.. note:: :class: sphx-glr-download-link-note Click :ref:`here ` to download the full example code or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_plot_johnson_lindenstrauss_bound.py: ===================================================================== The Johnson-Lindenstrauss bound for embedding with random projections ===================================================================== The `Johnson-Lindenstrauss lemma`_ states that any high dimensional dataset can be randomly projected into a lower dimensional Euclidean space while controlling the distortion in the pairwise distances. .. _`Johnson-Lindenstrauss lemma`: https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma .. code-block:: default print(__doc__) import sys from time import time import numpy as np import matplotlib import matplotlib.pyplot as plt from distutils.version import LooseVersion from sklearn.random_projection import johnson_lindenstrauss_min_dim from sklearn.random_projection import SparseRandomProjection from sklearn.datasets import fetch_20newsgroups_vectorized from sklearn.datasets import load_digits from sklearn.metrics.pairwise import euclidean_distances # `normed` is being deprecated in favor of `density` in histograms if LooseVersion(matplotlib.__version__) >= '2.1': density_param = {'density': True} else: density_param = {'normed': True} Theoretical bounds ================== The distortion introduced by a random projection `p` is asserted by the fact that `p` is defining an eps-embedding with good probability as defined by: .. math:: (1 - eps) \|u - v\|^2 < \|p(u) - p(v)\|^2 < (1 + eps) \|u - v\|^2 Where u and v are any rows taken from a dataset of shape [n_samples, n_features] and p is a projection by a random Gaussian N(0, 1) matrix with shape [n_components, n_features] (or a sparse Achlioptas matrix). The minimum number of components to guarantees the eps-embedding is given by: .. math:: n\_components >= 4 log(n\_samples) / (eps^2 / 2 - eps^3 / 3) The first plot shows that with an increasing number of samples ``n_samples``, the minimal number of dimensions ``n_components`` increased logarithmically in order to guarantee an ``eps``-embedding. .. code-block:: default # range of admissible distortions eps_range = np.linspace(0.1, 0.99, 5) colors = plt.cm.Blues(np.linspace(0.3, 1.0, len(eps_range))) # range of number of samples (observation) to embed n_samples_range = np.logspace(1, 9, 9) plt.figure() for eps, color in zip(eps_range, colors): min_n_components = johnson_lindenstrauss_min_dim(n_samples_range, eps=eps) plt.loglog(n_samples_range, min_n_components, color=color) plt.legend(["eps = %0.1f" % eps for eps in eps_range], loc="lower right") plt.xlabel("Number of observations to eps-embed") plt.ylabel("Minimum number of dimensions") plt.title("Johnson-Lindenstrauss bounds:\nn_samples vs n_components") plt.show() .. image:: /auto_examples/images/sphx_glr_plot_johnson_lindenstrauss_bound_001.png :class: sphx-glr-single-img The second plot shows that an increase of the admissible distortion ``eps`` allows to reduce drastically the minimal number of dimensions ``n_components`` for a given number of samples ``n_samples`` .. code-block:: default # range of admissible distortions eps_range = np.linspace(0.01, 0.99, 100) # range of number of samples (observation) to embed n_samples_range = np.logspace(2, 6, 5) colors = plt.cm.Blues(np.linspace(0.3, 1.0, len(n_samples_range))) plt.figure() for n_samples, color in zip(n_samples_range, colors): min_n_components = johnson_lindenstrauss_min_dim(n_samples, eps=eps_range) plt.semilogy(eps_range, min_n_components, color=color) plt.legend(["n_samples = %d" % n for n in n_samples_range], loc="upper right") plt.xlabel("Distortion eps") plt.ylabel("Minimum number of dimensions") plt.title("Johnson-Lindenstrauss bounds:\nn_components vs eps") plt.show() .. image:: /auto_examples/images/sphx_glr_plot_johnson_lindenstrauss_bound_002.png :class: sphx-glr-single-img Empirical validation ==================== We validate the above bounds on the 20 newsgroups text document (TF-IDF word frequencies) dataset or on the digits dataset: - for the 20 newsgroups dataset some 500 documents with 100k features in total are projected using a sparse random matrix to smaller euclidean spaces with various values for the target number of dimensions ``n_components``. - for the digits dataset, some 8x8 gray level pixels data for 500 handwritten digits pictures are randomly projected to spaces for various larger number of dimensions ``n_components``. The default dataset is the 20 newsgroups dataset. To run the example on the digits dataset, pass the ``--use-digits-dataset`` command line argument to this script. .. code-block:: default if '--use-digits-dataset' in sys.argv: data = load_digits().data[:500] else: data = fetch_20newsgroups_vectorized().data[:500] For each value of ``n_components``, we plot: - 2D distribution of sample pairs with pairwise distances in original and projected spaces as x and y axis respectively. - 1D histogram of the ratio of those distances (projected / original). .. code-block:: default n_samples, n_features = data.shape print("Embedding %d samples with dim %d using various random projections" % (n_samples, n_features)) n_components_range = np.array([300, 1000, 10000]) dists = euclidean_distances(data, squared=True).ravel() # select only non-identical samples pairs nonzero = dists != 0 dists = dists[nonzero] for n_components in n_components_range: t0 = time() rp = SparseRandomProjection(n_components=n_components) projected_data = rp.fit_transform(data) print("Projected %d samples from %d to %d in %0.3fs" % (n_samples, n_features, n_components, time() - t0)) if hasattr(rp, 'components_'): n_bytes = rp.components_.data.nbytes n_bytes += rp.components_.indices.nbytes print("Random matrix with size: %0.3fMB" % (n_bytes / 1e6)) projected_dists = euclidean_distances( projected_data, squared=True).ravel()[nonzero] plt.figure() min_dist = min(projected_dists.min(), dists.min()) max_dist = max(projected_dists.max(), dists.max()) plt.hexbin(dists, projected_dists, gridsize=100, cmap=plt.cm.PuBu, extent=[min_dist, max_dist, min_dist, max_dist]) plt.xlabel("Pairwise squared distances in original space") plt.ylabel("Pairwise squared distances in projected space") plt.title("Pairwise distances distribution for n_components=%d" % n_components) cb = plt.colorbar() cb.set_label('Sample pairs counts') rates = projected_dists / dists print("Mean distances rate: %0.2f (%0.2f)" % (np.mean(rates), np.std(rates))) plt.figure() plt.hist(rates, bins=50, range=(0., 2.), edgecolor='k', **density_param) plt.xlabel("Squared distances rate: projected / original") plt.ylabel("Distribution of samples pairs") plt.title("Histogram of pairwise distance rates for n_components=%d" % n_components) # TODO: compute the expected value of eps and add them to the previous plot # as vertical lines / region plt.show() .. rst-class:: sphx-glr-horizontal * .. image:: /auto_examples/images/sphx_glr_plot_johnson_lindenstrauss_bound_003.png :class: sphx-glr-multi-img * .. image:: /auto_examples/images/sphx_glr_plot_johnson_lindenstrauss_bound_004.png :class: sphx-glr-multi-img * .. image:: /auto_examples/images/sphx_glr_plot_johnson_lindenstrauss_bound_005.png :class: sphx-glr-multi-img * .. image:: /auto_examples/images/sphx_glr_plot_johnson_lindenstrauss_bound_006.png :class: sphx-glr-multi-img * .. image:: /auto_examples/images/sphx_glr_plot_johnson_lindenstrauss_bound_007.png :class: sphx-glr-multi-img * .. image:: /auto_examples/images/sphx_glr_plot_johnson_lindenstrauss_bound_008.png :class: sphx-glr-multi-img .. rst-class:: sphx-glr-script-out Out: .. code-block:: none Embedding 500 samples with dim 130107 using various random projections Projected 500 samples from 130107 to 300 in 0.632s Random matrix with size: 1.296MB Mean distances rate: 1.00 (0.19) Projected 500 samples from 130107 to 1000 in 2.118s Random matrix with size: 4.328MB Mean distances rate: 1.00 (0.10) Projected 500 samples from 130107 to 10000 in 22.211s Random matrix with size: 43.315MB Mean distances rate: 1.03 (0.04) We can see that for low values of ``n_components`` the distribution is wide with many distorted pairs and a skewed distribution (due to the hard limit of zero ratio on the left as distances are always positives) while for larger values of n_components the distortion is controlled and the distances are well preserved by the random projection. Remarks ======= According to the JL lemma, projecting 500 samples without too much distortion will require at least several thousands dimensions, irrespective of the number of features of the original dataset. Hence using random projections on the digits dataset which only has 64 features in the input space does not make sense: it does not allow for dimensionality reduction in this case. On the twenty newsgroups on the other hand the dimensionality can be decreased from 56436 down to 10000 while reasonably preserving pairwise distances. .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 28.844 seconds) **Estimated memory usage:** 96 MB .. _sphx_glr_download_auto_examples_plot_johnson_lindenstrauss_bound.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: binder-badge .. image:: https://mybinder.org/badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/0.22.X?urlpath=lab/tree/notebooks/auto_examples/plot_johnson_lindenstrauss_bound.ipynb :width: 150 px .. container:: sphx-glr-download :download:`Download Python source code: plot_johnson_lindenstrauss_bound.py ` .. container:: sphx-glr-download :download:`Download Jupyter notebook: plot_johnson_lindenstrauss_bound.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_