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Plot Digits Denoising

================================ Image denoising using kernel PCA

This example shows how to use :class:~sklearn.decomposition.KernelPCA to denoise images. In short, we take advantage of the approximation function learned during fit to reconstruct the original image.

We will compare the results with an exact reconstruction using

class:

~sklearn.decomposition.PCA.

We will use USPS digits dataset to reproduce presented in Sect. 4 of [1]_.

… rubric:: References

… [1] Bakır, Gökhan H., Jason Weston, and Bernhard Schölkopf.     "Learning to find pre-images."     Advances in neural information processing systems 16 (2004): 449-456.     <https://papers.nips.cc/paper/2003/file/ac1ad983e08ad3304a97e147f522747e-Paper.pdf>_

Imports for Image Denoising with PCA and Kernel PCA

KernelPCA with fit_inverse_transform=True learns a non-linear denoising function by projecting noisy images into a kernel-induced feature space and then reconstructing them via kernel ridge regression: Standard PCA denoises by projecting onto the top principal components and discarding low-variance directions that capture noise, but this linear projection cannot capture the non-linear manifold structure of digit images. KernelPCA with an RBF kernel maps images into a high-dimensional space where the principal components follow the curved manifold of clean digit images, and the inverse_transform uses kernel ridge regression (controlled by alpha=5e-3) to map from this space back to pixel space, producing smoother reconstructions that better preserve digit structure.

The USPS digits dataset (16x16 grayscale images) is normalized to [0,1] via MinMaxScaler and corrupted with Gaussian noise (scale=0.25) to create a realistic denoising benchmark: The train/test split strategy (1000 training, 100 testing) trains on noisy images and evaluates reconstruction quality on held-out noisy images against their clean originals using mean squared error (MSE). While PCA often achieves lower MSE due to its unbiased linear reconstruction, KernelPCA typically produces visually cleaner images with smoother backgrounds because the RBF kernel’s locality suppresses high-frequency noise that PCA’s global linear basis retains. The n_components parameter for each method (32 for PCA, 400 for KernelPCA) reflects the different dimensionality requirements of linear versus non-linear representations.

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

# %%
# Load the dataset via OpenML
# ---------------------------
#
# The USPS digits datasets is available in OpenML. We use
# :func:`~sklearn.datasets.fetch_openml` to get this dataset. In addition, we
# normalize the dataset such that all pixel values are in the range (0, 1).
import numpy as np

from sklearn.datasets import fetch_openml
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import MinMaxScaler

X, y = fetch_openml(data_id=41082, as_frame=False, return_X_y=True)
X = MinMaxScaler().fit_transform(X)

# %%
# The idea will be to learn a PCA basis (with and without a kernel) on
# noisy images and then use these models to reconstruct and denoise these
# images.
#
# Thus, we split our dataset into a training and testing set composed of 1,000
# samples for the training and 100 samples for testing. These images are
# noise-free and we will use them to evaluate the efficiency of the denoising
# approaches. In addition, we create a copy of the original dataset and add a
# Gaussian noise.
#
# The idea of this application, is to show that we can denoise corrupted images
# by learning a PCA basis on some uncorrupted images. We will use both a PCA
# and a kernel-based PCA to solve this problem.
X_train, X_test, y_train, y_test = train_test_split(
    X, y, stratify=y, random_state=0, train_size=1_000, test_size=100
)

rng = np.random.RandomState(0)
noise = rng.normal(scale=0.25, size=X_test.shape)
X_test_noisy = X_test + noise

noise = rng.normal(scale=0.25, size=X_train.shape)
X_train_noisy = X_train + noise

# %%
# In addition, we will create a helper function to qualitatively assess the
# image reconstruction by plotting the test images.
import matplotlib.pyplot as plt

Plot Digits

Small helper function to plot 100 digits.

def plot_digits(X, title):
    """Small helper function to plot 100 digits."""
    fig, axs = plt.subplots(nrows=10, ncols=10, figsize=(8, 8))
    for img, ax in zip(X, axs.ravel()):
        ax.imshow(img.reshape((16, 16)), cmap="Greys")
        ax.axis("off")
    fig.suptitle(title, fontsize=24)


# %%
# In addition, we will use the mean squared error (MSE) to quantitatively
# assess the image reconstruction.
#
# Let's first have a look to see the difference between noise-free and noisy
# images. We will check the test set in this regard.
plot_digits(X_test, "Uncorrupted test images")
plot_digits(
    X_test_noisy, f"Noisy test images\nMSE: {np.mean((X_test - X_test_noisy) ** 2):.2f}"
)

# %%
# Learn the `PCA` basis
# ---------------------
#
# We can now learn our PCA basis using both a linear PCA and a kernel PCA that
# uses a radial basis function (RBF) kernel.
from sklearn.decomposition import PCA, KernelPCA

pca = PCA(n_components=32, random_state=42)
kernel_pca = KernelPCA(
    n_components=400,
    kernel="rbf",
    gamma=1e-3,
    fit_inverse_transform=True,
    alpha=5e-3,
    random_state=42,
)

pca.fit(X_train_noisy)
_ = kernel_pca.fit(X_train_noisy)

# %%
# Reconstruct and denoise test images
# -----------------------------------
#
# Now, we can transform and reconstruct the noisy test set. Since we used less
# components than the number of original features, we will get an approximation
# of the original set. Indeed, by dropping the components explaining variance
# in PCA the least, we hope to remove noise. Similar thinking happens in kernel
# PCA; however, we expect a better reconstruction because we use a non-linear
# kernel to learn the PCA basis and a kernel ridge to learn the mapping
# function.
X_reconstructed_kernel_pca = kernel_pca.inverse_transform(
    kernel_pca.transform(X_test_noisy)
)
X_reconstructed_pca = pca.inverse_transform(pca.transform(X_test_noisy))

# %%
plot_digits(X_test, "Uncorrupted test images")
plot_digits(
    X_reconstructed_pca,
    f"PCA reconstruction\nMSE: {np.mean((X_test - X_reconstructed_pca) ** 2):.2f}",
)
plot_digits(
    X_reconstructed_kernel_pca,
    (
        "Kernel PCA reconstruction\n"
        f"MSE: {np.mean((X_test - X_reconstructed_kernel_pca) ** 2):.2f}"
    ),
)

# %%
# PCA has a lower MSE than kernel PCA. However, the qualitative analysis might
# not favor PCA instead of kernel PCA. We observe that kernel PCA is able to
# remove background noise and provide a smoother image.
#
# However, it should be noted that the results of the denoising with kernel PCA
# will depend of the parameters `n_components`, `gamma`, and `alpha`.