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.. rst-class:: sphx-glr-example-title
.. _sphx_glr_auto_examples_miscellaneous_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
.. GENERATED FROM PYTHON SOURCE LINES 15-29
.. code-block:: default
import sys
from time import time
import matplotlib.pyplot as plt
import numpy as np
from sklearn.datasets import fetch_20newsgroups_vectorized, load_digits
from sklearn.metrics.pairwise import euclidean_distances
from sklearn.random_projection import (
SparseRandomProjection,
johnson_lindenstrauss_min_dim,
)
.. GENERATED FROM PYTHON SOURCE LINES 30-53
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
of 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 \geq 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.
.. GENERATED FROM PYTHON SOURCE LINES 53-73
.. 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([f"eps = {eps:0.1f}" 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-sg:: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_001.png
:alt: Johnson-Lindenstrauss bounds: n_samples vs n_components
:srcset: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_001.png
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 74-77
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``
.. GENERATED FROM PYTHON SOURCE LINES 77-96
.. 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([f"n_samples = {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-sg:: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_002.png
:alt: Johnson-Lindenstrauss bounds: n_components vs eps
:srcset: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_002.png
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 97-115
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 300 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 300
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.
.. GENERATED FROM PYTHON SOURCE LINES 115-121
.. code-block:: default
if "--use-digits-dataset" in sys.argv:
data = load_digits().data[:300]
else:
data = fetch_20newsgroups_vectorized().data[:300]
.. GENERATED FROM PYTHON SOURCE LINES 122-128
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).
.. GENERATED FROM PYTHON SOURCE LINES 128-188
.. code-block:: default
n_samples, n_features = data.shape
print(
f"Embedding {n_samples} samples with dim {n_features} using various "
"random projections"
)
n_components_range = np.array([300, 1_000, 10_000])
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(
f"Projected {n_samples} samples from {n_features} to {n_components} in "
f"{time() - t0:0.3f}s"
)
if hasattr(rp, "components_"):
n_bytes = rp.components_.data.nbytes
n_bytes += rp.components_.indices.nbytes
print(f"Random matrix with size: {n_bytes / 1e6:0.3f} MB")
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(f"Mean distances rate: {np.mean(rates):.2f} ({np.std(rates):.2f})")
plt.figure()
plt.hist(rates, bins=50, range=(0.0, 2.0), edgecolor="k", density=True)
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-sg:: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_003.png
:alt: Pairwise distances distribution for n_components=300
:srcset: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_003.png
:class: sphx-glr-multi-img
*
.. image-sg:: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_004.png
:alt: Histogram of pairwise distance rates for n_components=300
:srcset: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_004.png
:class: sphx-glr-multi-img
*
.. image-sg:: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_005.png
:alt: Pairwise distances distribution for n_components=1000
:srcset: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_005.png
:class: sphx-glr-multi-img
*
.. image-sg:: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_006.png
:alt: Histogram of pairwise distance rates for n_components=1000
:srcset: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_006.png
:class: sphx-glr-multi-img
*
.. image-sg:: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_007.png
:alt: Pairwise distances distribution for n_components=10000
:srcset: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_007.png
:class: sphx-glr-multi-img
*
.. image-sg:: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_008.png
:alt: Histogram of pairwise distance rates for n_components=10000
:srcset: /auto_examples/miscellaneous/images/sphx_glr_plot_johnson_lindenstrauss_bound_008.png
:class: sphx-glr-multi-img
.. rst-class:: sphx-glr-script-out
.. code-block:: none
Embedding 300 samples with dim 130107 using various random projections
Projected 300 samples from 130107 to 300 in 0.266s
Random matrix with size: 1.293 MB
Mean distances rate: 1.01 (0.17)
Projected 300 samples from 130107 to 1000 in 0.849s
Random matrix with size: 4.323 MB
Mean distances rate: 1.00 (0.09)
Projected 300 samples from 130107 to 10000 in 8.441s
Random matrix with size: 43.268 MB
Mean distances rate: 1.01 (0.03)
.. GENERATED FROM PYTHON SOURCE LINES 189-209
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 300 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 56,436 down to 10,000 while reasonably preserving
pairwise distances.
.. rst-class:: sphx-glr-timing
**Total running time of the script:** (0 minutes 12.161 seconds)
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