def test_biject_to(constraint, shape):
transform = biject_to(constraint)
if isinstance(constraint, constraints._Interval):
assert transform.codomain.upper_bound == constraint.upper_bound
assert transform.codomain.lower_bound == constraint.lower_bound
elif isinstance(constraint, constraints._GreaterThan):
assert transform.codomain.lower_bound == constraint.lower_bound
if len(shape) < transform.event_dim:
return
rng = random.PRNGKey(0)
x = random.normal(rng, shape)
y = transform(x)
# test codomain
batch_shape = shape if transform.event_dim == 0 else shape[:-1]
assert_array_equal(transform.codomain(y),
np.ones(batch_shape, dtype=np.bool_))
# test inv
z = transform.inv(y)
assert_allclose(x, z, atol=1e-6, rtol=1e-6)
# test domain, currently all is constraints.real or constraints.real_vector
assert_array_equal(transform.domain(z), np.ones(batch_shape))
# test log_abs_det_jacobian
actual = transform.log_abs_det_jacobian(x, y)
assert np.shape(actual) == batch_shape
if len(shape) == transform.event_dim:
if constraint is constraints.simplex:
expected = onp.linalg.slogdet(
jax.jacobian(transform)(x)[:-1, :])[1]
inv_expected = onp.linalg.slogdet(
jax.jacobian(transform.inv)(y)[:, :-1])[1]
elif constraint is constraints.corr_cholesky:
vec_transform = lambda x: matrix_to_tril_vec(
transform(x), diagonal=-1) # noqa: E731
y_tril = matrix_to_tril_vec(y, diagonal=-1)
inv_vec_transform = lambda x: transform.inv(
vec_to_tril_matrix(x, diagonal=-1)) # noqa: E731
expected = onp.linalg.slogdet(jax.jacobian(vec_transform)(x))[1]
inv_expected = onp.linalg.slogdet(
jax.jacobian(inv_vec_transform)(y_tril))[1]
elif constraint is constraints.lower_cholesky:
vec_transform = lambda x: matrix_to_tril_vec(transform(x)
) # noqa: E731
y_tril = matrix_to_tril_vec(y)
inv_vec_transform = lambda x: transform.inv(vec_to_tril_matrix(x)
) # noqa: E731
expected = onp.linalg.slogdet(jax.jacobian(vec_transform)(x))[1]
inv_expected = onp.linalg.slogdet(
jax.jacobian(inv_vec_transform)(y_tril))[1]
else:
expected = np.log(np.abs(grad(transform)(x)))
inv_expected = np.log(np.abs(grad(transform.inv)(y)))
assert_allclose(actual, expected, atol=1e-6)
assert_allclose(actual, -inv_expected, atol=1e-6)
def _inverse(self, y):
# inverse stick-breaking
z1m_cumprod = 1 - jnp.cumsum(y * y, axis=-1)
pad_width = [(0, 0)] * y.ndim
pad_width[-1] = (1, 0)
z1m_cumprod_shifted = jnp.pad(z1m_cumprod[..., :-1], pad_width,
mode="constant", constant_values=1.)
t = matrix_to_tril_vec(y, diagonal=-1) / jnp.sqrt(
matrix_to_tril_vec(z1m_cumprod_shifted, diagonal=-1))
# inverse of tanh
x = jnp.log((1 + t) / (1 - t)) / 2
return x
def test_log_prob_LKJCholesky_uniform(dimension):
# When concentration=1, the distribution of correlation matrices is uniform.
# We will test that fact here.
d = dist.LKJCholesky(dimension=dimension, concentration=1)
N = 5
corr_log_prob = []
for i in range(N):
sample = d.sample(random.PRNGKey(i))
log_prob = d.log_prob(sample)
sample_tril = matrix_to_tril_vec(sample, diagonal=-1)
cholesky_to_corr_jac = onp.linalg.slogdet(
jax.jacobian(_tril_cholesky_to_tril_corr)(sample_tril))[1]
corr_log_prob.append(log_prob - cholesky_to_corr_jac)
corr_log_prob = np.array(corr_log_prob)
# test if they are constant
assert_allclose(corr_log_prob,
np.broadcast_to(corr_log_prob[0], corr_log_prob.shape),
rtol=1e-6)
if dimension == 2:
# when concentration = 1, LKJ gives a uniform distribution over correlation matrix,
# hence for the case dimension = 2,
# density of a correlation matrix will be Uniform(-1, 1) = 0.5.
# In addition, jacobian of the transformation from cholesky -> corr is 1 (hence its
# log value is 0) because the off-diagonal lower triangular element does not change
# in the transform.
# So target_log_prob = log(0.5)
assert_allclose(corr_log_prob[0], np.log(0.5), rtol=1e-6)
def log_abs_det_jacobian(self, x, y, intermediates=None):
# NB: because domain and codomain are two spaces with different dimensions, determinant of
# Jacobian is not well-defined. Here we return `log_abs_det_jacobian` of `x` and the
# flatten lower triangular part of `y`.
# stick_breaking_logdet = log(y / r) = log(z_cumprod) (modulo right shifted)
z1m_cumprod = 1 - jnp.cumsum(y * y, axis=-1)
# by taking diagonal=-2, we don't need to shift z_cumprod to the right
# NB: diagonal=-2 works fine for (2 x 2) matrix, where we get an empty array
z1m_cumprod_tril = matrix_to_tril_vec(z1m_cumprod, diagonal=-2)
stick_breaking_logdet = 0.5 * jnp.sum(jnp.log(z1m_cumprod_tril), axis=-1)
tanh_logdet = -2 * jnp.sum(x + softplus(-2 * x) - jnp.log(2.), axis=-1)
return stick_breaking_logdet + tanh_logdet
def __init__(self,
dimension,
concentration=1.,
sample_method='onion',
validate_args=None):
if dimension < 2:
raise ValueError("Dimension must be greater than or equal to 2.")
self.dimension = dimension
self.concentration = concentration
batch_shape = np.shape(concentration)
event_shape = (dimension, dimension)
# We construct base distributions to generate samples for each method.
# The purpose of this base distribution is to generate a distribution for
# correlation matrices which is propotional to `det(M)^{\eta - 1}`.
# (note that this is not a unique way to define base distribution)
# Both of the following methods have marginal distribution of each off-diagonal
# element of sampled correlation matrices is Beta(eta + (D-2) / 2, eta + (D-2) / 2)
# (up to a linear transform: x -> 2x - 1)
Dm1 = self.dimension - 1
marginal_concentration = concentration + 0.5 * (self.dimension - 2)
offset = 0.5 * np.arange(Dm1)
if sample_method == 'onion':
# The following construction follows from the algorithm in Section 3.2 of [1]:
# NB: in [1], the method for case k > 1 can also work for the case k = 1.
beta_concentration0 = np.expand_dims(marginal_concentration,
axis=-1) - offset
beta_concentration1 = offset + 0.5
self._beta = Beta(beta_concentration1, beta_concentration0)
elif sample_method == 'cvine':
# The following construction follows from the algorithm in Section 2.4 of [1]:
# offset_tril is [0, 1, 1, 2, 2, 2,...] / 2
offset_tril = matrix_to_tril_vec(
np.broadcast_to(offset, (Dm1, Dm1)))
beta_concentration = np.expand_dims(marginal_concentration,
axis=-1) - offset_tril
self._beta = Beta(beta_concentration, beta_concentration)
else:
raise ValueError("`method` should be one of 'cvine' or 'onion'.")
self.sample_method = sample_method
super(LKJCholesky, self).__init__(batch_shape=batch_shape,
event_shape=event_shape,
validate_args=validate_args)
def test_block_neural_arn(input_dim, hidden_factors, residual, batch_shape):
arn_init, arn = BlockNeuralAutoregressiveNN(input_dim, hidden_factors, residual)
rng = random.PRNGKey(0)
input_shape = batch_shape + (input_dim,)
out_shape, init_params = arn_init(rng, input_shape)
assert out_shape == input_shape
x = random.normal(random.PRNGKey(1), input_shape)
output, logdet = arn(init_params, x)
assert output.shape == input_shape
assert logdet.shape == input_shape
if len(batch_shape) == 1:
jac = vmap(jacfwd(lambda x: arn(init_params, x)[0]))(x)
else:
jac = jacfwd(lambda x: arn(init_params, x)[0])(x)
assert_allclose(logdet.sum(-1), np.linalg.slogdet(jac)[1], rtol=1e-6)
# make sure jacobians are lower triangular
assert onp.sum(onp.abs(onp.triu(jac, k=1))) == 0.0
assert onp.all(onp.abs(matrix_to_tril_vec(jac)) > 0)
def test_flows(flow_class, flow_args, input_dim, batch_shape):
transform = flow_class(*flow_args)
x = random.normal(random.PRNGKey(0), batch_shape + (input_dim, ))
# test inverse is correct
y = transform(x)
try:
inv = transform.inv(y)
assert_allclose(x, inv, atol=1e-5)
except NotImplementedError:
pass
# test jacobian shape
actual = transform.log_abs_det_jacobian(x, y)
assert onp.shape(actual) == batch_shape
if batch_shape == ():
# make sure transform.log_abs_det_jacobian is correct
jac = jacfwd(transform)(x)
expected = onp.linalg.slogdet(jac)[1]
assert_allclose(actual, expected, atol=1e-5)
# make sure jacobian is triangular, first permute jacobian as necessary
if isinstance(transform, InverseAutoregressiveTransform):
permuted_jac = onp.zeros(jac.shape)
_, rng_key_perm = random.split(random.PRNGKey(0))
perm = random.shuffle(rng_key_perm, onp.arange(input_dim))
for j in range(input_dim):
for k in range(input_dim):
permuted_jac[j, k] = jac[perm[j], perm[k]]
jac = permuted_jac
assert onp.sum(onp.abs(onp.triu(jac, 1))) == 0.00
assert onp.all(onp.abs(matrix_to_tril_vec(jac)) > 0)
def _inverse(self, y):
z = matrix_to_tril_vec(y, diagonal=-1)
diag = _softplus_inv(jnp.diagonal(y, axis1=-2, axis2=-1))
return jnp.concatenate([z, diag], axis=-1)
def _inverse(self, y):
z = matrix_to_tril_vec(y, diagonal=-1)
return jnp.concatenate([z, jnp.log(jnp.diagonal(y, axis1=-2, axis2=-1))], axis=-1)
def _tril_cholesky_to_tril_corr(x):
w = vec_to_tril_matrix(x, diagonal=-1)
diag = np.sqrt(1 - np.sum(w**2, axis=-1))
cholesky = w + np.expand_dims(diag, axis=-1) * np.identity(w.shape[-1])
corr = np.matmul(cholesky, cholesky.T)
return matrix_to_tril_vec(corr, diagonal=-1)
def inv(self, y):
z = matrix_to_tril_vec(y, diagonal=-1)
return np.concatenate(
[z, np.log(np.diagonal(y, axis1=-2, axis2=-1))], axis=-1)
def test_biject_to(constraint, shape):
transform = biject_to(constraint)
if transform.event_dim == 2:
event_dim = 1 # actual dim of unconstrained domain
else:
event_dim = transform.event_dim
if isinstance(constraint, constraints._Interval):
assert transform.codomain.upper_bound == constraint.upper_bound
assert transform.codomain.lower_bound == constraint.lower_bound
elif isinstance(constraint, constraints._GreaterThan):
assert transform.codomain.lower_bound == constraint.lower_bound
if len(shape) < event_dim:
return
rng_key = random.PRNGKey(0)
x = random.normal(rng_key, shape)
y = transform(x)
# test codomain
batch_shape = shape if event_dim == 0 else shape[:-1]
assert_array_equal(transform.codomain(y), np.ones(batch_shape, dtype=np.bool_))
# test inv
z = transform.inv(y)
assert_allclose(x, z, atol=1e-6, rtol=1e-6)
# test domain, currently all is constraints.real or constraints.real_vector
assert_array_equal(transform.domain(z), np.ones(batch_shape))
# test log_abs_det_jacobian
actual = transform.log_abs_det_jacobian(x, y)
assert np.shape(actual) == batch_shape
if len(shape) == event_dim:
if constraint is constraints.simplex:
expected = onp.linalg.slogdet(jax.jacobian(transform)(x)[:-1, :])[1]
inv_expected = onp.linalg.slogdet(jax.jacobian(transform.inv)(y)[:, :-1])[1]
elif constraint is constraints.ordered_vector:
expected = onp.linalg.slogdet(jax.jacobian(transform)(x))[1]
inv_expected = onp.linalg.slogdet(jax.jacobian(transform.inv)(y))[1]
elif constraint in [constraints.corr_cholesky, constraints.corr_matrix]:
vec_transform = lambda x: matrix_to_tril_vec(transform(x), diagonal=-1) # noqa: E731
y_tril = matrix_to_tril_vec(y, diagonal=-1)
def inv_vec_transform(y):
matrix = vec_to_tril_matrix(y, diagonal=-1)
if constraint is constraints.corr_matrix:
# fill the upper triangular part
matrix = matrix + np.swapaxes(matrix, -2, -1) + np.identity(matrix.shape[-1])
return transform.inv(matrix)
expected = onp.linalg.slogdet(jax.jacobian(vec_transform)(x))[1]
inv_expected = onp.linalg.slogdet(jax.jacobian(inv_vec_transform)(y_tril))[1]
elif constraint in [constraints.lower_cholesky, constraints.positive_definite]:
vec_transform = lambda x: matrix_to_tril_vec(transform(x)) # noqa: E731
y_tril = matrix_to_tril_vec(y)
def inv_vec_transform(y):
matrix = vec_to_tril_matrix(y)
if constraint is constraints.positive_definite:
# fill the upper triangular part
matrix = matrix + np.swapaxes(matrix, -2, -1) - np.diag(np.diag(matrix))
return transform.inv(matrix)
expected = onp.linalg.slogdet(jax.jacobian(vec_transform)(x))[1]
inv_expected = onp.linalg.slogdet(jax.jacobian(inv_vec_transform)(y_tril))[1]
else:
expected = np.log(np.abs(grad(transform)(x)))
inv_expected = np.log(np.abs(grad(transform.inv)(y)))
assert_allclose(actual, expected, atol=1e-6, rtol=1e-6)
assert_allclose(actual, -inv_expected, atol=1e-6, rtol=1e-6)
def _inverse(self, y):
diag = jnp.diagonal(y, axis1=-2, axis2=-1)
z = matrix_to_tril_vec(y / diag[..., None], diagonal=-1)
return jnp.concatenate([z, _softplus_inv(diag)], axis=-1)