def special_rbf_kernel(X, Y, gamma, norm_X, norm_Y):
"""
Rbf kernel expressed under the form f(x)f(u)f(xy^T)
Can handle X and Y as Sparse Factors.
:param X: n x d matrix
:param Y: n x d matrix
:return:
"""
assert len(X.shape) == len(Y.shape) == 2
if norm_X is None:
norm_X = get_squared_froebenius_norm_line_wise(X)
if norm_Y is None:
norm_Y = get_squared_froebenius_norm_line_wise(Y)
def f(norm_mat):
return np.exp(-gamma * norm_mat)
def g(scal):
return np.exp(2 * gamma * scal)
if isinstance(X, SparseFactors) and isinstance(Y, SparseFactors):
# xyt = SparseFactors(X.get_list_of_factors() + Y.transpose().get_list_of_factors()).compute_product(return_array=True)
S = SparseFactors(
lst_factors=X.get_list_of_factors() + Y.get_list_of_factors_H(),
lst_factors_H=X.get_list_of_factors_H() + Y.get_list_of_factors())
xyt = S.compute_product(return_array=True)
else:
xyt = X @ Y.transpose()
return f(norm_X).reshape(-1, 1) * g(xyt) * f(norm_Y).reshape(1, -1)
def make_batch_assignation_evaluation(X, centroids):
"""
Assign `size_batch` random samples of `X` to some of the centroids.
All the samples are assigned at the same time using a matrix-vector multiplication.
Time is recorded.
:param X: The input data from which to take the samples.
:param centroids: The centroids to which to assign the samples (must be of same dimension than `X`)
:param size_batch: The number of data points to assign
:return: None
"""
size_batch = paraman["--batch-assignation-time"]
if size_batch > X.shape[0]:
logger.warning("Batch size for batch assignation evaluation is bigger than data size. {} > {}. Using "
"data size instead.".format(size_batch, X.shape[0]))
size_batch = X.shape[0]
paraman["--batch-assignation-time"] = size_batch
precomputed_centroid_norms = get_squared_froebenius_norm_line_wise(centroids)
# precomputed_centroid_norms = None
indexes_batch = np.random.permutation(X.shape[0])[:size_batch]
start_time = time.process_time()
_ = get_distances(X[indexes_batch], centroids, precomputed_centroids_norm=precomputed_centroid_norms)
stop_time = time.process_time()
resprinter.add({
"batch_assignation_mean_time": (stop_time-start_time) / size_batch,
})
def make_assignation_evaluation(X, centroids):
nb_eval = paraman["--assignation-time"]
if nb_eval > X.shape[0]:
logger.warning(
"Batch size for assignation evaluation is bigger than data size. {} > {}. Using "
"data size instead.".format(nb_eval, X.shape[0]))
nb_eval = X.shape[0]
paraman["--assignation-time"] = nb_eval
times = []
precomputed_centroid_norms = get_squared_froebenius_norm_line_wise(
centroids)
for i in np.random.permutation(X.shape[0])[:nb_eval]:
start_time = time.time()
get_distances(X[i].reshape(1, -1),
centroids,
precomputed_centroids_norm=precomputed_centroid_norms)
stop_time = time.time()
times.append(stop_time - start_time)
mean_time = np.mean(times)
std_time = np.std(times)
resprinter.add({
"assignation_mean_time": mean_time,
"assignation_std_time": std_time
})
def kmean_tree_evaluation():
"""
Do the K-means partitioning version of nearest neighbor?=.
:return:
"""
# for each cluster, there is a sub nearest neighbor classifier for points in that cluster.
lst_clf_by_cluster = [KNeighborsClassifier(n_neighbors=1, algorithm="brute").fit(x_train[indicator_vector == i], y_train[indicator_vector == i]) for i in range(U_centroids.shape[0])]
log_memory_usage("Memory after definition of neighbors classifiers in kmean_tree_evaluation of make_1nn_evaluation")
precomputed_centroid_norms = get_squared_froebenius_norm_line_wise(U_centroids)
# precomputed_centroid_norms = None
start_inference_time = time.process_time()
distances = get_distances(x_test, U_centroids, precomputed_centroids_norm=precomputed_centroid_norms)
stop_get_distances_time = time.process_time()
get_distance_time = stop_get_distances_time - start_inference_time
indicator_vector_test = np.argmin(distances, axis=1)
predictions = np.empty_like(y_test)
for obs_idx, obs_test in enumerate(x_test):
# get the cluster to which belongs this data point and call the associated nearest neighbor classifier
idx_cluster = indicator_vector_test[obs_idx]
clf_cluster = lst_clf_by_cluster[idx_cluster]
predictions[obs_idx] = clf_cluster.predict(obs_test.reshape(1, -1))[0]
stop_inference_time = time.process_time()
log_memory_usage("Memory after label assignation in kmean_tree_evaluation of make_1nn_evaluation")
inference_time = (stop_inference_time - start_inference_time)
accuracy = np.sum(predictions == y_test) / y_test.shape[0]
results_1nn = {
"1nn_kmean_inference_time": inference_time,
"1nn_get_distance_time": get_distance_time / x_test.shape[0],
"1nn_kmean_accuracy": accuracy
}
resprinter.add(results_1nn)
return inference_time
def make_nystrom_evaluation(x_train, y_train, x_test, y_test, gamma, landmarks):
# verify sample size for evaluation
n_sample = paraman["--nystrom"]
if n_sample > x_train.shape[0]:
logger.warning("Batch size for nystrom evaluation is bigger than data size. {} > {}. Using "
"data size instead.".format(n_sample, x_train.shape[0]))
n_sample = x_train.shape[0]
paraman["--nystrom"] = n_sample
indexes_samples = np.random.permutation(x_train.shape[0])[:n_sample]
sample = x_train[indexes_samples]
samples_norm = None
# Make nystrom approximation
# nys_obj = Nystroem(gamma=gamma, n_components=landmarks.shape[0])
# nys_obj.fit(landmarks)
# nystrom_embedding = nys_obj.transform(sample)
landmarks_norm = get_squared_froebenius_norm_line_wise(landmarks)[:, np.newaxis]
metric = prepare_nystrom(landmarks, landmarks_norm, gamma=gamma)
nystrom_embedding = nystrom_transformation(sample, landmarks, metric, landmarks_norm, samples_norm, gamma=gamma)
nystrom_approx_kernel_value = nystrom_embedding @ nystrom_embedding.T
# Create real kernel matrix
real_kernel_special = special_rbf_kernel(sample, sample, gamma, norm_X=samples_norm, norm_Y=samples_norm)
# real_kernel = rbf_kernel(sample, sample, gamma)
real_kernel_norm = np.linalg.norm(real_kernel_special)
# evaluation reconstruction error
reconstruction_error_nystrom = np.linalg.norm(nystrom_approx_kernel_value - real_kernel_special) / real_kernel_norm
# start svm + nystrom classification
if x_test is not None:
logger.info("Start classification")
x_train_nystrom_embedding = nystrom_transformation(x_train, landmarks, metric, landmarks_norm, None, gamma=gamma)
x_test_nystrom_embedding = nystrom_transformation(x_test, landmarks, metric, landmarks_norm, None, gamma=gamma)
linear_svc_clf = LinearSVC(class_weight="balanced")
linear_svc_clf.fit(x_train_nystrom_embedding, y_train)
predictions = linear_svc_clf.predict(x_test_nystrom_embedding)
if paraman["--kddcup04"]:
# compute recall: nb_true_positive/real_nb_positive
recall = np.sum(predictions[y_test == 1])/np.sum(y_test[y_test == 1])
# compute precision: nb_true_positive/nb_positive
precision = np.sum(predictions[y_test == 1])/np.sum(predictions[predictions==1])
f1 = 2 * precision * recall / (precision + recall)
accuracy_nystrom_svm = f1
else:
accuracy_nystrom_svm = np.sum(predictions == y_test) / y_test.shape[0]
else:
accuracy_nystrom_svm = None
return reconstruction_error_nystrom, accuracy_nystrom_svm
def kmean_tree_evaluation():
"""
Do the K-means partitioning version of nearest neighbor?=.
:return:
"""
# for each cluster, there is a sub nearest neighbor classifier for points in that cluster.
lst_clf_by_cluster = []
indices_no_train_obs_in_cluster = []
for i in range(U_centroids.shape[0]):
try:
lst_clf_by_cluster.append(KNeighborsClassifier(n_neighbors=1, algorithm="brute").fit(x_train[indicator_vector == i], y_train[indicator_vector == i]))
except ValueError:
indices_no_train_obs_in_cluster.append(i)
lst_clf_by_cluster.append(None)
# lst_clf_by_cluster = [ for i in range(landmarks.shape[0])]
log_memory_usage("Memory after definition of neighbors classifiers in kmean_tree_evaluation of make_1nn_evaluation")
precomputed_centroid_norms = get_squared_froebenius_norm_line_wise(U_centroids)
# precomputed_centroid_norms = None
start_inference_time = time.process_time()
distances = get_distances(x_test, U_centroids, precomputed_centroids_norm=precomputed_centroid_norms)
stop_get_distances_time = time.process_time()
get_distance_time = stop_get_distances_time - start_inference_time
if len(indices_no_train_obs_in_cluster):
distances[:, np.array(indices_no_train_obs_in_cluster)] = np.inf
indicator_vector_test = np.argmin(distances, axis=1)
predictions = np.empty_like(y_test)
for obs_idx, obs_test in enumerate(x_test):
# get the cluster to which belongs this data point and call the associated nearest neighbor classifier
idx_cluster = indicator_vector_test[obs_idx]
clf_cluster = lst_clf_by_cluster[idx_cluster]
predictions[obs_idx] = clf_cluster.predict(obs_test.reshape(1, -1))[0]
stop_inference_time = time.process_time()
log_memory_usage("Memory after label assignation in kmean_tree_evaluation of make_1nn_evaluation")
inference_time = (stop_inference_time - start_inference_time)
if paraman["--kddcup04"]:
# compute recall: nb_true_positive/real_nb_positive
recall = np.sum(predictions[y_test == 1])/np.sum(y_test[y_test == 1])
# compute precision: nb_true_positive/nb_positive
precision = np.sum(predictions[y_test == 1])/np.sum(predictions[predictions==1])
f1 = 2 * precision * recall / (precision + recall)
accuracy = f1
else:
accuracy = np.sum(predictions == y_test) / y_test.shape[0]
results_1nn = {
"1nn_kmean_inference_time": inference_time,
"1nn_get_distance_time": get_distance_time / x_test.shape[0],
"1nn_kmean_accuracy": accuracy
}
resprinter.add(results_1nn)
return inference_time
def make_assignation_evaluation(X, centroids):
nb_eval = 100
times = []
precomputed_centroid_norms = get_squared_froebenius_norm_line_wise(
centroids)
for i in np.random.permutation(X.shape[0])[:nb_eval]:
start_time = time.time()
get_distances(X[i].reshape(1, -1),
centroids,
precomputed_centroids_norm=precomputed_centroid_norms)
stop_time = time.time()
times.append(stop_time - start_time)
mean_time = np.mean(times)
std_time = np.std(times)
resprinter.add({
"assignation_mean_time": mean_time,
"assignation_std_time": std_time
})
def setUp(self):
self.n_features = 2000
self.n_data = 100
sparsity = 2
self.sparse_data = create_sparse_factors(
shape=(self.n_data * 2, self.n_features),
n_factors=int(
np.ceil(np.log2(min(self.n_data * 2, self.n_features)))),
sparsity_level=sparsity)
self.data = self.sparse_data.compute_product(return_array=True)
self.data_verylittle = np.random.rand(*self.data.shape) * 1e2
self.data_norm = get_squared_froebenius_norm_line_wise(self.data)
self.sparse_data_norm = get_squared_froebenius_norm_line_wise(
self.sparse_data)
self.data_norm_verylittle = get_squared_froebenius_norm_line_wise(
self.data_verylittle)
self.gamma = compute_euristic_gamma(self.data)
self.random_data = np.random.rand(self.n_data, self.n_features)
self.mnist = mnist_dataset()["x_train"].astype(np.float64)
self.fashionmnist = fashion_mnist_dataset()["x_train"].astype(
np.float64)
# self.caltech = caltech_dataset(28)["x_train"].astype(np.float64)
self.pairs_data = {
"notSparse": self.data[:self.n_data],
"mnist": self.mnist[:self.n_data],
# "caltech": self.caltech[:self.n_data],
"fashmnist": self.fashionmnist[:self.n_data],
}
self.example_data = {
"notSparse": self.data[self.n_data:self.n_data * 2],
"mnist": self.mnist[self.n_data:self.n_data * 2],
# "caltech": self.caltech[self.n_data:self.n_data*2],
"fashmnist": self.fashionmnist[self.n_data:self.n_data * 2]
}
self.norm_data = {
"notSparse":
row_norms(self.pairs_data["notSparse"], squared=True)[:,
np.newaxis],
"mnist":
get_squared_froebenius_norm_line_wise(
self.pairs_data["mnist"])[:, np.newaxis],
# "caltech": row_norms(self.pairs_data["caltech"], squared=True)[:, np.newaxis],
"fashmnist":
row_norms(self.pairs_data["fashmnist"], squared=True)[:,
np.newaxis],
}
self.norm_example_data = {
"notSparse":
row_norms(self.example_data["notSparse"],
squared=True)[:, np.newaxis],
"mnist":
get_squared_froebenius_norm_line_wise(self.example_data["mnist"]),
# "caltech": row_norms(self.example_data["caltech"], squared=True)[:, np.newaxis],
"fashmnist":
row_norms(self.example_data["fashmnist"],
squared=True)[:, np.newaxis],
}
self.gamma_data = {
"notSparse": compute_euristic_gamma(self.pairs_data["notSparse"]),
"mnist": compute_euristic_gamma(self.pairs_data["mnist"]),
# "caltech": compute_euristic_gamma(self.pairs_data["caltech"]),
"fashmnist": compute_euristic_gamma(self.pairs_data["fashmnist"]),
}
def qmeans(X_data: np.ndarray,
K_nb_cluster: int,
nb_iter: int,
nb_factors: int,
params_palm4msa: dict,
initialization: np.ndarray,
hierarchical_inside=False,
delta_objective_error_threshold=1e-6,
hierarchical_init=False):
"""
:param X_data: The data matrix of n examples in dimensions d in shape (n, d).
:param K_nb_cluster: The number of clusters to look for.
:param nb_iter: The maximum number of iteration.
:param nb_factors: The number of factors for the decomposition.
:param initialization: The initial matrix of centroids not yet factorized.
:param params_palm4msa: The dictionnary of parameters for the palm4msa algorithm.
:param hierarchical_inside: Tell the algorithm if the hierarchical version of palm4msa should be used.
:param delta_objective_error_threshold:
:param hierarchical_init: Tells if the algorithm should make the initialization of sparse factors with the hierarchical version of palm or not.
:return:
"""
assert K_nb_cluster == initialization.shape[0], "The number of cluster {} is not equal to the number of centroids in the initialization {}.".format(K_nb_cluster, initialization.shape[0])
X_data_norms = get_squared_froebenius_norm_line_wise(X_data)
nb_examples = X_data.shape[0]
logger.info("Initializing Qmeans")
init_lambda = params_palm4msa["init_lambda"]
nb_iter_palm = params_palm4msa["nb_iter"]
lst_proj_op_by_fac_step = params_palm4msa["lst_constraint_sets"]
residual_on_right = params_palm4msa["residual_on_right"]
delta_objective_error_threshold_inner_palm = params_palm4msa["delta_objective_error_threshold"]
track_objective_palm = params_palm4msa["track_objective"]
X_centroids_hat = copy.deepcopy(initialization)
lst_factors = init_lst_factors(K_nb_cluster, X_centroids_hat.shape[1], nb_factors)
eye_norm = np.sqrt(K_nb_cluster)
if hierarchical_inside or hierarchical_init:
_lambda_tmp, op_factors, U_centroids, objective_palm, array_objective_hierarchical= \
hierarchical_palm4msa(
arr_X_target=np.eye(K_nb_cluster) @ X_centroids_hat,
lst_S_init=lst_factors,
lst_dct_projection_function=lst_proj_op_by_fac_step,
f_lambda_init=init_lambda * eye_norm,
nb_iter=nb_iter_palm,
update_right_to_left=True,
residual_on_right=residual_on_right,
track_objective_palm=track_objective_palm,
delta_objective_error_threshold_palm=delta_objective_error_threshold_inner_palm,
return_objective_function=track_objective_palm)
else:
_lambda_tmp, op_factors, U_centroids, objective_palm, nb_iter_palm = \
palm4msa(
arr_X_target=np.eye(K_nb_cluster) @ X_centroids_hat,
lst_S_init=lst_factors,
nb_factors=len(lst_factors),
lst_projection_functions=lst_proj_op_by_fac_step[-1][
"finetune"],
f_lambda_init=init_lambda * eye_norm,
nb_iter=nb_iter_palm,
update_right_to_left=True,
track_objective=track_objective_palm,
delta_objective_error_threshold=delta_objective_error_threshold_inner_palm)
lst_factors = None # safe assignment for debug
_lambda = _lambda_tmp / eye_norm
objective_function = np.ones(nb_iter) * -1
lst_all_objective_functions_palm = []
lst_all_objective_functions_palm.append(objective_palm)
i_iter = 0
delta_objective_error = np.inf
while ((i_iter < nb_iter) and (delta_objective_error > delta_objective_error_threshold)):
logger.info("Iteration Qmeans {}".format(i_iter))
lst_factors_ = op_factors.get_list_of_factors()
op_centroids = SparseFactors([lst_factors_[1] * _lambda] + lst_factors_[2:])
###########################
# Cluster assignment step #
###########################
indicator_vector, distances = assign_points_to_clusters(X_data, op_centroids, X_norms=X_data_norms)
#######################
# Cluster update step #
#######################
# get the number of observation in each cluster
cluster_names, counts = np.unique(indicator_vector, return_counts=True)
cluster_names_sorted = np.argsort(cluster_names)
# Update centroid location using the newly (it happens in the assess_cluster_integrity function)
# assigned data point classes
# and check if all clusters still have points
# and change the object X_centroids_hat in place if some cluster have lost points (biggest cluster)
counts, cluster_names_sorted = update_clusters_with_integrity_check(X_data,
X_data_norms,
X_centroids_hat, # in place changes
K_nb_cluster,
counts,
indicator_vector,
distances,
cluster_names,
cluster_names_sorted)
#################
# PALM4MSA step #
#################
# create the diagonal of the sqrt of those counts
diag_counts_sqrt_normalized = csr_matrix(
(np.sqrt(counts[cluster_names_sorted] / nb_examples),
(np.arange(K_nb_cluster), np.arange(K_nb_cluster))))
diag_counts_sqrt = np.sqrt(counts[cluster_names_sorted])
# set it as first factor
op_factors.set_factor(0, diag_counts_sqrt_normalized)
if hierarchical_inside:
_lambda_tmp, op_factors, _, objective_palm, array_objective_hierarchical = \
hierarchical_palm4msa(
arr_X_target=diag_counts_sqrt[:, None,] * X_centroids_hat,
lst_S_init=op_factors.get_list_of_factors(),
lst_dct_projection_function=lst_proj_op_by_fac_step,
f_lambda_init=_lambda * np.sqrt(nb_examples),
nb_iter=nb_iter_palm,
update_right_to_left=True,
residual_on_right=residual_on_right,
return_objective_function=track_objective_palm,
track_objective_palm=track_objective_palm,
delta_objective_error_threshold_palm=delta_objective_error_threshold_inner_palm)
else:
_lambda_tmp, op_factors, _, objective_palm, nb_iter_palm = \
palm4msa(arr_X_target=diag_counts_sqrt[:, None,] * X_centroids_hat,
lst_S_init=op_factors.get_list_of_factors(),
nb_factors=op_factors.n_factors,
lst_projection_functions=lst_proj_op_by_fac_step[-1][
"finetune"],
f_lambda_init=_lambda * np.sqrt(nb_examples),
nb_iter=nb_iter_palm,
update_right_to_left=True,
track_objective=track_objective_palm,
delta_objective_error_threshold=delta_objective_error_threshold_inner_palm)
lst_all_objective_functions_palm.append(objective_palm)
_lambda = _lambda_tmp / np.sqrt(nb_examples)
objective_function[i_iter] = compute_objective(X_data, op_centroids, indicator_vector)
if i_iter >= 1:
delta_objective_error = np.abs(objective_function[i_iter] - objective_function[i_iter-1]) / objective_function[i_iter-1]
# todo vérifier que l'erreur absolue est plus petite que le threshold plusieurs fois d'affilee
i_iter += 1
lst_factors_ = op_factors.get_list_of_factors()
op_centroids = SparseFactors([lst_factors_[1] * _lambda] + lst_factors_[2:])
return objective_function[:i_iter], op_centroids, indicator_vector, lst_all_objective_functions_palm
def kmeans_minibatch(X_data, K_nb_cluster, nb_iter, initialization,
batch_size):
"""
:param X_data: The data matrix of n examples in dimensions d in shape (n, d).
:param K_nb_cluster: The number of clusters to look for.
:param nb_iter: The maximum number of iteration.
:param initialization: The (K, d) matrix of centroids at initialization.
:param batch_size: The size of each batch.
:return:
"""
X_data_norms = get_squared_froebenius_norm_line_wise(X_data)
# Initialize our centroids by picking random data points
U_centroids_hat = copy.deepcopy(initialization)
U_centroids = U_centroids_hat
full_indicator_vector = np.zeros(X_data.shape[0], dtype=int)
full_count_vector = np.zeros(K_nb_cluster, dtype=int)
objective_function = np.empty((nb_iter, ))
# Loop for the maximum number of iterations
i_iter = 0
delta_objective_error_threshold = 1e-6
delta_objective_error = np.inf
while True:
for i_iter, example_batch_indexes in enumerate(
DataGenerator(X_data,
batch_size=batch_size,
return_indexes=True)):
if not (delta_objective_error > delta_objective_error_threshold):
logger.info(
"not (delta_objective_error {}-{}={} > delta_objective_error_threshold {})"
.format(objective_function[i_iter],
objective_function[i_iter - 1],
delta_objective_error,
delta_objective_error_threshold))
break
example_batch = X_data[example_batch_indexes]
logger.info("Iteration Kmeans {}".format(i_iter))
indicator_vector, distances = assign_points_to_clusters(
example_batch,
U_centroids,
X_norms=X_data_norms[example_batch_indexes])
full_indicator_vector[example_batch_indexes] = indicator_vector
cluster_names, counts = np.unique(indicator_vector,
return_counts=True)
# cluster_names_sorted = np.argsort(cluster_names)
#
count_vector = np.zeros(K_nb_cluster, dtype=int)
count_vector[cluster_names] = counts
full_count_vector += count_vector
# previous_full_count_vector = full_count_vector - count_vector
# Update centroid location using the newly
# assigned data point classes
# This way of updating the centroids (centroid index wise) is better than the one proposed in the paper "Web-Scale K-Means Clustering"
# as the number of update with always be <= batch_size
for c in range(K_nb_cluster):
if full_count_vector[c] != 0 and count_vector[c] != 0:
U_centroids_hat[c] += (1 / full_count_vector[c]) * np.sum(
example_batch[indicator_vector == c] -
U_centroids_hat[c],
axis=0)
# this is exactly equivalent to an update of the mean:
# U_centroids_hat[c] = (previous_full_count_vector[c] / full_count_vector[c]) * U_centroids_hat[c] + (1 / full_count_vector[c]) * np.sum(example_batch[indicator_vector == c], axis=0)
# for i_ex, ex in enumerate(example_batch):
# c = indicator_vector[i_ex]
# full_count_vector[c] += 1
# eta = 1./full_count_vector[c]
# U_centroids_hat[c] = (1-eta) * U_centroids_hat[c] + eta * ex
# counts, cluster_names_sorted = assess_clusters_integrity(X_data,
# X_data_norms,
# U_centroids_hat,
# K_nb_cluster,
# counts,
# indicator_vector,
# distances,
# cluster_names,
# cluster_names_sorted)
# check if all clusters still have points
# for c in range(K_nb_cluster):
# biggest_cluster_index = np.argmax(counts) # type: int
# biggest_cluster = cluster_names[biggest_cluster_index]
# biggest_cluster_data = X_data[indicator_vector == biggest_cluster]
#
# cluster_data = X_data[indicator_vector == c]
# if len(cluster_data) == 0:
# logger.warning("cluster has lost data, add new cluster. cluster idx: {}".format(c))
# U_centroids_hat[c] = biggest_cluster_data[np.random.randint(len(biggest_cluster_data))].reshape(1, -1)
# counts = list(counts)
# counts[biggest_cluster_index] -= 1
# counts.append(1)
# counts = np.array(counts)
# cluster_names_sorted = list(cluster_names_sorted)
# cluster_names_sorted.append(c)
# cluster_names_sorted = np.array(cluster_names_sorted)
# else:
# U_centroids_hat[c] = np.mean(X_data[indicator_vector == c], 0)
U_centroids = U_centroids_hat
objective_function[i_iter, ] = compute_objective(
X_data, U_centroids, full_indicator_vector)
if i_iter >= 1:
delta_objective_error = np.abs(
objective_function[i_iter] - objective_function[i_iter - 1]
) / objective_function[
i_iter -
1] # todo vérifier que l'erreur absolue est plus petite que le threshold plusieurs fois d'affilée
i_iter += 1
else:
continue
break
return objective_function[:i_iter], U_centroids, indicator_vector
def make_nystrom_evaluation(x_train, y_train, x_test, y_test, U_centroids):
"""
Evaluation Nystrom construction time and approximation precision.
The approximation is based on a subsample of size n_sample of the input data set.
:param x_train: Input dataset as ndarray.
:param U_centroids: The matrix of centroids as ndarray or SparseFactor object
:param n_sample: The number of sample to take into account in the reconstruction (can't be too large)
:return:
"""
n_sample = paraman["--nystrom"]
if n_sample > x_train.shape[0]:
logger.warning("Batch size for nystrom evaluation is bigger than data size. {} > {}. Using "
"data size instead.".format(n_sample, x_train.shape[0]))
n_sample = x_train.shape[0]
paraman["--nystrom"] = n_sample
# Compute euristic gamma as the mean of euclidian distance between example
gamma = compute_euristic_gamma(x_train)
log_memory_usage("Memory after euristic gamma computation in make_nystrom_evaluation")
# precompute the centroids norm for later use (optimization)
centroids_norm = get_squared_froebenius_norm_line_wise(U_centroids)[:, np.newaxis]
# centroids_norm = None
indexes_samples = np.random.permutation(x_train.shape[0])[:n_sample]
sample = x_train[indexes_samples]
samples_norm = None
log_memory_usage("Memory after sample selection in make_nystrom_evaluation")
########################
# Nystrom on centroids #
########################
logger.info("Build Nystrom on centroids")
## TIME: nystrom build time
# nystrom build time is Nystrom preparation time for later use.
## START
nystrom_build_start_time = time.process_time()
metric = prepare_nystrom(U_centroids, centroids_norm, gamma=gamma)
nystrom_build_stop_time = time.process_time()
log_memory_usage("Memory after SVD computation in make_nystrom_evaluation")
# STOP
nystrom_build_time = nystrom_build_stop_time - nystrom_build_start_time
## TIME: nystrom inference time
# Nystrom inference time is the time for Nystrom transformation for all the samples.
## START
nystrom_inference_time_start = time.process_time()
nystrom_embedding = nystrom_transformation(sample, U_centroids, metric, centroids_norm, samples_norm, gamma=gamma)
nystrom_approx_kernel_value = nystrom_embedding @ nystrom_embedding.T
nystrom_inference_time_stop = time.process_time()
log_memory_usage("Memory after kernel matrix approximation in make_nystrom_evaluation")
## STOP
nystrom_inference_time = (nystrom_inference_time_stop - nystrom_inference_time_start) / n_sample
################################################################
######################
# Nystrom on uniform #
######################
logger.info("Build Nystrom on uniform sampling")
indexes_uniform_samples = np.random.permutation(x_train.shape[0])[:U_centroids.shape[0]]
uniform_sample = x_train[indexes_uniform_samples]
uniform_sample_norm = get_squared_froebenius_norm_line_wise(uniform_sample)[:, np.newaxis]
log_memory_usage("Memory after uniform sample selection in make_nystrom_evaluation")
metric_uniform = prepare_nystrom(uniform_sample, uniform_sample_norm, gamma=gamma)
log_memory_usage("Memory after SVD computation in uniform part of make_nystrom_evaluation")
nystrom_embedding_uniform = nystrom_transformation(sample, uniform_sample, metric_uniform, uniform_sample_norm, samples_norm, gamma=gamma)
nystrom_approx_kernel_value_uniform = nystrom_embedding_uniform @ nystrom_embedding_uniform.T
#################################################################
###############
# Real Kernel #
###############
logger.info("Compute real kernel matrix")
real_kernel_special = special_rbf_kernel(sample, sample, gamma, norm_X=samples_norm, norm_Y=samples_norm)
# real_kernel = rbf_kernel(sample, sample, gamma)
real_kernel_norm = np.linalg.norm(real_kernel_special)
log_memory_usage("Memory after real kernel computation in make_nystrom_evaluation")
#################################
# Sklearn based Nystrom uniform #
#################################
# sklearn_nystrom = Nystroem(gamma=gamma, n_components=uniform_sample.shape[0])
# sklearn_nystrom = sklearn_nystrom.fit(uniform_sample)
# sklearn_transfo = sklearn_nystrom.transform(sample)
# kernel_sklearn_nys = sklearn_transfo @ sklearn_transfo.T
################################################################
####################
# Error evaluation #
####################
sampled_froebenius_norm = np.linalg.norm(nystrom_approx_kernel_value - real_kernel_special) / real_kernel_norm
sampled_froebenius_norm_uniform = np.linalg.norm(nystrom_approx_kernel_value_uniform - real_kernel_special) / real_kernel_norm
# svm evaluation
if x_test is not None:
logger.info("Start classification")
time_classification_start = time.process_time()
x_train_nystrom_embedding = nystrom_transformation(x_train, U_centroids, metric, centroids_norm, None, gamma=gamma)
x_test_nystrom_embedding = nystrom_transformation(x_test, U_centroids, metric, centroids_norm, None, gamma=gamma)
linear_svc_clf = LinearSVC(class_weight="balanced")
linear_svc_clf.fit(x_train_nystrom_embedding, y_train)
predictions = linear_svc_clf.predict(x_test_nystrom_embedding)
time_classification_stop = time.process_time()
if paraman["--kddcup04"]:
# compute recall: nb_true_positive/real_nb_positive
recall = np.sum(predictions[y_test == 1])/np.sum(y_test[y_test == 1])
# compute precision: nb_true_positive/nb_positive
precision = np.sum(predictions[y_test == 1])/np.sum(predictions[predictions==1])
f1 = 2 * precision * recall / (precision + recall)
accuracy_nystrom_svm = f1
else:
accuracy_nystrom_svm = np.sum(predictions == y_test) / y_test.shape[0]
delta_time_classification = time_classification_stop - time_classification_start
else:
accuracy_nystrom_svm = None
delta_time_classification = None
nystrom_results = {
"nystrom_build_time": nystrom_build_time,
"nystrom_inference_time": nystrom_inference_time,
"nystrom_sampled_error_reconstruction": sampled_froebenius_norm,
"nystrom_sampled_error_reconstruction_uniform": sampled_froebenius_norm_uniform,
"nystrom_svm_accuracy": accuracy_nystrom_svm,
"nystrom_svm_time": delta_time_classification
}
resprinter.add(nystrom_results)
def make_nystrom_evaluation(x_train, y_train, x_test, y_test, U_centroids):
"""
Evaluation Nystrom construction time and approximation precision.
The approximation is based on a subsample of size n_sample of the input data set.
:param x_train: Input dataset as ndarray.
:param U_centroids: The matrix of centroids as ndarray or SparseFactor object
:param n_sample: The number of sample to take into account in the reconstruction (can't be too large)
:return:
"""
def prepare_nystrom(landmarks, landmarks_norm):
basis_kernel_W = special_rbf_kernel(landmarks, landmarks, gamma,
landmarks_norm, landmarks_norm)
U, S, V = np.linalg.svd(basis_kernel_W)
S = np.maximum(S, 1e-12)
normalization_ = np.dot(U / np.sqrt(S), V)
return normalization_
def nystrom_transformation(x_input, landmarks, p_metric, landmarks_norm,
x_input_norm):
nystrom_embedding = special_rbf_kernel(landmarks, x_input, gamma,
landmarks_norm,
x_input_norm).T @ p_metric
return nystrom_embedding
n_sample = paraman["--nystrom"]
if n_sample > x_train.shape[0]:
logger.warning(
"Batch size for nystrom evaluation is bigger than data size. {} > {}. Using "
"data size instead.".format(n_sample, x_train.shape[0]))
n_sample = x_train.shape[0]
paraman["--nystrom"] = n_sample
# Compute euristic gamma as the mean of euclidian distance between example
gamma = compute_euristic_gamma(x_train)
log_memory_usage(
"Memory after euristic gamma computation in make_nystrom_evaluation")
# precompute the centroids norm for later use (optimization)
centroids_norm = get_squared_froebenius_norm_line_wise(U_centroids)
# centroids_norm = None
indexes_samples = np.random.permutation(x_train.shape[0])[:n_sample]
sample = x_train[indexes_samples]
samples_norm = None
log_memory_usage(
"Memory after sample selection in make_nystrom_evaluation")
########################
# Nystrom on centroids #
########################
logger.info("Build Nystrom on centroids")
## TIME: nystrom build time
# nystrom build time is Nystrom preparation time for later use.
## START
nystrom_build_start_time = time.process_time()
metric = prepare_nystrom(U_centroids, centroids_norm)
nystrom_build_stop_time = time.process_time()
log_memory_usage("Memory after SVD computation in make_nystrom_evaluation")
# STOP
nystrom_build_time = nystrom_build_stop_time - nystrom_build_start_time
## TIME: nystrom inference time
# Nystrom inference time is the time for Nystrom transformation for all the samples.
## START
nystrom_inference_time_start = time.process_time()
nystrom_embedding = nystrom_transformation(sample, U_centroids, metric,
centroids_norm, samples_norm)
nystrom_approx_kernel_value = nystrom_embedding @ nystrom_embedding.T
nystrom_inference_time_stop = time.process_time()
log_memory_usage(
"Memory after kernel matrix approximation in make_nystrom_evaluation")
## STOP
nystrom_inference_time = (nystrom_inference_time_stop -
nystrom_inference_time_start) / n_sample
################################################################
######################
# Nystrom on uniform #
######################
logger.info("Build Nystrom on uniform sampling")
indexes_uniform_samples = np.random.permutation(
x_train.shape[0])[:U_centroids.shape[0]]
uniform_sample = x_train[indexes_uniform_samples]
uniform_sample_norm = None
log_memory_usage(
"Memory after uniform sample selection in make_nystrom_evaluation")
metric_uniform = prepare_nystrom(uniform_sample, uniform_sample_norm)
log_memory_usage(
"Memory after SVD computation in uniform part of make_nystrom_evaluation"
)
nystrom_embedding_uniform = nystrom_transformation(sample, uniform_sample,
metric_uniform,
uniform_sample_norm,
samples_norm)
nystrom_approx_kernel_value_uniform = nystrom_embedding_uniform @ nystrom_embedding_uniform.T
#################################################################
###############
# Real Kernel #
###############
logger.info("Compute real kernel matrix")
real_kernel = special_rbf_kernel(sample, sample, gamma, samples_norm,
samples_norm)
real_kernel_norm = np.linalg.norm(real_kernel)
log_memory_usage(
"Memory after real kernel computation in make_nystrom_evaluation")
################################################################
####################
# Error evaluation #
####################
sampled_froebenius_norm = np.linalg.norm(nystrom_approx_kernel_value -
real_kernel) / real_kernel_norm
sampled_froebenius_norm_uniform = np.linalg.norm(
nystrom_approx_kernel_value_uniform - real_kernel) / real_kernel_norm
# svm evaluation
if x_test is not None:
logger.info("Start classification")
time_classification_start = time.process_time()
x_train_nystrom_embedding = nystrom_transformation(
x_train, U_centroids, metric, centroids_norm, None)
x_test_nystrom_embedding = nystrom_transformation(
x_test, U_centroids, metric, centroids_norm, None)
linear_svc_clf = LinearSVC()
linear_svc_clf.fit(x_train_nystrom_embedding, y_train)
accuracy_nystrom_svm = linear_svc_clf.score(x_test_nystrom_embedding,
y_test)
time_classification_stop = time.process_time()
delta_time_classification = time_classification_stop - time_classification_start
else:
accuracy_nystrom_svm = None
delta_time_classification = None
nystrom_results = {
"nystrom_build_time": nystrom_build_time,
"nystrom_inference_time": nystrom_inference_time,
"nystrom_sampled_error_reconstruction": sampled_froebenius_norm,
"nystrom_sampled_error_reconstruction_uniform":
sampled_froebenius_norm_uniform,
"nystrom_svm_accuracy": accuracy_nystrom_svm,
"nystrom_svm_time": delta_time_classification
}
resprinter.add(nystrom_results)
def qmeans(X_data: np.ndarray,
K_nb_cluster: int,
nb_iter: int,
nb_factors: int,
params_palm4msa: dict,
initialization: np.ndarray,
hierarchical_inside=False,
graphical_display=False):
"""
:param X_data: The data matrix of n examples in dimensions d in shape (n, d).
:param K_nb_cluster: The number of clusters to look for.
:param nb_iter: The maximum number of iteration.
:param nb_factors: The number of factors for the decomposition.
:param initialization: The initial matrix of centroids not yet factorized.
:param params_palm4msa: The dictionnary of parameters for the palm4msa algorithm.
:param hierarchical_inside: Tell the algorithm if the hierarchical version of palm4msa should be used.
:param graphical_display: Tell the algorithm to display the results.
:return:
"""
assert K_nb_cluster == initialization.shape[0]
X_data_norms = get_squared_froebenius_norm_line_wise(X_data)
init_lambda = params_palm4msa["init_lambda"]
nb_iter_palm = params_palm4msa["nb_iter"]
lst_proj_op_by_fac_step = params_palm4msa["lst_constraint_sets"]
residual_on_right = params_palm4msa["residual_on_right"]
X_centroids_hat = copy.deepcopy(initialization)
min_K_d = min(X_centroids_hat.shape)
lst_factors = [np.eye(min_K_d) for _ in range(nb_factors)]
eye_norm = np.sqrt(K_nb_cluster)
lst_factors[0] = np.eye(K_nb_cluster) / eye_norm
lst_factors[1] = np.eye(K_nb_cluster, min_K_d)
lst_factors[-1] = np.zeros((min_K_d, X_centroids_hat.shape[1]))
if graphical_display:
lst_factors_init = copy.deepcopy(lst_factors)
_lambda_tmp, lst_factors, U_centroids, nb_iter_by_factor, objective_palm = hierarchical_palm4msa(
arr_X_target=np.eye(K_nb_cluster) @ X_centroids_hat,
lst_S_init=lst_factors,
lst_dct_projection_function=lst_proj_op_by_fac_step,
f_lambda_init=init_lambda * eye_norm,
nb_iter=nb_iter_palm,
update_right_to_left=True,
residual_on_right=residual_on_right,
graphical_display=False)
_lambda = _lambda_tmp / eye_norm
if graphical_display:
if hierarchical_inside:
plt.figure()
plt.yscale("log")
plt.scatter(np.arange(len(objective_palm) * 3, step=3),
objective_palm[:, 0],
marker="x",
label="before split")
plt.scatter(np.arange(len(objective_palm) * 3, step=3) + 1,
objective_palm[:, 1],
marker="x",
label="between")
plt.scatter(np.arange(len(objective_palm) * 3, step=3) + 2,
objective_palm[:, 2],
marker="x",
label="after finetune")
plt.plot(np.arange(len(objective_palm) * 3),
objective_palm.flatten(),
color="k")
plt.legend()
plt.show()
visual_evaluation_palm4msa(
np.eye(K_nb_cluster) @ X_centroids_hat, lst_factors_init,
lst_factors, _lambda * multi_dot(lst_factors))
objective_function = np.empty((nb_iter, 2))
# Loop for the maximum number of iterations
i_iter = 0
delta_objective_error_threshold = 1e-6
delta_objective_error = np.inf
while (i_iter <= 1) or (
(i_iter < nb_iter) and
(delta_objective_error > delta_objective_error_threshold)):
logger.info("Iteration Qmeans {}".format(i_iter))
U_centroids = _lambda * multi_dot(lst_factors[1:])
if i_iter > 0:
objective_function[i_iter,
0] = compute_objective(X_data, U_centroids,
indicator_vector)
# Assign all points to the nearest centroid
# first get distance from all points to all centroids
distances = get_distances(X_data,
U_centroids,
precomputed_data_points_norm=X_data_norms)
# then, Determine class membership of each point
# by picking the closest centroid
indicator_vector = np.argmin(distances, axis=1)
objective_function[i_iter,
1] = compute_objective(X_data, U_centroids,
indicator_vector)
# Update centroid location using the newly
# assigned data point classes
for c in range(K_nb_cluster):
X_centroids_hat[c] = np.mean(X_data[indicator_vector == c], 0)
# get the number of observation in each cluster
cluster_names, counts = np.unique(indicator_vector, return_counts=True)
cluster_names_sorted = np.argsort(cluster_names)
if len(counts) < K_nb_cluster:
raise ValueError(
"Some clusters have no point. Aborting iteration {}".format(
i_iter))
diag_counts_sqrt = np.diag(np.sqrt(
counts[cluster_names_sorted])) # todo use sparse matrix object
diag_counts_sqrt_norm = np.linalg.norm(
diag_counts_sqrt
) # todo analytic sqrt(n) instead of cumputing it with norm
diag_counts_sqrt_normalized = diag_counts_sqrt / diag_counts_sqrt_norm
# set it as first factor
lst_factors[0] = diag_counts_sqrt_normalized
if graphical_display:
lst_factors_init = copy.deepcopy(lst_factors)
if hierarchical_inside:
_lambda_tmp, lst_factors, _, nb_iter_by_factor, objective_palm = hierarchical_palm4msa(
arr_X_target=diag_counts_sqrt @ X_centroids_hat,
lst_S_init=lst_factors,
lst_dct_projection_function=lst_proj_op_by_fac_step,
# f_lambda_init=_lambda,
f_lambda_init=_lambda * diag_counts_sqrt_norm,
nb_iter=nb_iter_palm,
update_right_to_left=True,
residual_on_right=residual_on_right,
graphical_display=False)
loss_palm_before = objective_palm[0, 0]
loss_palm_after = objective_palm[-1, -1]
else:
_lambda_tmp, lst_factors, _, objective_palm, nb_iter_palm = palm4msa(
arr_X_target=diag_counts_sqrt @ X_centroids_hat,
lst_S_init=lst_factors,
nb_factors=len(lst_factors),
lst_projection_functions=lst_proj_op_by_fac_step[-1]
["finetune"],
f_lambda_init=_lambda * diag_counts_sqrt_norm,
nb_iter=nb_iter_palm,
update_right_to_left=True,
graphical_display=False)
loss_palm_before = objective_palm[0, -1]
loss_palm_after = objective_palm[-1, -1]
logger.debug("Loss palm before: {}".format(loss_palm_before))
logger.debug("Loss palm after: {}".format(loss_palm_after))
if graphical_display:
if hierarchical_inside:
plt.figure()
plt.yscale("log")
plt.scatter(np.arange(len(objective_palm) * 3, step=3),
objective_palm[:, 0],
marker="x",
label="before split")
plt.scatter(np.arange(len(objective_palm) * 3, step=3) + 1,
objective_palm[:, 1],
marker="x",
label="between")
plt.scatter(np.arange(len(objective_palm) * 3, step=3) + 2,
objective_palm[:, 2],
marker="x",
label="after finetune")
plt.plot(np.arange(len(objective_palm) * 3),
objective_palm.flatten(),
color="k")
plt.legend()
plt.show()
visual_evaluation_palm4msa(diag_counts_sqrt @ X_centroids_hat,
lst_factors_init, lst_factors,
_lambda_tmp * multi_dot(lst_factors))
_lambda = _lambda_tmp / diag_counts_sqrt_norm
logger.debug("Returned loss (with diag) palm: {}".format(
objective_palm[-1, 0]))
if i_iter >= 2:
delta_objective_error = np.abs(
objective_function[i_iter, 0] -
objective_function[i_iter - 1, 0]
) / objective_function[
i_iter - 1,
0] # todo vérifier que l'erreur absolue est plus petite que le threshold plusieurs fois d'affilée
i_iter += 1
U_centroids = _lambda * multi_dot(lst_factors[1:])
distances = get_distances(X_data,
U_centroids,
precomputed_data_points_norm=X_data_norms)
indicator_vector = np.argmin(distances, axis=1)
return objective_function[:i_iter], U_centroids, indicator_vector
def make_nystrom_evaluation(x_train, U_centroids):
"""
Evaluation Nystrom construction time and approximation precision.
The approximation is based on a subsample of size n_sample of the input data set.
:param x_train: Input dataset as ndarray.
:param U_centroids: The matrix of centroids as ndarray or SparseFactor object
:param n_sample: The number of sample to take into account in the reconstruction (can't be too large)
:return:
"""
n_sample = paraman["--nystrom"]
if n_sample > x_train.shape[0]:
logger.warning(
"Batch size for nystrom evaluation is bigger than data size. {} > {}. Using "
"data size instead.".format(n_sample, x_train.shape[0]))
n_sample = x_train.shape[0]
paraman["--nystrom"] = n_sample
# Compute euristic gamma as the mean of euclidian distance between example
gamma = compute_euristic_gamma(x_train)
# precompute the centroids norm for later use (optimization)
centroids_norm = get_squared_froebenius_norm_line_wise(U_centroids)
## TIME: nystrom build time
# nystrom build time is Nystrom preparation time for later use.
## START
nystrom_build_start_time = time.time()
basis_kernel_W = special_rbf_kernel(U_centroids, U_centroids, gamma,
centroids_norm, centroids_norm)
U, S, V = np.linalg.svd(basis_kernel_W)
S = np.maximum(S, 1e-12)
normalization_ = np.dot(U / np.sqrt(S), V)
nystrom_build_stop_time = time.time()
# STOP
nystrom_build_time = nystrom_build_stop_time - nystrom_build_start_time
indexes_samples = np.random.permutation(x_train.shape[0])[:n_sample]
sample = x_train[indexes_samples]
samples_norm = np.linalg.norm(sample, axis=1)**2
real_kernel = special_rbf_kernel(sample, sample, gamma, samples_norm,
samples_norm)
## TIME: nystrom inference time
# Nystrom inference time is the time for Nystrom transformation for all the samples.
## START
nystrom_inference_time_start = time.time()
nystrom_embedding = special_rbf_kernel(U_centroids, sample, gamma,
centroids_norm,
samples_norm).T @ normalization_
nystrom_approx_kernel_value = nystrom_embedding @ nystrom_embedding.T
nystrom_inference_time_stop = time.time()
## STOP
nystrom_inference_time = (nystrom_inference_time_stop -
nystrom_inference_time_start) / n_sample
sampled_froebenius_norm = np.linalg.norm(nystrom_approx_kernel_value -
real_kernel)
nystrom_results = {
"nystrom_build_time": nystrom_build_time,
"nystrom_inference_time": nystrom_inference_time,
"nystrom_sampled_error_reconstruction": sampled_froebenius_norm
}
resprinter.add(nystrom_results)
def kmeans(X_data,
K_nb_cluster,
nb_iter,
initialization,
delta_objective_error_threshold=1e-6,
proj_l1=False,
_lambda=None,
epsilon=None):
"""
:param X_data: The data matrix of n examples in dimensions d in shape (n, d).
:param K_nb_cluster: The number of clusters to look for.
:param nb_iter: The maximum number of iteration.
:param initialization: The (K, d) matrix of centroids at initialization.
:param delta_objective_error_threshold: The normalized difference between the error criterion at 2 successive step must be greater or equal to that value.
:return:
"""
X_data_norms = get_squared_froebenius_norm_line_wise(X_data)
# Initialize our centroids by picking random data points
U_centroids_hat = copy.deepcopy(initialization)
U_centroids = U_centroids_hat
objective_function = np.empty((nb_iter, ))
# Loop for the maximum number of iterations
i_iter = 0
delta_objective_error = np.inf
while (i_iter == 0) or (
(i_iter < nb_iter) and
(delta_objective_error > delta_objective_error_threshold)):
logger.info("Iteration Kmeans {}".format(i_iter))
indicator_vector, distances = assign_points_to_clusters(
X_data, U_centroids, X_norms=X_data_norms)
cluster_names, counts = np.unique(indicator_vector, return_counts=True)
cluster_names_sorted = np.argsort(cluster_names)
# Update centroid location using the new indicator vector
counts, cluster_names_sorted = update_clusters_with_integrity_check(
X_data, X_data_norms, U_centroids_hat, K_nb_cluster, counts,
indicator_vector, distances, cluster_names, cluster_names_sorted)
U_centroids = U_centroids_hat
if proj_l1:
if _lambda is None or epsilon is None:
raise ValueError(
"epsilon and lambda must be set if proj_l1 is True")
for i_centroid, centroid in enumerate(U_centroids):
U_centroids[i_centroid, :] = proj_onto_l1_ball(
_lambda=_lambda, epsilon_tol=epsilon, vec=centroid)
objective_function[i_iter, ] = compute_objective(
X_data, U_centroids, indicator_vector)
if i_iter >= 1:
delta_objective_error = np.abs(objective_function[i_iter] -
objective_function[i_iter - 1]
) / objective_function[i_iter - 1]
i_iter += 1
return objective_function[:i_iter], U_centroids, indicator_vector
