# Author: Alexander Fabisch -- # License: BSD 3 clause (C) 2014 # This is the standard t-SNE implementation. There are faster modifications of # the algorithm: # * Barnes-Hut-SNE: reduces the complexity of the gradient computation from # N^2 to N log N (http://arxiv.org/abs/1301.3342) # * Fast Optimization for t-SNE: # http://cseweb.ucsd.edu/~lvdmaaten/workshops/nips2010/papers/vandermaaten.pdf import numpy as np from scipy import linalg from scipy.spatial.distance import pdist from scipy.spatial.distance import squareform from ..base import BaseEstimator from ..utils import check_array from ..utils import check_random_state from ..utils.extmath import _ravel from ..decomposition import RandomizedPCA from ..metrics.pairwise import pairwise_distances from . import _utils 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 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 _kl_divergence(params, P, alpha, n_samples, n_components): """t-SNE objective function: 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. alpha : 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 /= alpha n **= (alpha + 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(n_samples): np.dot(_ravel(PQd[i]), X_embedded[i] - X_embedded, out=grad[i]) grad = grad.ravel() c = 2.0 * (alpha + 1.0) / alpha grad *= c return kl_divergence, grad def _gradient_descent(objective, p0, it, n_iter, n_iter_without_progress=30, momentum=0.5, learning_rate=1000.0, min_gain=0.01, min_grad_norm=1e-7, min_error_diff=1e-7, verbose=0, args=[]): """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. 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_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. Returns ------- p : array, shape (n_params,) Optimum parameters. error : float Optimum. i : int Last iteration. """ 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) error_diff = np.abs(new_error - error) error = new_error grad_norm = linalg.norm(grad) 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 min_grad_norm >= grad_norm: if verbose >= 2: print("[t-SNE] Iteration %d: gradient norm %f. Finished." % (i + 1, grad_norm)) break if min_error_diff >= error_diff: if verbose >= 2: print("[t-SNE] Iteration %d: error difference %f. Finished." % (i + 1, error_diff)) break 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 verbose >= 2 and (i+1) % 10 == 0: print("[t-SNE] Iteration %d: error = %.7f, gradient norm = %.7f" % (i + 1, error, grad_norm)) 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]. 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 selcting 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. metric : string or callable, (default: "euclidean") 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. init : string, optional (default: "random") Initialization of embedding. Possible options are 'random' and 'pca'. 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. Attributes ---------- embedding_ : array-like, shape (n_samples, n_components) Stores the embedding vectors. training_data_ : array-like, shape (n_samples, n_features) Stores the training data. 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) >>> model.fit_transform(X) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE array([[ 887.28..., 238.61...], [ -714.79..., 3243.34...], [ 957.30..., -2505.78...], [-1130.28..., -974.78...]) 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 """ def __init__(self, n_components=2, perplexity=30.0, early_exaggeration=4.0, learning_rate=1000.0, n_iter=1000, metric="euclidean", init="random", verbose=0, random_state=None): if init not in ["pca", "random"]: raise ValueError("'init' must be either 'pca' or 'random'") self.n_components = n_components self.perplexity = perplexity self.early_exaggeration = early_exaggeration self.learning_rate = learning_rate self.n_iter = n_iter self.metric = metric self.init = init self.verbose = verbose self.random_state = random_state def _fit(self, X): """Fit the model using X as training data. 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 = check_array(X, accept_sparse=['csr', 'csc', 'coo']) 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 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...") distances = pairwise_distances(X, metric=self.metric, squared=True) # Degrees of freedom of the Student's t-distribution. The suggestion # alpha = n_components - 1 comes from "Learning a Parametric Embedding # by Preserving Local Structure" Laurens van der Maaten, 2009. alpha = self.n_components - 1.0 n_samples = X.shape[0] self.training_data_ = X P = _joint_probabilities(distances, self.perplexity, self.verbose) if self.init == 'pca': pca = RandomizedPCA(n_components=self.n_components, 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) self.embedding_ = self._tsne(P, alpha, n_samples, random_state, X_embedded=X_embedded) def _tsne(self, P, alpha, n_samples, random_state, X_embedded=None): """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() # Early exaggeration P *= self.early_exaggeration params, error, it = _gradient_descent( _kl_divergence, params, it=0, n_iter=50, momentum=0.5, min_grad_norm=0.0, min_error_diff=0.0, learning_rate=self.learning_rate, verbose=self.verbose, args=[P, alpha, n_samples, self.n_components]) params, error, it = _gradient_descent( _kl_divergence, params, it=it + 1, n_iter=100, momentum=0.8, min_grad_norm=0.0, min_error_diff=0.0, learning_rate=self.learning_rate, verbose=self.verbose, args=[P, alpha, n_samples, self.n_components]) if self.verbose: print("[t-SNE] Error after %d iterations with early " "exaggeration: %f" % (it + 1, error)) # Final optimization P /= self.early_exaggeration params, error, it = _gradient_descent( _kl_divergence, params, it=it + 1, n_iter=self.n_iter, momentum=0.8, learning_rate=self.learning_rate, verbose=self.verbose, args=[P, alpha, n_samples, self.n_components]) if self.verbose: print("[t-SNE] Error after %d iterations: %f" % (it + 1, error)) X_embedded = params.reshape(n_samples, self.n_components) return X_embedded def fit_transform(self, X): """Transform X to the 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. Returns ------- X_new : array, shape (n_samples, n_components) Embedding of the training data in low-dimensional space. """ self._fit(X) return self.embedding_