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/ manifold / _t_sne.py
# Author: Alexander Fabisch -- <afabisch@informatik.uni-bremen.de>
# Author: Christopher Moody <chrisemoody@gmail.com>
# Author: Nick Travers <nickt@squareup.com>
# License: BSD 3 clause (C) 2014
# This is the exact and Barnes-Hut t-SNE implementation. There are other
# modifications of the algorithm:
# * Fast Optimization for t-SNE:
# https://cseweb.ucsd.edu/~lvdmaaten/workshops/nips2010/papers/vandermaaten.pdf
from time import time
import numpy as np
from scipy import linalg
from scipy.spatial.distance import pdist
from scipy.spatial.distance import squareform
from scipy.sparse import csr_matrix, issparse
from ..neighbors import NearestNeighbors
from ..base import BaseEstimator
from ..utils import check_array
from ..utils import check_random_state
from ..utils._openmp_helpers import _openmp_effective_n_threads
from ..utils.validation import check_non_negative
from ..decomposition import PCA
from ..metrics.pairwise import pairwise_distances
from . import _utils
from . import _barnes_hut_tsne
MACHINE_EPSILON = np.finfo(np.double).eps
def _joint_probabilities(distances, desired_perplexity, verbose):
"""Compute joint probabilities p_ij from distances.
Parameters
----------
distances : array, shape (n_samples * (n_samples-1) / 2,)
Distances of samples are stored as condensed matrices, i.e.
we omit the diagonal and duplicate entries and store everything
in a one-dimensional array.
desired_perplexity : float
Desired perplexity of the joint probability distributions.
verbose : int
Verbosity level.
Returns
-------
P : array, shape (n_samples * (n_samples-1) / 2,)
Condensed joint probability matrix.
"""
# Compute conditional probabilities such that they approximately match
# the desired perplexity
distances = distances.astype(np.float32, copy=False)
conditional_P = _utils._binary_search_perplexity(
distances, desired_perplexity, verbose)
P = conditional_P + conditional_P.T
sum_P = np.maximum(np.sum(P), MACHINE_EPSILON)
P = np.maximum(squareform(P) / sum_P, MACHINE_EPSILON)
return P
def _joint_probabilities_nn(distances, desired_perplexity, verbose):
"""Compute joint probabilities p_ij from distances using just nearest
neighbors.
This method is approximately equal to _joint_probabilities. The latter
is O(N), but limiting the joint probability to nearest neighbors improves
this substantially to O(uN).
Parameters
----------
distances : CSR sparse matrix, shape (n_samples, n_samples)
Distances of samples to its n_neighbors nearest neighbors. All other
distances are left to zero (and are not materialized in memory).
desired_perplexity : float
Desired perplexity of the joint probability distributions.
verbose : int
Verbosity level.
Returns
-------
P : csr sparse matrix, shape (n_samples, n_samples)
Condensed joint probability matrix with only nearest neighbors.
"""
t0 = time()
# Compute conditional probabilities such that they approximately match
# the desired perplexity
distances.sort_indices()
n_samples = distances.shape[0]
distances_data = distances.data.reshape(n_samples, -1)
distances_data = distances_data.astype(np.float32, copy=False)
conditional_P = _utils._binary_search_perplexity(
distances_data, desired_perplexity, verbose)
assert np.all(np.isfinite(conditional_P)), \
"All probabilities should be finite"
# Symmetrize the joint probability distribution using sparse operations
P = csr_matrix((conditional_P.ravel(), distances.indices,
distances.indptr),
shape=(n_samples, n_samples))
P = P + P.T
# Normalize the joint probability distribution
sum_P = np.maximum(P.sum(), MACHINE_EPSILON)
P /= sum_P
assert np.all(np.abs(P.data) <= 1.0)
if verbose >= 2:
duration = time() - t0
print("[t-SNE] Computed conditional probabilities in {:.3f}s"
.format(duration))
return P
def _kl_divergence(params, P, degrees_of_freedom, n_samples, n_components,
skip_num_points=0, compute_error=True):
"""t-SNE objective function: gradient of the KL divergence
of p_ijs and q_ijs and the absolute error.
Parameters
----------
params : array, shape (n_params,)
Unraveled embedding.
P : array, shape (n_samples * (n_samples-1) / 2,)
Condensed joint probability matrix.
degrees_of_freedom : int
Degrees of freedom of the Student's-t distribution.
n_samples : int
Number of samples.
n_components : int
Dimension of the embedded space.
skip_num_points : int (optional, default:0)
This does not compute the gradient for points with indices below
`skip_num_points`. This is useful when computing transforms of new
data where you'd like to keep the old data fixed.
compute_error: bool (optional, default:True)
If False, the kl_divergence is not computed and returns NaN.
Returns
-------
kl_divergence : float
Kullback-Leibler divergence of p_ij and q_ij.
grad : array, shape (n_params,)
Unraveled gradient of the Kullback-Leibler divergence with respect to
the embedding.
"""
X_embedded = params.reshape(n_samples, n_components)
# Q is a heavy-tailed distribution: Student's t-distribution
dist = pdist(X_embedded, "sqeuclidean")
dist /= degrees_of_freedom
dist += 1.
dist **= (degrees_of_freedom + 1.0) / -2.0
Q = np.maximum(dist / (2.0 * np.sum(dist)), MACHINE_EPSILON)
# Optimization trick below: np.dot(x, y) is faster than
# np.sum(x * y) because it calls BLAS
# Objective: C (Kullback-Leibler divergence of P and Q)
if compute_error:
kl_divergence = 2.0 * np.dot(
P, np.log(np.maximum(P, MACHINE_EPSILON) / Q))
else:
kl_divergence = np.nan
# Gradient: dC/dY
# pdist always returns double precision distances. Thus we need to take
grad = np.ndarray((n_samples, n_components), dtype=params.dtype)
PQd = squareform((P - Q) * dist)
for i in range(skip_num_points, n_samples):
grad[i] = np.dot(np.ravel(PQd[i], order='K'),
X_embedded[i] - X_embedded)
grad = grad.ravel()
c = 2.0 * (degrees_of_freedom + 1.0) / degrees_of_freedom
grad *= c
return kl_divergence, grad
def _kl_divergence_bh(params, P, degrees_of_freedom, n_samples, n_components,
angle=0.5, skip_num_points=0, verbose=False,
compute_error=True, num_threads=1):
"""t-SNE objective function: KL divergence of p_ijs and q_ijs.
Uses Barnes-Hut tree methods to calculate the gradient that
runs in O(NlogN) instead of O(N^2)
Parameters
----------
params : array, shape (n_params,)
Unraveled embedding.
P : csr sparse matrix, shape (n_samples, n_sample)
Sparse approximate joint probability matrix, computed only for the
k nearest-neighbors and symmetrized.
degrees_of_freedom : int
Degrees of freedom of the Student's-t distribution.
n_samples : int
Number of samples.
n_components : int
Dimension of the embedded space.
angle : float (default: 0.5)
This is the trade-off between speed and accuracy for Barnes-Hut T-SNE.
'angle' is the angular size (referred to as theta in [3]) of a distant
node as measured from a point. If this size is below 'angle' then it is
used as a summary node of all points contained within it.
This method is not very sensitive to changes in this parameter
in the range of 0.2 - 0.8. Angle less than 0.2 has quickly increasing
computation time and angle greater 0.8 has quickly increasing error.
skip_num_points : int (optional, default:0)
This does not compute the gradient for points with indices below
`skip_num_points`. This is useful when computing transforms of new
data where you'd like to keep the old data fixed.
verbose : int
Verbosity level.
compute_error: bool (optional, default:True)
If False, the kl_divergence is not computed and returns NaN.
num_threads : int (optional, default:1)
Number of threads used to compute the gradient. This is set here to
avoid calling _openmp_effective_n_threads for each gradient step.
Returns
-------
kl_divergence : float
Kullback-Leibler divergence of p_ij and q_ij.
grad : array, shape (n_params,)
Unraveled gradient of the Kullback-Leibler divergence with respect to
the embedding.
"""
params = params.astype(np.float32, copy=False)
X_embedded = params.reshape(n_samples, n_components)
val_P = P.data.astype(np.float32, copy=False)
neighbors = P.indices.astype(np.int64, copy=False)
indptr = P.indptr.astype(np.int64, copy=False)
grad = np.zeros(X_embedded.shape, dtype=np.float32)
error = _barnes_hut_tsne.gradient(val_P, X_embedded, neighbors, indptr,
grad, angle, n_components, verbose,
dof=degrees_of_freedom,
compute_error=compute_error,
num_threads=num_threads)
c = 2.0 * (degrees_of_freedom + 1.0) / degrees_of_freedom
grad = grad.ravel()
grad *= c
return error, grad
def _gradient_descent(objective, p0, it, n_iter,
n_iter_check=1, n_iter_without_progress=300,
momentum=0.8, learning_rate=200.0, min_gain=0.01,
min_grad_norm=1e-7, verbose=0, args=None, kwargs=None):
"""Batch gradient descent with momentum and individual gains.
Parameters
----------
objective : function or callable
Should return a tuple of cost and gradient for a given parameter
vector. When expensive to compute, the cost can optionally
be None and can be computed every n_iter_check steps using
the objective_error function.
p0 : array-like, shape (n_params,)
Initial parameter vector.
it : int
Current number of iterations (this function will be called more than
once during the optimization).
n_iter : int
Maximum number of gradient descent iterations.
n_iter_check : int
Number of iterations before evaluating the global error. If the error
is sufficiently low, we abort the optimization.
n_iter_without_progress : int, optional (default: 300)
Maximum number of iterations without progress before we abort the
optimization.
momentum : float, within (0.0, 1.0), optional (default: 0.8)
The momentum generates a weight for previous gradients that decays
exponentially.
learning_rate : float, optional (default: 200.0)
The learning rate for t-SNE is usually in the range [10.0, 1000.0]. If
the learning rate is too high, the data may look like a 'ball' with any
point approximately equidistant from its nearest neighbours. If the
learning rate is too low, most points may look compressed in a dense
cloud with few outliers.
min_gain : float, optional (default: 0.01)
Minimum individual gain for each parameter.
min_grad_norm : float, optional (default: 1e-7)
If the gradient norm is below this threshold, the optimization will
be aborted.
verbose : int, optional (default: 0)
Verbosity level.
args : sequence
Arguments to pass to objective function.
kwargs : dict
Keyword arguments to pass to objective function.
Returns
-------
p : array, shape (n_params,)
Optimum parameters.
error : float
Optimum.
i : int
Last iteration.
"""
if args is None:
args = []
if kwargs is None:
kwargs = {}
p = p0.copy().ravel()
update = np.zeros_like(p)
gains = np.ones_like(p)
error = np.finfo(np.float).max
best_error = np.finfo(np.float).max
best_iter = i = it
tic = time()
for i in range(it, n_iter):
check_convergence = (i + 1) % n_iter_check == 0
# only compute the error when needed
kwargs['compute_error'] = check_convergence or i == n_iter - 1
error, grad = objective(p, *args, **kwargs)
grad_norm = linalg.norm(grad)
inc = update * grad < 0.0
dec = np.invert(inc)
gains[inc] += 0.2
gains[dec] *= 0.8
np.clip(gains, min_gain, np.inf, out=gains)
grad *= gains
update = momentum * update - learning_rate * grad
p += update
if check_convergence:
toc = time()
duration = toc - tic
tic = toc
if verbose >= 2:
print("[t-SNE] Iteration %d: error = %.7f,"
" gradient norm = %.7f"
" (%s iterations in %0.3fs)"
% (i + 1, error, grad_norm, n_iter_check, duration))
if error < best_error:
best_error = error
best_iter = i
elif i - best_iter > n_iter_without_progress:
if verbose >= 2:
print("[t-SNE] Iteration %d: did not make any progress "
"during the last %d episodes. Finished."
% (i + 1, n_iter_without_progress))
break
if grad_norm <= min_grad_norm:
if verbose >= 2:
print("[t-SNE] Iteration %d: gradient norm %f. Finished."
% (i + 1, grad_norm))
break
return p, error, i
def trustworthiness(X, X_embedded, n_neighbors=5, metric='euclidean'):
r"""Expresses to what extent the local structure is retained.
The trustworthiness is within [0, 1]. It is defined as
.. math::
T(k) = 1 - \frac{2}{nk (2n - 3k - 1)} \sum^n_{i=1}
\sum_{j \in \mathcal{N}_{i}^{k}} \max(0, (r(i, j) - k))
where for each sample i, :math:`\mathcal{N}_{i}^{k}` are its k nearest
neighbors in the output space, and every sample j is its :math:`r(i, j)`-th
nearest neighbor in the input space. In other words, any unexpected nearest
neighbors in the output space are penalised in proportion to their rank in
the input space.
* "Neighborhood Preservation in Nonlinear Projection Methods: An
Experimental Study"
J. Venna, S. Kaski
* "Learning a Parametric Embedding by Preserving Local Structure"
L.J.P. van der Maaten
Parameters
----------
X : array, shape (n_samples, n_features) or (n_samples, n_samples)
If the metric is 'precomputed' X must be a square distance
matrix. Otherwise it contains a sample per row.
X_embedded : array, shape (n_samples, n_components)
Embedding of the training data in low-dimensional space.
n_neighbors : int, optional (default: 5)
Number of neighbors k that will be considered.
metric : string, or callable, optional, default 'euclidean'
Which metric to use for computing pairwise distances between samples
from the original input space. If metric is 'precomputed', X must be a
matrix of pairwise distances or squared distances. Otherwise, see the
documentation of argument metric in sklearn.pairwise.pairwise_distances
for a list of available metrics.
Returns
-------
trustworthiness : float
Trustworthiness of the low-dimensional embedding.
"""
dist_X = pairwise_distances(X, metric=metric)
if metric == 'precomputed':
dist_X = dist_X.copy()
# we set the diagonal to np.inf to exclude the points themselves from
# their own neighborhood
np.fill_diagonal(dist_X, np.inf)
ind_X = np.argsort(dist_X, axis=1)
# `ind_X[i]` is the index of sorted distances between i and other samples
ind_X_embedded = NearestNeighbors(n_neighbors).fit(X_embedded).kneighbors(
return_distance=False)
# We build an inverted index of neighbors in the input space: For sample i,
# we define `inverted_index[i]` as the inverted index of sorted distances:
# inverted_index[i][ind_X[i]] = np.arange(1, n_sample + 1)
n_samples = X.shape[0]
inverted_index = np.zeros((n_samples, n_samples), dtype=int)
ordered_indices = np.arange(n_samples + 1)
inverted_index[ordered_indices[:-1, np.newaxis],
ind_X] = ordered_indices[1:]
ranks = inverted_index[ordered_indices[:-1, np.newaxis],
ind_X_embedded] - n_neighbors
t = np.sum(ranks[ranks > 0])
t = 1.0 - t * (2.0 / (n_samples * n_neighbors *
(2.0 * n_samples - 3.0 * n_neighbors - 1.0)))
return t
class TSNE(BaseEstimator):
"""t-distributed Stochastic Neighbor Embedding.
t-SNE [1] is a tool to visualize high-dimensional data. It converts
similarities between data points to joint probabilities and tries
to minimize the Kullback-Leibler divergence between the joint
probabilities of the low-dimensional embedding and the
high-dimensional data. t-SNE has a cost function that is not convex,
i.e. with different initializations we can get different results.
It is highly recommended to use another dimensionality reduction
method (e.g. PCA for dense data or TruncatedSVD for sparse data)
to reduce the number of dimensions to a reasonable amount (e.g. 50)
if the number of features is very high. This will suppress some
noise and speed up the computation of pairwise distances between
samples. For more tips see Laurens van der Maaten's FAQ [2].
Read more in the :ref:`User Guide <t_sne>`.
Parameters
----------
n_components : int, optional (default: 2)
Dimension of the embedded space.
perplexity : float, optional (default: 30)
The perplexity is related to the number of nearest neighbors that
is used in other manifold learning algorithms. Larger datasets
usually require a larger perplexity. Consider selecting a value
between 5 and 50. Different values can result in significanlty
different results.
early_exaggeration : float, optional (default: 12.0)
Controls how tight natural clusters in the original space are in
the embedded space and how much space will be between them. For
larger values, the space between natural clusters will be larger
in the embedded space. Again, the choice of this parameter is not
very critical. If the cost function increases during initial
optimization, the early exaggeration factor or the learning rate
might be too high.
learning_rate : float, optional (default: 200.0)
The learning rate for t-SNE is usually in the range [10.0, 1000.0]. If
the learning rate is too high, the data may look like a 'ball' with any
point approximately equidistant from its nearest neighbours. If the
learning rate is too low, most points may look compressed in a dense
cloud with few outliers. If the cost function gets stuck in a bad local
minimum increasing the learning rate may help.
n_iter : int, optional (default: 1000)
Maximum number of iterations for the optimization. Should be at
least 250.
n_iter_without_progress : int, optional (default: 300)
Maximum number of iterations without progress before we abort the
optimization, used after 250 initial iterations with early
exaggeration. Note that progress is only checked every 50 iterations so
this value is rounded to the next multiple of 50.
.. versionadded:: 0.17
parameter *n_iter_without_progress* to control stopping criteria.
min_grad_norm : float, optional (default: 1e-7)
If the gradient norm is below this threshold, the optimization will
be stopped.
metric : string or callable, optional
The metric to use when calculating distance between instances in a
feature array. If metric is a string, it must be one of the options
allowed by scipy.spatial.distance.pdist for its metric parameter, or
a metric listed in pairwise.PAIRWISE_DISTANCE_FUNCTIONS.
If metric is "precomputed", X is assumed to be a distance matrix.
Alternatively, if metric is a callable function, it is called on each
pair of instances (rows) and the resulting value recorded. The callable
should take two arrays from X as input and return a value indicating
the distance between them. The default is "euclidean" which is
interpreted as squared euclidean distance.
init : string or numpy array, optional (default: "random")
Initialization of embedding. Possible options are 'random', 'pca',
and a numpy array of shape (n_samples, n_components).
PCA initialization cannot be used with precomputed distances and is
usually more globally stable than random initialization.
verbose : int, optional (default: 0)
Verbosity level.
random_state : int, RandomState instance or None, optional (default: None)
If int, random_state is the seed used by the random number generator;
If RandomState instance, random_state is the random number generator;
If None, the random number generator is the RandomState instance used
by `np.random`. Note that different initializations might result in
different local minima of the cost function.
method : string (default: 'barnes_hut')
By default the gradient calculation algorithm uses Barnes-Hut
approximation running in O(NlogN) time. method='exact'
will run on the slower, but exact, algorithm in O(N^2) time. The
exact algorithm should be used when nearest-neighbor errors need
to be better than 3%. However, the exact method cannot scale to
millions of examples.
.. versionadded:: 0.17
Approximate optimization *method* via the Barnes-Hut.
angle : float (default: 0.5)
Only used if method='barnes_hut'
This is the trade-off between speed and accuracy for Barnes-Hut T-SNE.
'angle' is the angular size (referred to as theta in [3]) of a distant
node as measured from a point. If this size is below 'angle' then it is
used as a summary node of all points contained within it.
This method is not very sensitive to changes in this parameter
in the range of 0.2 - 0.8. Angle less than 0.2 has quickly increasing
computation time and angle greater 0.8 has quickly increasing error.
n_jobs : int or None, optional (default=None)
The number of parallel jobs to run for neighbors search. This parameter
has no impact when ``metric="precomputed"`` or
(``metric="euclidean"`` and ``method="exact"``).
``None`` means 1 unless in a :obj:`joblib.parallel_backend` context.
``-1`` means using all processors. See :term:`Glossary <n_jobs>`
for more details.
.. versionadded:: 0.22
Attributes
----------
embedding_ : array-like, shape (n_samples, n_components)
Stores the embedding vectors.
kl_divergence_ : float
Kullback-Leibler divergence after optimization.
n_iter_ : int
Number of iterations run.
Examples
--------
>>> import numpy as np
>>> from sklearn.manifold import TSNE
>>> X = np.array([[0, 0, 0], [0, 1, 1], [1, 0, 1], [1, 1, 1]])
>>> X_embedded = TSNE(n_components=2).fit_transform(X)
>>> X_embedded.shape
(4, 2)
References
----------
[1] van der Maaten, L.J.P.; Hinton, G.E. Visualizing High-Dimensional Data
Using t-SNE. Journal of Machine Learning Research 9:2579-2605, 2008.
[2] van der Maaten, L.J.P. t-Distributed Stochastic Neighbor Embedding
https://lvdmaaten.github.io/tsne/
[3] L.J.P. van der Maaten. Accelerating t-SNE using Tree-Based Algorithms.
Journal of Machine Learning Research 15(Oct):3221-3245, 2014.
https://lvdmaaten.github.io/publications/papers/JMLR_2014.pdf
"""
# Control the number of exploration iterations with early_exaggeration on
_EXPLORATION_N_ITER = 250
# Control the number of iterations between progress checks
_N_ITER_CHECK = 50
def __init__(self, n_components=2, perplexity=30.0,
early_exaggeration=12.0, learning_rate=200.0, n_iter=1000,
n_iter_without_progress=300, min_grad_norm=1e-7,
metric="euclidean", init="random", verbose=0,
random_state=None, method='barnes_hut', angle=0.5,
n_jobs=None):
self.n_components = n_components
self.perplexity = perplexity
self.early_exaggeration = early_exaggeration
self.learning_rate = learning_rate
self.n_iter = n_iter
self.n_iter_without_progress = n_iter_without_progress
self.min_grad_norm = min_grad_norm
self.metric = metric
self.init = init
self.verbose = verbose
self.random_state = random_state
self.method = method
self.angle = angle
self.n_jobs = n_jobs
def _fit(self, X, skip_num_points=0):
"""Private function to fit the model using X as training data."""
if self.method not in ['barnes_hut', 'exact']:
raise ValueError("'method' must be 'barnes_hut' or 'exact'")
if self.angle < 0.0 or self.angle > 1.0:
raise ValueError("'angle' must be between 0.0 - 1.0")
if self.method == 'barnes_hut':
X = check_array(X, accept_sparse=['csr'], ensure_min_samples=2,
dtype=[np.float32, np.float64])
else:
X = check_array(X, accept_sparse=['csr', 'csc', 'coo'],
dtype=[np.float32, np.float64])
if self.metric == "precomputed":
if isinstance(self.init, str) and self.init == 'pca':
raise ValueError("The parameter init=\"pca\" cannot be "
"used with metric=\"precomputed\".")
if X.shape[0] != X.shape[1]:
raise ValueError("X should be a square distance matrix")
check_non_negative(X, "TSNE.fit(). With metric='precomputed', X "
"should contain positive distances.")
if self.method == "exact" and issparse(X):
raise TypeError(
'TSNE with method="exact" does not accept sparse '
'precomputed distance matrix. Use method="barnes_hut" '
'or provide the dense distance matrix.')
if self.method == 'barnes_hut' and self.n_components > 3:
raise ValueError("'n_components' should be inferior to 4 for the "
"barnes_hut algorithm as it relies on "
"quad-tree or oct-tree.")
random_state = check_random_state(self.random_state)
if self.early_exaggeration < 1.0:
raise ValueError("early_exaggeration must be at least 1, but is {}"
.format(self.early_exaggeration))
if self.n_iter < 250:
raise ValueError("n_iter should be at least 250")
n_samples = X.shape[0]
neighbors_nn = None
if self.method == "exact":
# Retrieve the distance matrix, either using the precomputed one or
# computing it.
if self.metric == "precomputed":
distances = X
else:
if self.verbose:
print("[t-SNE] Computing pairwise distances...")
if self.metric == "euclidean":
distances = pairwise_distances(X, metric=self.metric,
squared=True)
else:
distances = pairwise_distances(X, metric=self.metric,
n_jobs=self.n_jobs)
if np.any(distances < 0):
raise ValueError("All distances should be positive, the "
"metric given is not correct")
# compute the joint probability distribution for the input space
P = _joint_probabilities(distances, self.perplexity, self.verbose)
assert np.all(np.isfinite(P)), "All probabilities should be finite"
assert np.all(P >= 0), "All probabilities should be non-negative"
assert np.all(P <= 1), ("All probabilities should be less "
"or then equal to one")
else:
# Compute the number of nearest neighbors to find.
# LvdM uses 3 * perplexity as the number of neighbors.
# In the event that we have very small # of points
# set the neighbors to n - 1.
n_neighbors = min(n_samples - 1, int(3. * self.perplexity + 1))
if self.verbose:
print("[t-SNE] Computing {} nearest neighbors..."
.format(n_neighbors))
# Find the nearest neighbors for every point
knn = NearestNeighbors(algorithm='auto',
n_jobs=self.n_jobs,
n_neighbors=n_neighbors,
metric=self.metric)
t0 = time()
knn.fit(X)
duration = time() - t0
if self.verbose:
print("[t-SNE] Indexed {} samples in {:.3f}s...".format(
n_samples, duration))
t0 = time()
distances_nn = knn.kneighbors_graph(mode='distance')
duration = time() - t0
if self.verbose:
print("[t-SNE] Computed neighbors for {} samples "
"in {:.3f}s...".format(n_samples, duration))
# Free the memory used by the ball_tree
del knn
if self.metric == "euclidean":
# knn return the euclidean distance but we need it squared
# to be consistent with the 'exact' method. Note that the
# the method was derived using the euclidean method as in the
# input space. Not sure of the implication of using a different
# metric.
distances_nn.data **= 2
# compute the joint probability distribution for the input space
P = _joint_probabilities_nn(distances_nn, self.perplexity,
self.verbose)
if isinstance(self.init, np.ndarray):
X_embedded = self.init
elif self.init == 'pca':
pca = PCA(n_components=self.n_components, svd_solver='randomized',
random_state=random_state)
X_embedded = pca.fit_transform(X).astype(np.float32, copy=False)
elif self.init == 'random':
# The embedding is initialized with iid samples from Gaussians with
# standard deviation 1e-4.
X_embedded = 1e-4 * random_state.randn(
n_samples, self.n_components).astype(np.float32)
else:
raise ValueError("'init' must be 'pca', 'random', or "
"a numpy array")
# Degrees of freedom of the Student's t-distribution. The suggestion
# degrees_of_freedom = n_components - 1 comes from
# "Learning a Parametric Embedding by Preserving Local Structure"
# Laurens van der Maaten, 2009.
degrees_of_freedom = max(self.n_components - 1, 1)
return self._tsne(P, degrees_of_freedom, n_samples,
X_embedded=X_embedded,
neighbors=neighbors_nn,
skip_num_points=skip_num_points)
def _tsne(self, P, degrees_of_freedom, n_samples, X_embedded,
neighbors=None, skip_num_points=0):
"""Runs t-SNE."""
# t-SNE minimizes the Kullback-Leiber divergence of the Gaussians P
# and the Student's t-distributions Q. The optimization algorithm that
# we use is batch gradient descent with two stages:
# * initial optimization with early exaggeration and momentum at 0.5
# * final optimization with momentum at 0.8
params = X_embedded.ravel()
opt_args = {
"it": 0,
"n_iter_check": self._N_ITER_CHECK,
"min_grad_norm": self.min_grad_norm,
"learning_rate": self.learning_rate,
"verbose": self.verbose,
"kwargs": dict(skip_num_points=skip_num_points),
"args": [P, degrees_of_freedom, n_samples, self.n_components],
"n_iter_without_progress": self._EXPLORATION_N_ITER,
"n_iter": self._EXPLORATION_N_ITER,
"momentum": 0.5,
}
if self.method == 'barnes_hut':
obj_func = _kl_divergence_bh
opt_args['kwargs']['angle'] = self.angle
# Repeat verbose argument for _kl_divergence_bh
opt_args['kwargs']['verbose'] = self.verbose
# Get the number of threads for gradient computation here to
# avoid recomputing it at each iteration.
opt_args['kwargs']['num_threads'] = _openmp_effective_n_threads()
else:
obj_func = _kl_divergence
# Learning schedule (part 1): do 250 iteration with lower momentum but
# higher learning rate controlled via the early exaggeration parameter
P *= self.early_exaggeration
params, kl_divergence, it = _gradient_descent(obj_func, params,
**opt_args)
if self.verbose:
print("[t-SNE] KL divergence after %d iterations with early "
"exaggeration: %f" % (it + 1, kl_divergence))
# Learning schedule (part 2): disable early exaggeration and finish
# optimization with a higher momentum at 0.8
P /= self.early_exaggeration
remaining = self.n_iter - self._EXPLORATION_N_ITER
if it < self._EXPLORATION_N_ITER or remaining > 0:
opt_args['n_iter'] = self.n_iter
opt_args['it'] = it + 1
opt_args['momentum'] = 0.8
opt_args['n_iter_without_progress'] = self.n_iter_without_progress
params, kl_divergence, it = _gradient_descent(obj_func, params,
**opt_args)
# Save the final number of iterations
self.n_iter_ = it
if self.verbose:
print("[t-SNE] KL divergence after %d iterations: %f"
% (it + 1, kl_divergence))
X_embedded = params.reshape(n_samples, self.n_components)
self.kl_divergence_ = kl_divergence
return X_embedded
def fit_transform(self, X, y=None):
"""Fit X into an embedded space and return that transformed
output.
Parameters
----------
X : array, shape (n_samples, n_features) or (n_samples, n_samples)
If the metric is 'precomputed' X must be a square distance
matrix. Otherwise it contains a sample per row. If the method
is 'exact', X may be a sparse matrix of type 'csr', 'csc'
or 'coo'. If the method is 'barnes_hut' and the metric is
'precomputed', X may be a precomputed sparse graph.
y : Ignored
Returns
-------
X_new : array, shape (n_samples, n_components)
Embedding of the training data in low-dimensional space.
"""
embedding = self._fit(X)
self.embedding_ = embedding
return self.embedding_
def fit(self, X, y=None):
"""Fit X into an embedded space.
Parameters
----------
X : array, shape (n_samples, n_features) or (n_samples, n_samples)
If the metric is 'precomputed' X must be a square distance
matrix. Otherwise it contains a sample per row. If the method
is 'exact', X may be a sparse matrix of type 'csr', 'csc'
or 'coo'. If the method is 'barnes_hut' and the metric is
'precomputed', X may be a precomputed sparse graph.
y : Ignored
"""
self.fit_transform(X)
return self