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# 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.

    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,
    )

## 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:

\[(1 - eps) \|u - v\|^2 < \|p(u) - p(v)\|^2 < (1 + eps) \|u - v\|^2\]

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:

\[n\_components \geq 4 log(n\_samples) / (eps^2 / 2 - eps^3 / 3)\]

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.

    # 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()

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`

    # 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()

## 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.

    if "--use-digits-dataset" in sys.argv:
        data = load_digits().data[:300]
    else:
        data = fetch_20newsgroups_vectorized().data[:300]

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).

    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()

    Embedding 300 samples with dim 130107 using various random projections
    Projected 300 samples from 130107 to 300 in 0.239s
    Random matrix with size: 1.293 MB
    Mean distances rate: 1.01 (0.17)
    Projected 300 samples from 130107 to 1000 in 0.784s
    Random matrix with size: 4.323 MB
    Mean distances rate: 1.00 (0.09)
    Projected 300 samples from 130107 to 10000 in 7.776s
    Random matrix with size: 43.268 MB
    Mean distances rate: 1.01 (0.03)

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.

Total running time of the script: (0 minutes 10.608 seconds)

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