The distances between points in a linear Euclidean manifold are the same before and after dimensionality reduction with principal component analysis.

pca_experiment.py

import numpy as np
from sklearn.decomposition import PCA
from scipy.spatial.distance import pdist


def get_points_from_square(num_points: int, ambient_dim: int) -> np.ndarray:
    """Get points from a grid of a linear manifold (a square in this case) endowed with the Euclidean metric.

    """
    aux = np.linspace(-1, 1, int(np.sqrt(num_points)))
    x1, x2 = np.meshgrid(aux, aux)
    X = np.zeros((num_points, ambient_dim))
    X[:, :2] = np.stack((x1.flatten(), x2.flatten())).T
    return X


def make_random_rotation(dim: int) -> np.ndarray:
    """Create a random rotation matrix.

    """
    random_matrix = np.random.randn(dim, dim)
    rotation_matrix, _ = np.linalg.qr(random_matrix)
    return rotation_matrix


N: int = int(100**2)            # Number of samples
d: int = 200                    # Dimension of ambient space
threshold: float = 1e4 * np.finfo(np.float).eps  # Threshold for identifying relevant coordinates.

# Generate points on square and scramble their coordinates.
Q = make_random_rotation(d)
X = get_points_from_square(N, d) @ Q

# Do dimensionality reduction using principal component analysis.
pca = PCA(n_components=10)      # Pretend we don't know it's a square just for fun.
Y = pca.fit_transform(X)
Y = Y[:, pca.singular_values_ > threshold]               # Get just the relevant coordinates.

assert np.allclose(pdist(X), pdist(Y)), 'Something went awry.'
print('Everything went well.')