plot_faces_decomposition.py

"""
============================
Faces dataset decompositions
============================

This example applies to :ref:`olivetti_faces_dataset` different unsupervised
matrix decomposition (dimension reduction) methods from the module
:mod:`sklearn.decomposition` (see the documentation chapter
:ref:`decompositions`).


- Authors: Vlad Niculae, Alexandre Gramfort
- License: BSD 3 clause
"""

# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause

# %%
# Dataset preparation
# -------------------
#
# Loading and preprocessing the Olivetti faces dataset.

import logging

import matplotlib.pyplot as plt
from numpy.random import RandomState

from sklearn import cluster, decomposition
from sklearn.datasets import fetch_olivetti_faces

rng = RandomState(0)

# Display progress logs on stdout
logging.basicConfig(level=logging.INFO, format="%(asctime)s %(levelname)s %(message)s")

faces, _ = fetch_olivetti_faces(return_X_y=True, shuffle=True, random_state=rng)
n_samples, n_features = faces.shape

# Global centering (focus on one feature, centering all samples)
faces_centered = faces - faces.mean(axis=0)

# Local centering (focus on one sample, centering all features)
faces_centered -= faces_centered.mean(axis=1).reshape(n_samples, -1)

print("Dataset consists of %d faces" % n_samples)

# %%
# Define a base function to plot the gallery of faces.

n_row, n_col = 2, 3
n_components = n_row * n_col
image_shape = (64, 64)


def plot_gallery(title, images, n_col=n_col, n_row=n_row, cmap=plt.cm.gray):
    fig, axs = plt.subplots(
        nrows=n_row,
        ncols=n_col,
        figsize=(2.0 * n_col, 2.3 * n_row),
        facecolor="white",
        constrained_layout=True,
    )
    fig.set_constrained_layout_pads(w_pad=0.01, h_pad=0.02, hspace=0, wspace=0)
    fig.set_edgecolor("black")
    fig.suptitle(title, size=16)
    for ax, vec in zip(axs.flat, images):
        vmax = max(vec.max(), -vec.min())
        im = ax.imshow(
            vec.reshape(image_shape),
            cmap=cmap,
            interpolation="nearest",
            vmin=-vmax,
            vmax=vmax,
        )
        ax.axis("off")

    fig.colorbar(im, ax=axs, orientation="horizontal", shrink=0.99, aspect=40, pad=0.01)
    plt.show()


# %%
# Let's take a look at our data. Gray color indicates negative values,
# white indicates positive values.

plot_gallery("Faces from dataset", faces_centered[:n_components])

# %%
# Decomposition
# -------------
#
# Initialise different estimators for decomposition and fit each
# of them on all images and plot some results. Each estimator extracts
# 6 components as vectors :math:`h \in \mathbb{R}^{4096}`.
# We just displayed these vectors in human-friendly visualisation as 64x64 pixel images.
#
# Read more in the :ref:`User Guide <decompositions>`.

# %%
# Eigenfaces - PCA using randomized SVD
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
# Linear dimensionality reduction using Singular Value Decomposition (SVD) of the data
# to project it to a lower dimensional space.
#
#
# .. note::
#
#     The Eigenfaces estimator, via the :py:mod:`sklearn.decomposition.PCA`,
#     also provides a scalar `noise_variance_` (the mean of pixelwise variance)
#     that cannot be displayed as an image.

# %%
pca_estimator = decomposition.PCA(
    n_components=n_components, svd_solver="randomized", whiten=True
)
pca_estimator.fit(faces_centered)
plot_gallery(
    "Eigenfaces - PCA using randomized SVD", pca_estimator.components_[:n_components]
)

# %%
# Non-negative components - NMF
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# Estimate non-negative original data as production of two non-negative matrices.

# %%
nmf_estimator = decomposition.NMF(n_components=n_components, tol=5e-3)
nmf_estimator.fit(faces)  # original non- negative dataset
plot_gallery("Non-negative components - NMF", nmf_estimator.components_[:n_components])

# %%
# Independent components - FastICA
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
# Independent component analysis separates a multivariate vectors into additive
# subcomponents that are maximally independent.

# %%
ica_estimator = decomposition.FastICA(
    n_components=n_components, max_iter=400, whiten="arbitrary-variance", tol=15e-5
)
ica_estimator.fit(faces_centered)
plot_gallery(
    "Independent components - FastICA", ica_estimator.components_[:n_components]
)

# %%
# Sparse components - MiniBatchSparsePCA
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# Mini-batch sparse PCA (:class:`~sklearn.decomposition.MiniBatchSparsePCA`)
# extracts the set of sparse components that best reconstruct the data. This
# variant is faster but less accurate than the similar
# :class:`~sklearn.decomposition.SparsePCA`.

# %%
batch_pca_estimator = decomposition.MiniBatchSparsePCA(
    n_components=n_components, alpha=0.1, max_iter=100, batch_size=3, random_state=rng
)
batch_pca_estimator.fit(faces_centered)
plot_gallery(
    "Sparse components - MiniBatchSparsePCA",
    batch_pca_estimator.components_[:n_components],
)

# %%
# Dictionary learning
# ^^^^^^^^^^^^^^^^^^^
#
# By default, :class:`~sklearn.decomposition.MiniBatchDictionaryLearning`
# divides the data into mini-batches and optimizes in an online manner by
# cycling over the mini-batches for the specified number of iterations.

# %%
batch_dict_estimator = decomposition.MiniBatchDictionaryLearning(
    n_components=n_components, alpha=0.1, max_iter=50, batch_size=3, random_state=rng
)
batch_dict_estimator.fit(faces_centered)
plot_gallery("Dictionary learning", batch_dict_estimator.components_[:n_components])

# %%
# Cluster centers - MiniBatchKMeans
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# :class:`sklearn.cluster.MiniBatchKMeans` is computationally efficient and
# implements on-line learning with a
# :meth:`~sklearn.cluster.MiniBatchKMeans.partial_fit` method. That is
# why it could be beneficial to enhance some time-consuming algorithms with
# :class:`~sklearn.cluster.MiniBatchKMeans`.

# %%
kmeans_estimator = cluster.MiniBatchKMeans(
    n_clusters=n_components,
    tol=1e-3,
    batch_size=20,
    max_iter=50,
    random_state=rng,
)
kmeans_estimator.fit(faces_centered)
plot_gallery(
    "Cluster centers - MiniBatchKMeans",
    kmeans_estimator.cluster_centers_[:n_components],
)


# %%
# Factor Analysis components - FA
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# :class:`~sklearn.decomposition.FactorAnalysis` is similar to
# :class:`~sklearn.decomposition.PCA` but has the advantage of modelling the
# variance in every direction of the input space independently (heteroscedastic
# noise). Read more in the :ref:`User Guide <FA>`.

# %%
fa_estimator = decomposition.FactorAnalysis(n_components=n_components, max_iter=20)
fa_estimator.fit(faces_centered)
plot_gallery("Factor Analysis (FA)", fa_estimator.components_[:n_components])

# --- Pixelwise variance
plt.figure(figsize=(3.2, 3.6), facecolor="white", tight_layout=True)
vec = fa_estimator.noise_variance_
vmax = max(vec.max(), -vec.min())
plt.imshow(
    vec.reshape(image_shape),
    cmap=plt.cm.gray,
    interpolation="nearest",
    vmin=-vmax,
    vmax=vmax,
)
plt.axis("off")
plt.title("Pixelwise variance from \n Factor Analysis (FA)", size=16, wrap=True)
plt.colorbar(orientation="horizontal", shrink=0.8, pad=0.03)
plt.show()

# %%
# Decomposition: Dictionary learning
# ----------------------------------
#
# In the further section, let's consider :ref:`DictionaryLearning` more precisely.
# Dictionary learning is a problem that amounts to finding a sparse representation
# of the input data as a combination of simple elements. These simple elements form
# a dictionary. It is possible to constrain the dictionary and/or coding coefficients
# to be positive to match constraints that may be present in the data.
#
# :class:`~sklearn.decomposition.MiniBatchDictionaryLearning` implements a
# faster, but less accurate version of the dictionary learning algorithm that
# is better suited for large datasets. Read more in the :ref:`User Guide
# <MiniBatchDictionaryLearning>`.

# %%
# Plot the same samples from our dataset but with another colormap.
# Red indicates negative values, blue indicates positive values,
# and white represents zeros.

plot_gallery("Faces from dataset", faces_centered[:n_components], cmap=plt.cm.RdBu)

# %%
# Similar to the previous examples, we change parameters and train
# :class:`~sklearn.decomposition.MiniBatchDictionaryLearning` estimator on all
# images. Generally, the dictionary learning and sparse encoding decompose
# input data into the dictionary and the coding coefficients matrices. :math:`X
# \approx UV`, where :math:`X = [x_1, . . . , x_n]`, :math:`X \in
# \mathbb{R}^{m×n}`, dictionary :math:`U \in \mathbb{R}^{m×k}`, coding
# coefficients :math:`V \in \mathbb{R}^{k×n}`.
#
# Also below are the results when the dictionary and coding
# coefficients are positively constrained.

# %%
# Dictionary learning - positive dictionary
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# In the following section we enforce positivity when finding the dictionary.

# %%
dict_pos_dict_estimator = decomposition.MiniBatchDictionaryLearning(
    n_components=n_components,
    alpha=0.1,
    max_iter=50,
    batch_size=3,
    random_state=rng,
    positive_dict=True,
)
dict_pos_dict_estimator.fit(faces_centered)
plot_gallery(
    "Dictionary learning - positive dictionary",
    dict_pos_dict_estimator.components_[:n_components],
    cmap=plt.cm.RdBu,
)

# %%
# Dictionary learning - positive code
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# Below we constrain the coding coefficients as a positive matrix.

# %%
dict_pos_code_estimator = decomposition.MiniBatchDictionaryLearning(
    n_components=n_components,
    alpha=0.1,
    max_iter=50,
    batch_size=3,
    fit_algorithm="cd",
    random_state=rng,
    positive_code=True,
)
dict_pos_code_estimator.fit(faces_centered)
plot_gallery(
    "Dictionary learning - positive code",
    dict_pos_code_estimator.components_[:n_components],
    cmap=plt.cm.RdBu,
)

# %%
# Dictionary learning - positive dictionary & code
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# Also below are the results if the dictionary values and coding
# coefficients are positively constrained.

# %%
dict_pos_estimator = decomposition.MiniBatchDictionaryLearning(
    n_components=n_components,
    alpha=0.1,
    max_iter=50,
    batch_size=3,
    fit_algorithm="cd",
    random_state=rng,
    positive_dict=True,
    positive_code=True,
)
dict_pos_estimator.fit(faces_centered)
plot_gallery(
    "Dictionary learning - positive dictionary & code",
    dict_pos_estimator.components_[:n_components],
    cmap=plt.cm.RdBu,
)