
# predictive mean
inp = iL.dot(zeta.T).transpose(0, 2, 1)
iLdotSx = iL.dot(Sx)
B = (iLdotSx[:, :, None, :] *
iL[:, None, :, :]).sum(-1) + tt.eye(idims)
t = tt.stack([solve(B[i].T, inp[i].T).T for i in range(odims)])
c = sf2 / tt.sqrt(tt.stack([det(B[i]) for i in range(odims)]))
l = tt.exp(-0.5 * tt.sum(inp * t, 2))
lb = l * self.beta
M = tt.sum(lb, 1) * c
# input output covariance
tiL = tt.stack([t[i].dot(iL[i]) for i in range(odims)])
V = tt.stack([tiL[i].T.dot(lb[i]) for i in range(odims)]).T * c
# predictive covariance
logk = (tt.log(sf2))[:, None] - 0.5 * tt.sum(inp * inp, 2)
logk_r = logk.dimshuffle(0, 'x', 1)
logk_c = logk.dimshuffle(0, 1, 'x')
Lambda = tt.square(iL)
LL = (Lambda.dimshuffle(0, 'x', 1, 2) + Lambda).transpose(0, 1, 3, 2)
R = tt.dot(LL, Sx).transpose(0, 1, 3, 2) + tt.eye(idims)
z_ = Lambda.dot(zeta.T).transpose(0, 2, 1)
M2 = tt.zeros((odims, odims))
# initialize indices
triu_indices = np.triu_indices(odims)
indices = [tt.as_index_variable(idx) for idx in triu_indices]
def second_moments(i, j, M2, beta, R, logk_c, logk_r, z_, Sx, *args):
# This comes from Deisenroth's thesis ( Eqs 2.51- 2.54 )
Rij = R[i, j]
n2 = logk_c[i] + logk_r[j]
n2 += utils.maha(z_[i], -z_[j], 0.5 * solve(Rij, Sx))
Q = tt.exp(n2) / tt.sqrt(det(Rij))
# Eq 2.55
m2 = matrix_dot(beta[i], Q, beta[j])
m2 = theano.ifelse.ifelse(tt.eq(i, j), m2 + 1e-6, m2)
M2 = tt.set_subtensor(M2[i, j], m2)
return M2
nseq = [self.beta, R, logk_c, logk_r, z_, Sx, self.iK, self.L]
if unroll_scan:
from lasagne.utils import unroll_scan
[M2_] = unroll_scan(second_moments, indices, [M2], nseq,
len(triu_indices[0]))
updts = {}
else:
M2_, updts = theano.scan(fn=second_moments,
sequences=indices,
outputs_info=[M2],
non_sequences=nseq,
allow_gc=False,
strict=True,
name="%s>M2_scan" % (self.name))
M2 = M2_[-1]
M2 = M2 + tt.triu(M2, k=1).T
S = M2 - tt.outer(M, M)
# apply saturating function to the output if available
if self.sat_func is not None:
# saturate the output
M, S, U = self.sat_func(M, S)
# compute the joint input output covariance
V = V.dot(U)
return M, S, V
def inverse_map(self, y):
return slinalg.solve(self.weights, (y - self.biases).T).T
#theano.config.exception_verbosity = 'high'
from scipy import optimize
with pm.Model() as model:
# 2 state model
P = pm.Dirichlet('P', a=np.ones((N_states,N_states)), shape=(N_states,N_states))
A1 = pm.Normal('A1',mu=0, sd=0.3)
A2 = pm.Normal('A2',mu=1, sd=0.3)
S1 = pm.InverseGamma('S1',alpha=alphaS, beta=betaS)
S2 = pm.InverseGamma('S2',alpha=alphaS, beta=betaS)
AA = tt.dmatrix('AA')
AA = tt.eye(N_states) - P + tt.ones(shape=(N_states,N_states))
PA = pm.Deterministic('PA',sla.solve(AA.T,tt.ones(shape=(N_states))))
states1 = HMMStatesN('states1',P=P,PA=PA, shape=len(dataset[4]))
emission1 = HMMGaussianEmissions('emission1',
A1=A1,
A2=A1,
S1=S1,
S2=S2,
states=states1,
observed = dataset[4])
states2 = HMMStatesN('states2',P=P,PA=PA, shape=len(dataset[205]))
emission2 = HMMGaussianEmissions('emission2',
A1=A1,