# 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: # http://cseweb.ucsd.edu/~lvdmaaten/workshops/nips2010/papers/vandermaaten.pdf import numpy as np from scipy import linalg import scipy.sparse as sp from scipy.spatial.distance import pdist from scipy.spatial.distance import squareform from ..neighbors import BallTree from ..base import BaseEstimator from ..utils import check_array from ..utils import check_random_state from ..utils.extmath import _ravel from ..decomposition import PCA from ..metrics.pairwise import pairwise_distances from . import _utils from . import _barnes_hut_tsne from ..utils.fixes import astype from ..externals.six import string_types 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 = astype(distances, np.float32, copy=False) conditional_P = _utils._binary_search_perplexity( distances, None, 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, neighbors, 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 : 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 = astype(distances, np.float32, copy=False) neighbors = astype(neighbors, np.int64, copy=False) conditional_P = _utils._binary_search_perplexity( distances, neighbors, desired_perplexity, verbose) m = "All probabilities should be finite" assert np.all(np.isfinite(conditional_P)), m P = conditional_P + conditional_P.T sum_P = np.maximum(np.sum(P), MACHINE_EPSILON) P = np.maximum(squareform(P) / sum_P, MACHINE_EPSILON) assert np.all(np.abs(P) <= 1.0) return P def _kl_divergence(params, P, degrees_of_freedom, n_samples, n_components, skip_num_points=0): """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 : float 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. 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 n = pdist(X_embedded, "sqeuclidean") n += 1. n /= degrees_of_freedom n **= (degrees_of_freedom + 1.0) / -2.0 Q = np.maximum(n / (2.0 * np.sum(n)), 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) kl_divergence = 2.0 * np.dot(P, np.log(P / Q)) # Gradient: dC/dY grad = np.ndarray((n_samples, n_components)) PQd = squareform((P - Q) * n) for i in range(skip_num_points, n_samples): np.dot(_ravel(PQd[i]), X_embedded[i] - X_embedded, out=grad[i]) grad = grad.ravel() c = 2.0 * (degrees_of_freedom + 1.0) / degrees_of_freedom grad *= c return kl_divergence, grad def _kl_divergence_error(params, P, neighbors, degrees_of_freedom, n_samples, n_components): """t-SNE objective function: the absolute error of the KL divergence of p_ijs and q_ijs. Parameters ---------- params : array, shape (n_params,) Unraveled embedding. P : array, shape (n_samples * (n_samples-1) / 2,) Condensed joint probability matrix. neighbors : array (n_samples, K) The neighbors is not actually required to calculate the divergence, but is here to match the signature of the gradient function degrees_of_freedom : float Degrees of freedom of the Student's-t distribution. n_samples : int Number of samples. n_components : int Dimension of the embedded space. 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 n = pdist(X_embedded, "sqeuclidean") n += 1. n /= degrees_of_freedom n **= (degrees_of_freedom + 1.0) / -2.0 Q = np.maximum(n / (2.0 * np.sum(n)), 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 len(P.shape) == 2: P = squareform(P) kl_divergence = 2.0 * np.dot(P, np.log(P / Q)) return kl_divergence def _kl_divergence_bh(params, P, neighbors, degrees_of_freedom, n_samples, n_components, angle=0.5, skip_num_points=0, verbose=False): """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 : array, shape (n_samples * (n_samples-1) / 2,) Condensed joint probability matrix. neighbors: int64 array, shape (n_samples, K) Array with element [i, j] giving the index for the jth closest neighbor to point i. degrees_of_freedom : float 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. 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 = astype(params, np.float32, copy=False) X_embedded = params.reshape(n_samples, n_components) neighbors = astype(neighbors, np.int64, copy=False) if len(P.shape) == 1: sP = squareform(P).astype(np.float32) else: sP = P.astype(np.float32) grad = np.zeros(X_embedded.shape, dtype=np.float32) error = _barnes_hut_tsne.gradient(sP, X_embedded, neighbors, grad, angle, n_components, verbose, dof=degrees_of_freedom) 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, objective_error=None, n_iter_check=1, n_iter_without_progress=50, momentum=0.5, learning_rate=1000.0, min_gain=0.01, min_grad_norm=1e-7, min_error_diff=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. objective_error : function or callable Should return a tuple of cost and gradient for a given parameter vector. n_iter_without_progress : int, optional (default: 30) Maximum number of iterations without progress before we abort the optimization. momentum : float, within (0.0, 1.0), optional (default: 0.5) The momentum generates a weight for previous gradients that decays exponentially. learning_rate : float, optional (default: 1000.0) The learning rate should be extremely high for t-SNE! Values in the range [100.0, 1000.0] are common. 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. min_error_diff : float, optional (default: 1e-7) If the absolute difference of two successive cost function values 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 = 0 for i in range(it, n_iter): new_error, grad = objective(p, *args, **kwargs) grad_norm = linalg.norm(grad) inc = update * grad >= 0.0 dec = np.invert(inc) gains[inc] += 0.05 gains[dec] *= 0.95 np.clip(gains, min_gain, np.inf) grad *= gains update = momentum * update - learning_rate * grad p += update if (i + 1) % n_iter_check == 0: if new_error is None: new_error = objective_error(p, *args) error_diff = np.abs(new_error - error) error = new_error if verbose >= 2: m = "[t-SNE] Iteration %d: error = %.7f, gradient norm = %.7f" print(m % (i + 1, error, grad_norm)) 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 if error_diff <= min_error_diff: if verbose >= 2: m = "[t-SNE] Iteration %d: error difference %f. Finished." print(m % (i + 1, error_diff)) break if new_error is not None: error = new_error return p, error, i def trustworthiness(X, X_embedded, n_neighbors=5, precomputed=False): """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 U^{(k)}_i (r(i, j) - k)} where :math:`r(i, j)` is the rank of the embedded datapoint j according to the pairwise distances between the embedded datapoints, :math:`U^{(k)}_i` is the set of points that are in the k nearest neighbors in the embedded space but not in the original 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. precomputed : bool, optional (default: False) Set this flag if X is a precomputed square distance matrix. Returns ------- trustworthiness : float Trustworthiness of the low-dimensional embedding. """ if precomputed: dist_X = X else: dist_X = pairwise_distances(X, squared=True) dist_X_embedded = pairwise_distances(X_embedded, squared=True) ind_X = np.argsort(dist_X, axis=1) ind_X_embedded = np.argsort(dist_X_embedded, axis=1)[:, 1:n_neighbors + 1] n_samples = X.shape[0] t = 0.0 ranks = np.zeros(n_neighbors) for i in range(n_samples): for j in range(n_neighbors): ranks[j] = np.where(ind_X[i] == ind_X_embedded[i, j])[0][0] ranks -= 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. The choice is not extremely critical since t-SNE is quite insensitive to this parameter. early_exaggeration : float, optional (default: 4.0) 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: 1000) The learning rate can be a critical parameter. It should be between 100 and 1000. If the cost function increases during initial optimization, the early exaggeration factor or the learning rate might be too high. If the cost function gets stuck in a bad local minimum increasing the learning rate helps sometimes. n_iter : int, optional (default: 1000) Maximum number of iterations for the optimization. Should be at least 200. n_iter_without_progress : int, optional (default: 30) Maximum number of iterations without progress before we abort the optimization. .. 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 aborted. 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 or RandomState instance or None (default) Pseudo Random Number generator seed control. If None, use the numpy.random singleton. 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. Attributes ---------- embedding_ : array-like, shape (n_samples, n_components) Stores the embedding vectors. kl_divergence_ : float Kullback-Leibler divergence after optimization. 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]]) >>> model = TSNE(n_components=2, random_state=0) >>> np.set_printoptions(suppress=True) >>> model.fit_transform(X) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE array([[ 0.00017599, 0.00003993], [ 0.00009891, 0.00021913], [ 0.00018554, -0.00009357], [ 0.00009528, -0.00001407]]) 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 http://homepage.tudelft.nl/19j49/t-SNE.html [3] L.J.P. van der Maaten. Accelerating t-SNE using Tree-Based Algorithms. Journal of Machine Learning Research 15(Oct):3221-3245, 2014. http://lvdmaaten.github.io/publications/papers/JMLR_2014.pdf """ def __init__(self, n_components=2, perplexity=30.0, early_exaggeration=4.0, learning_rate=1000.0, n_iter=1000, n_iter_without_progress=30, min_grad_norm=1e-7, metric="euclidean", init="random", verbose=0, random_state=None, method='barnes_hut', angle=0.5): if not ((isinstance(init, string_types) and init in ["pca", "random"]) or isinstance(init, np.ndarray)): msg = "'init' must be 'pca', 'random', or a numpy array" raise ValueError(msg) 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.embedding_ = None def _fit(self, X, skip_num_points=0): """Fit the model using X as training data. Note that sparse arrays can only be handled by method='exact'. It is recommended that you convert your sparse array to dense (e.g. `X.toarray()`) if it fits in memory, or otherwise using a dimensionality reduction technique (e.g. TruncatedSVD). 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. Note that this when method='barnes_hut', X cannot be a sparse array and if need be will be converted to a 32 bit float array. Method='exact' allows sparse arrays and 64bit floating point inputs. 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. """ 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' and sp.issparse(X): raise TypeError('A sparse matrix was passed, but dense ' 'data is required for method="barnes_hut". Use ' 'X.toarray() to convert to a dense numpy array if ' 'the array is small enough for it to fit in ' 'memory. Otherwise consider dimensionality ' 'reduction techniques (e.g. TruncatedSVD)') else: X = check_array(X, accept_sparse=['csr', 'csc', 'coo'], dtype=np.float64) 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 " "%f" % self.early_exaggeration) if self.n_iter < 200: raise ValueError("n_iter should be at least 200") if self.metric == "precomputed": if isinstance(self.init, string_types) 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") 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) if not np.all(distances >= 0): raise ValueError("All distances should be positive, either " "the metric or precomputed distances given " "as X are not correct") # 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 - 1.0, 1) n_samples = X.shape[0] # the number of nearest neighbors to find k = min(n_samples - 1, int(3. * self.perplexity + 1)) neighbors_nn = None if self.method == 'barnes_hut': if self.verbose: print("[t-SNE] Computing %i nearest neighbors..." % k) if self.metric == 'precomputed': # Use the precomputed distances to find # the k nearest neighbors and their distances neighbors_nn = np.argsort(distances, axis=1)[:, :k] else: # Find the nearest neighbors for every point bt = BallTree(X) # LvdM uses 3 * perplexity as the number of neighbors # And we add one to not count the data point itself # In the event that we have very small # of points # set the neighbors to n - 1 distances_nn, neighbors_nn = bt.query(X, k=k + 1) neighbors_nn = neighbors_nn[:, 1:] P = _joint_probabilities_nn(distances, neighbors_nn, self.perplexity, self.verbose) else: 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 zero or positive" assert np.all(P <= 1), ("All probabilities should be less " "or then equal to one") 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) elif self.init == 'random': X_embedded = None else: raise ValueError("Unsupported initialization scheme: %s" % self.init) return self._tsne(P, degrees_of_freedom, n_samples, random_state, X_embedded=X_embedded, neighbors=neighbors_nn, skip_num_points=skip_num_points) def _tsne(self, P, degrees_of_freedom, n_samples, random_state, X_embedded=None, 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 three stages: # * early exaggeration with momentum 0.5 # * early exaggeration with momentum 0.8 # * final optimization with momentum 0.8 # The embedding is initialized with iid samples from Gaussians with # standard deviation 1e-4. if X_embedded is None: # Initialize embedding randomly X_embedded = 1e-4 * random_state.randn(n_samples, self.n_components) params = X_embedded.ravel() opt_args = {} opt_args = {"n_iter": 50, "momentum": 0.5, "it": 0, "learning_rate": self.learning_rate, "verbose": self.verbose, "n_iter_check": 25, "kwargs": dict(skip_num_points=skip_num_points)} if self.method == 'barnes_hut': m = "Must provide an array of neighbors to use Barnes-Hut" assert neighbors is not None, m obj_func = _kl_divergence_bh objective_error = _kl_divergence_error sP = squareform(P).astype(np.float32) neighbors = neighbors.astype(np.int64) args = [sP, neighbors, degrees_of_freedom, n_samples, self.n_components] opt_args['args'] = args opt_args['min_grad_norm'] = 1e-3 opt_args['n_iter_without_progress'] = 30 # Don't always calculate the cost since that calculation # can be nearly as expensive as the gradient opt_args['objective_error'] = objective_error opt_args['kwargs']['angle'] = self.angle opt_args['kwargs']['verbose'] = self.verbose else: obj_func = _kl_divergence opt_args['args'] = [P, degrees_of_freedom, n_samples, self.n_components] opt_args['min_error_diff'] = 0.0 opt_args['min_grad_norm'] = 0.0 # Early exaggeration P *= self.early_exaggeration params, kl_divergence, it = _gradient_descent(obj_func, params, **opt_args) opt_args['n_iter'] = 100 opt_args['momentum'] = 0.8 opt_args['it'] = it + 1 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)) # Save the final number of iterations self.n_iter_final = it # Final optimization P /= self.early_exaggeration opt_args['n_iter'] = self.n_iter opt_args['it'] = it + 1 params, error, it = _gradient_descent(obj_func, params, **opt_args) if self.verbose: print("[t-SNE] Error 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. 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'. """ self.fit_transform(X) return self