Manifold Learning on Handwritten Digits

Introduction

In this lab, we will explore various manifold embedding techniques on the digits dataset. We will use different techniques to embed the digits dataset, plot the projection of the original data onto each embedding, and compare the results obtained from different embedding methods.

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Load the Digits Dataset

We will load the digits dataset and only use six of the ten available classes. We will also plot the first hundred digits from this dataset.

## Load digits dataset
from sklearn.datasets import load_digits

digits = load_digits(n_class=6)
X, y = digits.data, digits.target
n_samples, n_features = X.shape
n_neighbors = 30

## Plot first hundred digits
import matplotlib.pyplot as plt

fig, axs = plt.subplots(nrows=10, ncols=10, figsize=(6, 6))
for idx, ax in enumerate(axs.ravel()):
    ax.imshow(X[idx].reshape((8, 8)), cmap=plt.cm.binary)
    ax.axis("off")
_ = fig.suptitle("A selection from the 64-dimensional digits dataset", fontsize=16)

Plot Embedding Function

We will define a helper function to plot the embedding. The function takes the embedding data and the title for the plot as input. The function will plot every digit on the embedding and show an annotation box for a group of digits.

import numpy as np
from matplotlib import offsetbox
from sklearn.preprocessing import MinMaxScaler

def plot_embedding(X, title):
    _, ax = plt.subplots()
    X = MinMaxScaler().fit_transform(X)

    for digit in digits.target_names:
        ax.scatter(
            *X[y == digit].T,
            marker=f"${digit}$",
            s=60,
            color=plt.cm.Dark2(digit),
            alpha=0.425,
            zorder=2,
        )
    shown_images = np.array([[1.0, 1.0]])  ## just something big
    for i in range(X.shape[0]):
        ## plot every digit on the embedding
        ## show an annotation box for a group of digits
        dist = np.sum((X[i] - shown_images) ** 2, 1)
        if np.min(dist) < 4e-3:
            ## don't show points that are too close
            continue
        shown_images = np.concatenate([shown_images, [X[i]]], axis=0)
        imagebox = offsetbox.AnnotationBbox(
            offsetbox.OffsetImage(digits.images[i], cmap=plt.cm.gray_r), X[i]
        )
        imagebox.set(zorder=1)
        ax.add_artist(imagebox)

    ax.set_title(title)
    ax.axis("off")

Compare Embedding Techniques

We will compare different embedding techniques using different methods. We will store the projected data as well as the computational time needed to perform each projection.

from sklearn.decomposition import TruncatedSVD
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
from sklearn.ensemble import RandomTreesEmbedding
from sklearn.manifold import (
    Isomap,
    LocallyLinearEmbedding,
    MDS,
    SpectralEmbedding,
    TSNE,
)
from sklearn.neighbors import NeighborhoodComponentsAnalysis
from sklearn.pipeline import make_pipeline
from sklearn.random_projection import SparseRandomProjection

embeddings = {
    "Random projection embedding": SparseRandomProjection(
        n_components=2, random_state=42
    ),
    "Truncated SVD embedding": TruncatedSVD(n_components=2),
    "Linear Discriminant Analysis embedding": LinearDiscriminantAnalysis(
        n_components=2
    ),
    "Isomap embedding": Isomap(n_neighbors=n_neighbors, n_components=2),
    "Standard LLE embedding": LocallyLinearEmbedding(
        n_neighbors=n_neighbors, n_components=2, method="standard"
    ),
    "Modified LLE embedding": LocallyLinearEmbedding(
        n_neighbors=n_neighbors, n_components=2, method="modified"
    ),
    "Hessian LLE embedding": LocallyLinearEmbedding(
        n_neighbors=n_neighbors, n_components=2, method="hessian"
    ),
    "LTSA LLE embedding": LocallyLinearEmbedding(
        n_neighbors=n_neighbors, n_components=2, method="ltsa"
    ),
    "MDS embedding": MDS(
        n_components=2, n_init=1, max_iter=120, n_jobs=2, normalized_stress="auto"
    ),
    "Random Trees embedding": make_pipeline(
        RandomTreesEmbedding(n_estimators=200, max_depth=5, random_state=0),
        TruncatedSVD(n_components=2),
    ),
    "Spectral embedding": SpectralEmbedding(
        n_components=2, random_state=0, eigen_solver="arpack"
    ),
    "t-SNE embeedding": TSNE(
        n_components=2,
        n_iter=500,
        n_iter_without_progress=150,
        n_jobs=2,
        random_state=0,
    ),
    "NCA embedding": NeighborhoodComponentsAnalysis(
        n_components=2, init="pca", random_state=0
    ),
}

projections, timing = {}, {}
for name, transformer in embeddings.items():
    if name.startswith("Linear Discriminant Analysis"):
        data = X.copy()
        data.flat[:: X.shape[1] + 1] += 0.01  ## Make X invertible
    else:
        data = X

    print(f"Computing {name}...")
    start_time = time()
    projections[name] = transformer.fit_transform(data, y)
    timing[name] = time() - start_time

Plot Results

We will plot the resulting projection given by each method.

for name in timing:
    title = f"{name} (time {timing[name]:.3f}s)"
    plot_embedding(projections[name], title)

plt.show()

Summary

In this lab, we explored various manifold embedding techniques on the digits dataset. We used different techniques to embed the digits dataset, plotted the projection of the original data onto each embedding, and compared the results obtained from different embedding methods. The results provide insight into the effectiveness of each embedding method in grouping similar digits together in the embedding space.