# Author: Alexander Fabisch -- # Author: Christopher Moody # Author: Nick Travers # 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_random_state from ..utils._openmp_helpers import _openmp_effective_n_threads from ..utils.validation import check_non_negative from ..utils.validation import _deprecate_positional_args from ..decomposition import PCA from ..metrics.pairwise import pairwise_distances from . import _utils # mypy error: Module 'sklearn.manifold' has no attribute '_barnes_hut_tsne' from . import _barnes_hut_tsne # type: ignore 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 @_deprecate_positional_args 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. .. versionadded:: 0.20 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=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 `. 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, default=None Determines the random number generator. Pass an int for reproducible results across multiple function calls. Note that different initializations might result in different local minima of the cost function. See :term: `Glossary `. 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 ` 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 @_deprecate_positional_args 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 = self._validate_data(X, accept_sparse=['csr'], ensure_min_samples=2, dtype=[np.float32, np.float64]) else: X = self._validate_data(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