Source code for gmmvi.models.gmm

import tensorflow as tf
from math import pi


[docs]class GMM: """An abstract class for Gaussian mixture models (GMMs). This class stores the parameters of a GMM (weights, means and Cholesky matrices) and provides functionality that is common for different types of GMMs (e.g., GMMs with full covariance matrices, and those with diagonal covariance matrices). For example, this class provides methods for sampling the *GMM*, evaluating its probability density function, and entropy, while relying on the subclass for sampling the *components*, etc. Parameters: log_weights: tf.Variable(float) A tensorflow Variable for storing the log-probabilities of the component weights. means: tf.Variable(float) A tensorflow Variable for storing the component means (number of components X number of dimensions) chol_covar: tf.Variable(float) A tensorflow Variable for storing the Cholesky matrix of the component's covariance matrix. The first dimension specifies the index of the components. The rank may vary depending on the subclass. For example, when storing the Cholesky matrix for a diagonal covariance matrix, it is possible to use a rank-2 Tensor (number of components X number of dimensions) for better memory efficiency. """ def __init__(self, log_weights: tf.Variable, means: tf.Variable, chol_covs: tf.Variable): self.diagonal_covs = tf.rank(chol_covs) == 2 self.num_dimensions = int(tf.shape(means)[1]) self._const_log_det = tf.constant(0.5 * self.num_dimensions * tf.math.log(2 * pi)) self.log_weights = log_weights self.means = means self.chol_cov = chol_covs self.replace_weights(self.log_weights)
[docs] def sample_from_component(self, index: int, num_samples: int) -> tf.Tensor: """ draw samples from the specified components Parameters: index: int The index of the component from which we want to sample. num_samples: int The number of samples to be drawn. Returns: tf.Tensor: The drawn samples, tensor of size num_samples x dimensions. """ raise NotImplementedError
[docs] def component_log_density(self, index: int, samples: tf.Tensor) -> tf.Tensor: """ Use the specified component to evaluate the Gaussian log-density at the given samples. Parameters: index: int The index of the component of which we want to compute the log densities. samples: tf.Tensor A two-dimensional tensor of shape number_of_samples x num_dimensions, which we want to evaluate. Returns: tf.Tensor: a one-dimensional tensor of size number_of_samples, containing the log-densities. """ raise NotImplementedError
[docs] def component_log_densities(self, samples: tf.Tensor) -> tf.Tensor: """ Evaluate the log densities for each mixture component on the given samples. Parameters: samples: tf.Tensor A two-dimensional tensor of shape number_of_samples x num_dimensions, which we want to evaluate. Returns: tf.Tensor: a two-dimensional tensor of size number_of_components x number_of_samples, containing the log-densities for each component. """ raise NotImplementedError
[docs] def component_marginal_log_densities(self, samples: tf.Tensor, dimension: int) -> tf.Tensor: """ Evaluate the marginal log densities for each mixture component along the given dimension for each sample. Parameters: samples: tf.Tensor A two-dimensional tensor of shape number_of_samples x num_dimensions, which we want to evaluate. Note that for providing an easier interface, each sample has the number of dimensions compatible with this GMM, although only a single entry is actually used for evaluating the marginal density. dimension: int The dimension of interest. Returns: tf.Tensor: a two-dimensional tensor of size number_of_components x number_of_samples, containing the marginal log-densities for each component. """ raise NotImplementedError
[docs] def gaussian_entropy(self, chol: tf.Tensor) -> tf.Tensor: """ Computes the entropy of Gaussian distribution with the given Cholesky matrix. Parameters: chol: tf.Tensor A two-dimensional tensor of shape number_of_dimensions x number_of_dimensions specifying the Cholesky matrix Returns: tf.float32: The entropy """ raise NotImplementedError
[docs] def add_component(self, initial_weight: tf.Tensor, initial_mean: tf.Tensor, initial_cov: tf.Tensor): """ Add a component to the mixture model. The weights will be automatically normalized. Parameters: initial_weight: tf.Tensor The weight of the new component (before re-normalization) initial_mean: tf.Tensor The mean of the new component initial_cov: tf.Tensor The covariance matrix of the new component """ raise NotImplementedError
[docs] def sample_categorical(self, num_samples: int) -> tf.Tensor: """ Sample components according to the weights Parameters: num_samples : int The number of components to be drawn Returns: tf.Tensor: a one-dimensional tensor of int, containing the component indices. """ thresholds = tf.expand_dims(tf.cumsum(self.weights), 0) eps = tf.random.uniform(shape=[num_samples, 1]) samples = tf.argmax(eps < thresholds, axis=-1, output_type=tf.int32) return samples
[docs] def sample(self, num_samples: int) -> [tf.Tensor, tf.Tensor]: """ Draw samples from this GMM, also returns for every sample, the index of the component that was used for sampling it. Parameters: num_samples: int The number of samples to be drawn Returns: tuple(tf.Tensor, tf.Tensor): **drawn_samples** - a two-dimensional tensor of shape num_samples x num_dimensions, containing the drawn samples. **component_indices** - a one-dimensional tensor of int, containing the component indices. """ sampled_components = self.sample_categorical(num_samples=num_samples) samples = tf.TensorArray(tf.float32, size=self.num_components, infer_shape=False) for i in range(self.num_components): n_samples = tf.reduce_sum(tf.cast(sampled_components == i, tf.int32)) this_samples = self.sample_from_component(i, n_samples) samples = samples.write(i, this_samples) samples = samples.concat() return samples, sampled_components
@property def weights(self) -> tf.Tensor: """ Returns: tf.Tensor: a one-dimensional tensor of size num_components, containing the component weights. """ return tf.math.exp(self.log_weights)
[docs] def replace_weights(self, new_log_weights: tf.Tensor): """ Overwrites the component log(weights). This method will take care of normalization. Parameters: new_log_weights: tf.Tensor a one-dimensional tensor of size num_components, containing the new log(weights) """ self.log_weights.assign(new_log_weights - tf.reduce_logsumexp(new_log_weights))
[docs] def log_densities_also_individual(self, samples: tf.Tensor) -> [tf.Tensor, tf.Tensor]: """ Evaluates the given samples on this GMM, but also returns individual log-densities for each Gaussian component. Parameters: samples: tf.Tensor A two-dimensional tensor of shape number_of_samples x num_dimensions, which we want to evaluate Returns: tuple(tf.Tensor, tf.Tensor): **model_log_densities** - a one-dimensional tnsor of shape num_samples containing the model log-densities. **component_log_densities** - a two-dimensional tensor of shape num_components x num_samples containing the component log-densities """ component_log_densities = self.component_log_densities(samples) weighted_densities = component_log_densities + tf.expand_dims(self.log_weights, axis=1) return tf.reduce_logsumexp(weighted_densities, axis=0), component_log_densities
[docs] def log_density(self, samples: tf.Tensor) -> tf.Tensor: """ Evaluates the given samples on this GMM. Parameters: samples: tf.Tensor A two-dimensional tensor of shape number_of_samples x num_dimensions, which we want to evaluate. Returns: tf.Tensor: a one-dimensional tensor of shape num_samples containing the model log-densities. """ log_densities = self.component_log_densities(samples) weighted_densities = log_densities + tf.expand_dims(self.log_weights, axis=1) return tf.reduce_logsumexp(weighted_densities, axis=0)
[docs] def marginal_log_density(self, samples: tf.Tensor, dimension: int) -> tf.Tensor: """ Evaluates this GMM on the given samples with respect to the marginal log-density along the given dimensions Parameters: samples: tf.Tensor A two-dimensional tensor of shape number_of_samples x num_dimensions, which we want to evaluate dimension: int The dimension of interest Returns: tf.Tensor: a one-dimensional tensor of shape num_samples containing the marginal log-densities """ log_densities = self.component_marginal_log_densities(samples, dimension) weighted_densities = log_densities + tf.expand_dims(self.log_weights, axis=1) return tf.reduce_logsumexp(weighted_densities, axis=0)
[docs] def density(self, samples: tf.Tensor) -> tf.Tensor: """ Evaluates the given samples on this GMM. Parameters: samples: tf.Tensor A two-dimensional tensor of shape number_of_samples x num_dimensions, which we want to evaluate Returns: tf.Tensor: a one-dimensional tensor of shape num_samples containing the model densities. """ return tf.exp(self.log_density(samples))
[docs] def component_entropies(self) -> tf.Tensor: """ Computes the entropy of each component. Returns: tf.Tensor: a one-dimensional tensor of shape num_components containing the component entropies. """ entropies = tf.TensorArray(tf.float32, size=self.num_components) for i in range(self.num_components): this_entropy = self.gaussian_entropy(self.chol_cov[i]) entropies = entropies.write(i, this_entropy) return entropies.stack()
[docs] def get_average_entropy(self) -> tf.float32: """ Averages the entropies of the individual components based on their respective weights. Returns: tf.float32: the average component entropy """ avg_entropy = 0. for i in range(self.num_components): avg_entropy += tf.exp(self.log_weights[i]) * self.gaussian_entropy(self.chol_cov[i]) return avg_entropy
[docs] def log_density_and_grad(self, samples: tf.Tensor) -> [tf.Tensor, tf.Tensor, tf.Tensor]: """ Evaluates the given samples on this GMM, returns the log-densities of the whole model, their gradients, and also the individual log-densities for each Gaussian component. Parameters: samples: tf.Tensor A two-dimensional tensor of shape number_of_samples x num_dimensions, which we want to evaluate. Returns: tuple(tf.Tensor, tf.Tensor): **model_log_densities** - a one-dimensional tensor of shape num_samples containing the model log-densities. **model_log_density_grads** - a two-dimensional tf.Tensor of shape num_samples x num_dimensions containing the gradients of the model log-densities. **component_log_densities** - a two-dimensional tf.Tensor of shape num_components x num_samples containing the component log-densities. """ with tf.GradientTape() as gfg: gfg.watch(samples) log_component_densities = self.component_log_densities(samples) log_densities = log_component_densities + tf.expand_dims(self.log_weights, axis=1) log_densities = tf.reduce_logsumexp(log_densities, axis=0) log_densities_grad = gfg.gradient(log_densities, samples) return log_densities, log_densities_grad, log_component_densities
[docs] def component_log_density_and_grad(self, index: int, samples: tf.Tensor) -> [tf.Tensor, tf.Tensor]: """ Evaluates for the given component the log-density and its gradient. Parameters: samples: tf.Tensor A two-dimensional tensor of shape number_of_samples x num_dimensions, which we want to evaluate. Returns: **component_log_densities** - a one-dimensional tf.Tensor of shape num_samples containing the log-densities of the given component. **component_log_density_grads** - a two-dimensional tf.Tensor of shape num_samples x num_dimensions containing the gradients of the component's log-densities. """ with tf.GradientTape() as gfg: gfg.watch(samples) log_component_density = self.component_log_density(index, samples) log_density_grad = gfg.gradient(log_component_density, samples) return log_component_density, log_density_grad
@property def covs(self) -> tf.Tensor: """ Returns: tf.Tensor: the covariance matrices as a 3-dimensional tensor of shape num_components x num_dimensions x num_dimensions """ raise NotImplementedError @property def num_components(self) -> int: """ Returns: int: the number of components of this GMM """ return tf.shape(self.log_weights)[0]
[docs] def sample_from_components(self, samples_per_component: tf.Tensor) -> tf.Tensor: """ Draws from each component the corresponding number of samples (provided as a one-dimensional tensor). Parameters: samples_per_component : tf.Tensor a one-dimensional tensor of size number_of_component, containint for each component the number of samples to be drawn. Returns: tf.Tensor: a tensor of shape sum(samples_per_component) x num_dimensions containing the samples (shuffled). """ samples = tf.TensorArray(tf.float32, size=self.num_components) for i in range(self.num_components): this_samples = self.sample_from_component(i, samples_per_component[i]) samples = samples.write(i, this_samples) samples = tf.reshape(samples.stack(), [-1, self.num_dimensions]) chosen_indices = tf.random.shuffle(tf.range(tf.shape(samples)[0])) return tf.gather(samples, chosen_indices)
[docs] @tf.function(input_signature=[tf.TensorSpec(shape=[None], dtype=tf.int32)]) def sample_from_components_no_shuffle(self, samples_per_component: tf.Tensor) -> [tf.Tensor, tf.Tensor]: """ Draws from each component the corresponding number of samples (provided as a one-dimensional tensor). Similar to :py:meth:`sample_from_components<gmmvi.models.gmm.GMM.sample_from_components>`, but the returned samples are not shuffled. Parameters: samples_per_component: tf.Tensor a one-dimensional tensor of size number_of_component, containint for each component the number of samples to be drawn. Returns: tf.Tensor: a tensor of shape sum(samples_per_component) x num_dimensions containing the samples (not shuffled). """ mapping = tf.repeat( tf.range(self.num_components), samples_per_component ) samples = tf.TensorArray(tf.float32, size=self.num_components) for i in range(self.num_components): this_samples = self.sample_from_component(i, samples_per_component[i]) samples = samples.write(i, this_samples) samples = samples.concat() return samples, mapping
[docs] def remove_component(self, idx: int): """ Removes the specified component, and renormalizes the weights afterwards. Parameters: idx: int The idx of the component to be removed. """ self.replace_weights(tf.concat((self.log_weights[:idx], self.log_weights[idx + 1:]), axis=0)) self.means.assign(tf.concat((self.means[:idx], self.means[idx + 1:]), axis=0)) self.chol_cov.assign(tf.concat((self.chol_cov[:idx], self.chol_cov[idx + 1:]), axis=0))
[docs] def replace_components(self, new_means: tf.Tensor, new_chols: tf.Tensor): """ Updates the means and covariances matrices (Cholesky) of the GMM. The weights and, therefore, the number of components can not be changed with this method. Parameters: new_means: tf.Tensor a two-dimensional tensor of shape current_number_of_components x dimensions, specifying the updated means. new_chols: tf.Tensor a three-dimensional tensor of shape current_number_of_components x dimensions x dimensions, specifying the updated Cholesky matrix. """ new_means = tf.stack(new_means, axis=0) new_chols = tf.stack(new_chols, axis=0) self.means.assign(new_means) self.chol_cov.assign(new_chols)