""" ============================================================================= Manifold learning on handwritten digits: Locally Linear Embedding, Isomap... ============================================================================= An illustration of various embeddings on the digits dataset. The RandomTreesEmbedding, from the :mod:`sklearn.ensemble` module, is not technically a manifold embedding method, as it learn a high-dimensional representation on which we apply a dimensionality reduction method. However, it is often useful to cast a dataset into a representation in which the classes are linearly-separable. t-SNE will be initialized with the embedding that is generated by PCA in this example, which is not the default setting. It ensures global stability of the embedding, i.e., the embedding does not depend on random initialization. Linear Discriminant Analysis, from the :mod:`sklearn.discriminant_analysis` module, and Neighborhood Components Analysis, from the :mod:`sklearn.neighbors` module, are supervised dimensionality reduction method, i.e. they make use of the provided labels, contrary to other methods. Adapted from `<https://scikit-learn.org/stable/auto_examples/manifold/plot_lle_digits.html>`_ """ # Authors: Fabian Pedregosa <fabian.pedregosa@inria.fr> # Olivier Grisel <olivier.grisel@ensta.org> # Mathieu Blondel <mathieu@mblondel.org> # Gael Varoquaux # Roman Feldbauer # License: BSD 3 clause (C) INRIA 2011 print(__doc__) from time import time import numpy as np import matplotlib.pyplot as plt from matplotlib import offsetbox from sklearn import (manifold, datasets, decomposition, ensemble, discriminant_analysis, random_projection) from skhubness import neighbors digits = datasets.load_digits(n_class=6) X = digits.data y = digits.target n_samples, n_features = X.shape n_neighbors = 30 # ---------------------------------------------------------------------- # Scale and visualize the embedding vectors def plot_embedding(X, title=None): x_min, x_max = np.min(X, 0), np.max(X, 0) X = (X - x_min) / (x_max - x_min) plt.figure() ax = plt.subplot(111) for i in range(X.shape[0]): plt.text(X[i, 0], X[i, 1], str(y[i]), color=plt.cm.Set1(y[i] / 10.), fontdict={'weight': 'bold', 'size': 9}) if hasattr(offsetbox, 'AnnotationBbox'): # only print thumbnails with matplotlib > 1.0 shown_images = np.array([[1., 1.]]) # just something big for i in range(X.shape[0]): dist = np.sum((X[i] - shown_images) ** 2, 1) if np.min(dist) < 4e-3: # don't show points that are too close continue shown_images = np.r_[shown_images, [X[i]]] imagebox = offsetbox.AnnotationBbox( offsetbox.OffsetImage(digits.images[i], cmap=plt.cm.gray_r), X[i]) ax.add_artist(imagebox) plt.xticks([]), plt.yticks([]) if title is not None: plt.title(title) # ---------------------------------------------------------------------- # Plot images of the digits n_img_per_row = 20 img = np.zeros((10 * n_img_per_row, 10 * n_img_per_row)) for i in range(n_img_per_row): ix = 10 * i + 1 for j in range(n_img_per_row): iy = 10 * j + 1 img[ix:ix + 8, iy:iy + 8] = X[i * n_img_per_row + j].reshape((8, 8)) plt.imshow(img, cmap=plt.cm.binary) plt.xticks([]) plt.yticks([]) plt.title('A selection from the 64-dimensional digits dataset') # ---------------------------------------------------------------------- # Random 2D projection using a random unitary matrix print("Computing random projection") rp = random_projection.SparseRandomProjection(n_components=2, random_state=42) X_projected = rp.fit_transform(X) plot_embedding(X_projected, "Random Projection of the digits") #---------------------------------------------------------------------- # Projection on to the first 2 principal components print("Computing PCA projection") t0 = time() X_pca = decomposition.TruncatedSVD(n_components=2).fit_transform(X) plot_embedding(X_pca, "Principal Components projection of the digits (time %.2fs)" % (time() - t0)) # ---------------------------------------------------------------------- # Projection on to the first 2 linear discriminant components print("Computing Linear Discriminant Analysis projection") X2 = X.copy() X2.flat[::X.shape[1] + 1] += 0.01 # Make X invertible t0 = time() X_lda = discriminant_analysis.LinearDiscriminantAnalysis(n_components=2).fit_transform(X2, y) plot_embedding(X_lda, "Linear Discriminant projection of the digits (time %.2fs)" % (time() - t0)) # ---------------------------------------------------------------------- # Isomap projection of the digits dataset print("Computing Isomap projection") t0 = time() X_iso = manifold.Isomap(n_neighbors, n_components=2).fit_transform(X) print("Done.") plot_embedding(X_iso, "Isomap projection of the digits (time %.2fs)" % (time() - t0)) # ---------------------------------------------------------------------- # Locally linear embedding of the digits dataset print("Computing LLE embedding") clf = manifold.LocallyLinearEmbedding(n_neighbors, n_components=2, method='standard') t0 = time() X_lle = clf.fit_transform(X) print("Done. Reconstruction error: %g" % clf.reconstruction_error_) plot_embedding(X_lle, "Locally Linear Embedding of the digits (time %.2fs)" % (time() - t0)) # ---------------------------------------------------------------------- # Modified Locally linear embedding of the digits dataset print("Computing modified LLE embedding") clf = manifold.LocallyLinearEmbedding(n_neighbors, n_components=2, method='modified') t0 = time() X_mlle = clf.fit_transform(X) print("Done. Reconstruction error: %g" % clf.reconstruction_error_) plot_embedding(X_mlle, "Modified Locally Linear Embedding of the digits (time %.2fs)" % (time() - t0)) # ---------------------------------------------------------------------- # HLLE embedding of the digits dataset print("Computing Hessian LLE embedding") clf = manifold.LocallyLinearEmbedding(n_neighbors, n_components=2, method='hessian') t0 = time() X_hlle = clf.fit_transform(X) print("Done. Reconstruction error: %g" % clf.reconstruction_error_) plot_embedding(X_hlle, "Hessian Locally Linear Embedding of the digits (time %.2fs)" % (time() - t0)) # ---------------------------------------------------------------------- # LTSA embedding of the digits dataset print("Computing LTSA embedding") clf = manifold.LocallyLinearEmbedding(n_neighbors, n_components=2, method='ltsa') t0 = time() X_ltsa = clf.fit_transform(X) print("Done. Reconstruction error: %g" % clf.reconstruction_error_) plot_embedding(X_ltsa, "Local Tangent Space Alignment of the digits (time %.2fs)" % (time() - t0)) # ---------------------------------------------------------------------- # MDS embedding of the digits dataset print("Computing MDS embedding") clf = manifold.MDS(n_components=2, n_init=1, max_iter=2000, dissimilarity='euclidean', metric=True, ) t0 = time() X_mds = clf.fit_transform(X) print("Done. Stress: %f" % clf.stress_) plot_embedding(X_mds, "MDS embedding of the digits (time %.2fs)" % (time() - t0)) # ---------------------------------------------------------------------- # Hubness reduction (LS) + MDS embedding of the digits dataset print("Computing MDS embedding from local scaling neighbors graph") clf = manifold.MDS(n_components=2, n_init=1, max_iter=2000, dissimilarity='precomputed', metric=True, ) t0 = time() graph = neighbors.graph.kneighbors_graph( X, n_neighbors=X.shape[0]-1, mode='distance', hubness='local_scaling').toarray() X_mds = clf.fit_transform(graph) print("Done. Stress: %f" % clf.stress_) plot_embedding(X_mds, "Hubness reduction (LS) - MDS embedding (time %.2fs)" % (time() - t0)) # ---------------------------------------------------------------------- # Hubness reduction (MP) + MDS embedding of the digits dataset print("Computing MDS embedding from mutual proximity graph") clf = manifold.MDS(n_components=2, n_init=1, max_iter=2000, dissimilarity='precomputed', metric=True, ) t0 = time() graph = neighbors.graph.kneighbors_graph( X, n_neighbors=1082, mode='distance', hubness='mp').toarray() X_mds = clf.fit_transform(graph) print("Done. Stress: %f" % clf.stress_) plot_embedding(X_mds, "Hubness reduction (MP) - MDS embedding (time %.2fs)" % (time() - t0)) # ---------------------------------------------------------------------- # Random Trees embedding of the digits dataset print("Computing Totally Random Trees embedding") hasher = ensemble.RandomTreesEmbedding(n_estimators=200, random_state=0, max_depth=5) t0 = time() X_transformed = hasher.fit_transform(X) pca = decomposition.TruncatedSVD(n_components=2) X_reduced = pca.fit_transform(X_transformed) plot_embedding(X_reduced, "Random forest embedding of the digits (time %.2fs)" % (time() - t0)) # ---------------------------------------------------------------------- # Spectral embedding of the digits dataset print("Computing Spectral embedding") embedder = manifold.SpectralEmbedding(n_components=2, random_state=0, eigen_solver="arpack") t0 = time() X_se = embedder.fit_transform(X) plot_embedding(X_se, "Spectral embedding of the digits (time %.2fs)" % (time() - t0)) # ---------------------------------------------------------------------- # t-SNE embedding of the digits dataset print("Computing t-SNE embedding") tsne = manifold.TSNE(n_components=2, init='pca', random_state=0) t0 = time() X_tsne = tsne.fit_transform(X) plot_embedding(X_tsne, "t-SNE embedding of the digits (time %.2fs)" % (time() - t0)) # ---------------------------------------------------------------------- # NCA projection of the digits dataset print("Computing NCA projection") nca = neighbors.NeighborhoodComponentsAnalysis(n_components=2, random_state=0) t0 = time() X_nca = nca.fit_transform(X, y) plot_embedding(X_nca, "NCA embedding of the digits (time %.2fs)" % (time() - t0)) plt.show()