def test_multivariate_gaussian() -> None:
num_samples = 2000
dim = 2
mu = np.arange(0, dim) / float(dim)
L_diag = np.ones((dim, ))
L_low = 0.1 * np.ones((dim, dim)) * np.tri(dim, k=-1)
L = np.diag(L_diag) + L_low
Sigma = L.dot(L.transpose())
distr = MultivariateGaussian(mu=mx.nd.array(mu), L=mx.nd.array(L))
samples = distr.sample(num_samples)
mu_hat, L_hat = maximum_likelihood_estimate_sgd(
MultivariateGaussianOutput(dim=dim),
samples,
init_biases=
None, # todo we would need to rework biases a bit to use it in the multivariate case
hybridize=False,
learning_rate=PositiveFloat(0.01),
num_epochs=PositiveInt(10),
)
distr = MultivariateGaussian(mu=mx.nd.array([mu_hat]),
L=mx.nd.array([L_hat]))
Sigma_hat = distr.variance[0].asnumpy()
assert np.allclose(
mu_hat, mu, atol=0.1,
rtol=0.1), f"mu did not match: mu = {mu}, mu_hat = {mu_hat}"
assert np.allclose(
Sigma_hat, Sigma, atol=0.1, rtol=0.1
), f"Sigma did not match: sigma = {Sigma}, sigma_hat = {Sigma_hat}"
def sample(self,
num_samples: Optional[int] = None,
scale: Optional[Tensor] = None) -> Tensor:
r"""
Generates samples from the LDS: p(z_1, z_2, \ldots, z_{`seq_length`}).
Parameters
----------
num_samples
Number of samples to generate
scale
Scale of each sequence in x, shape (batch_size, output_dim)
Returns
-------
Tensor
Samples, shape (num_samples, batch_size, seq_length, output_dim)
"""
F = self.F
# Note on shapes: here we work with tensors of the following shape
# in each time step t: (num_samples, batch_size, dim, dim),
# where dim can be obs_dim or latent_dim or a constant 1 to facilitate
# generalized matrix multiplication (gemm2)
# Sample observation noise for all time steps
# noise_std: (batch_size, seq_length, obs_dim, 1)
noise_std = F.stack(*self.noise_std, axis=1).expand_dims(axis=-1)
# samples_eps_obs[t]: (num_samples, batch_size, obs_dim, 1)
samples_eps_obs = (Gaussian(noise_std.zeros_like(),
noise_std).sample(num_samples).split(
axis=-3,
num_outputs=self.seq_length,
squeeze_axis=True))
# Sample standard normal for all time steps
# samples_eps_std_normal[t]: (num_samples, batch_size, obs_dim, 1)
samples_std_normal = (Gaussian(
noise_std.zeros_like(),
noise_std.ones_like()).sample(num_samples).split(
axis=-3, num_outputs=self.seq_length, squeeze_axis=True))
# Sample the prior state.
# samples_lat_state: (num_samples, batch_size, latent_dim, 1)
# The prior covariance is observed to be slightly negative definite whenever there is
# excessive zero padding at the beginning of the time series.
# We add positive tolerance to the diagonal to avoid numerical issues.
# Note that `jitter_cholesky` adds positive tolerance only if the decomposition without jitter fails.
state = MultivariateGaussian(
self.prior_mean,
jitter_cholesky(F,
self.prior_cov,
self.latent_dim,
float_type=np.float32),
)
samples_lat_state = state.sample(num_samples).expand_dims(axis=-1)
samples_seq = []
for t in range(self.seq_length):
# Expand all coefficients to include samples in axis 0
# emission_coeff_t: (num_samples, batch_size, obs_dim, latent_dim)
# transition_coeff_t:
# (num_samples, batch_size, latent_dim, latent_dim)
# innovation_coeff_t: (num_samples, batch_size, 1, latent_dim)
emission_coeff_t, transition_coeff_t, innovation_coeff_t = [
_broadcast_param(coeff, axes=[0], sizes=[num_samples])
if num_samples is not None else coeff for coeff in [
self.emission_coeff[t],
self.transition_coeff[t],
self.innovation_coeff[t],
]
]
# Expand residuals as well
# residual_t: (num_samples, batch_size, obs_dim, 1)
residual_t = (_broadcast_param(
self.residuals[t].expand_dims(axis=-1),
axes=[0],
sizes=[num_samples],
) if num_samples is not None else self.residuals[t].expand_dims(
axis=-1))
# (num_samples, batch_size, 1, obs_dim)
samples_t = (F.linalg_gemm2(emission_coeff_t, samples_lat_state) +
residual_t + samples_eps_obs[t])
samples_t = (samples_t.swapaxes(dim1=2, dim2=3) if num_samples
is not None else samples_t.swapaxes(dim1=1, dim2=2))
samples_seq.append(samples_t)
# sample next state: (num_samples, batch_size, latent_dim, 1)
samples_lat_state = F.linalg_gemm2(
transition_coeff_t, samples_lat_state) + F.linalg_gemm2(
innovation_coeff_t,
samples_std_normal[t],
transpose_a=True)
# (num_samples, batch_size, seq_length, obs_dim)
samples = F.concat(*samples_seq, dim=-2)
return (samples if scale is None else F.broadcast_mul(
samples,
scale.expand_dims(axis=1).expand_dims(
axis=0) if num_samples is not None else scale.expand_dims(
axis=1),
))
def sample_marginals(self,
num_samples: Optional[int] = None,
scale: Optional[Tensor] = None) -> Tensor:
r"""
Generates samples from the marginals p(z_t),
t = 1, \ldots, `seq_length`.
Parameters
----------
num_samples
Number of samples to generate
scale
Scale of each sequence in x, shape (batch_size, output_dim)
Returns
-------
Tensor
Samples, shape (num_samples, batch_size, seq_length, output_dim)
"""
F = self.F
state_mean = self.prior_mean.expand_dims(axis=-1)
state_cov = self.prior_cov
output_mean_seq = []
output_cov_seq = []
for t in range(self.seq_length):
# compute and store observation mean at time t
output_mean = F.linalg_gemm2(
self.emission_coeff[t],
state_mean) + self.residuals[t].expand_dims(axis=-1)
output_mean_seq.append(output_mean)
# compute and store observation cov at time t
output_cov = F.linalg_gemm2(
self.emission_coeff[t],
F.linalg_gemm2(
state_cov, self.emission_coeff[t], transpose_b=True),
) + make_nd_diag(F=F,
x=self.noise_std[t] * self.noise_std[t],
d=self.output_dim)
output_cov_seq.append(output_cov.expand_dims(axis=1))
state_mean = F.linalg_gemm2(self.transition_coeff[t], state_mean)
state_cov = F.linalg_gemm2(
self.transition_coeff[t],
F.linalg_gemm2(
state_cov, self.transition_coeff[t], transpose_b=True),
) + F.linalg_gemm2(
self.innovation_coeff[t],
self.innovation_coeff[t],
transpose_a=True,
)
output_mean = F.concat(*output_mean_seq, dim=1)
output_cov = F.concat(*output_cov_seq, dim=1)
L = F.linalg_potrf(output_cov)
output_distribution = MultivariateGaussian(output_mean, L)
samples = output_distribution.sample(num_samples=num_samples)
return (samples if scale is None else F.broadcast_mul(
samples, scale.expand_dims(axis=1)))
def sample(
self, num_samples: Optional[int] = None, scale: Optional[Tensor] = None
) -> Tensor:
r"""
Generates samples from the LDS: p(z_1, z_2, \ldots, z_{`seq_length`}).
Parameters
----------
num_samples
Number of samples to generate
scale
Scale of each sequence in x, shape (batch_size, output_dim)
Returns
-------
Tensor
Samples, shape (num_samples, batch_size, seq_length, output_dim)
"""
F = self.F
# Note on shapes: here we work with tensors of the following shape
# in each time step t: (num_samples, batch_size, dim, dim),
# where dim can be obs_dim or latent_dim or a constant 1 to facilitate
# generalized matrix multiplication (gemm2)
# Sample observation noise for all time steps
# noise_std: (batch_size, seq_length, obs_dim, 1)
noise_std = F.stack(*self.noise_std, axis=1).expand_dims(axis=-1)
# samples_eps_obs[t]: (num_samples, batch_size, obs_dim, 1)
samples_eps_obs = (
Gaussian(noise_std.zeros_like(), noise_std)
.sample(num_samples)
.split(axis=2, num_outputs=self.seq_length, squeeze_axis=True)
)
# Sample standard normal for all time steps
# samples_eps_std_normal[t]: (num_samples, batch_size, obs_dim, 1)
samples_std_normal = (
Gaussian(noise_std.zeros_like(), noise_std.ones_like())
.sample(num_samples)
.split(axis=2, num_outputs=self.seq_length, squeeze_axis=True)
)
# Sample the prior state.
# samples_lat_state: (num_samples, batch_size, latent_dim, 1)
state = MultivariateGaussian(
self.prior_mean, F.linalg_potrf(self.prior_cov)
)
samples_lat_state = state.sample(num_samples).expand_dims(axis=-1)
samples_seq = []
for t in range(self.seq_length):
# Expand all coefficients to include samples in axis 0
# emission_coeff_t: (num_samples, batch_size, obs_dim, latent_dim)
# transition_coeff_t:
# (num_samples, batch_size, latent_dim, latent_dim)
# innovation_coeff_t: (num_samples, batch_size, 1, latent_dim)
emission_coeff_t, transition_coeff_t, innovation_coeff_t = [
_broadcast_param(coeff, axes=[0], sizes=[num_samples])
for coeff in [
self.emission_coeff[t],
self.transition_coeff[t],
self.innovation_coeff[t],
]
]
# Expand residuals as well
# residual_t: (num_samples, batch_size, obs_dim, 1)
residual_t = _broadcast_param(
self.residuals[t].expand_dims(axis=-1),
axes=[0],
sizes=[num_samples],
)
# (num_samples, batch_size, 1, obs_dim)
samples_t = (
F.linalg_gemm2(emission_coeff_t, samples_lat_state)
+ residual_t
+ samples_eps_obs[t]
).swapaxes(dim1=2, dim2=3)
samples_seq.append(samples_t)
# sample next state: (num_samples, batch_size, latent_dim, 1)
samples_lat_state = F.linalg_gemm2(
transition_coeff_t, samples_lat_state
) + F.linalg_gemm2(
innovation_coeff_t, samples_std_normal[t], transpose_a=True
)
# (num_samples, batch_size, seq_length, obs_dim)
samples = F.concat(*samples_seq, dim=2)
return (
samples
if scale is None
else F.broadcast_mul(samples, scale.expand_dims(axis=1))
)