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Geometric Analysis

Geometric estimators calculate the "Intrinsic Dimension" (ID) based on distances between points, rather than variance of global projections. This is crucial for manifolds that are non-linear (e.g., a Swiss Roll).

The Swiss Roll Problem

A "Swiss Roll" is a 2D plane rolled up in 3D. * PCA will see it as 3D (because variance exists in x, y, z). * Geometric ID should see it as 2D (locally, it's a plane).

import numpy as np
import effdim
from sklearn.datasets import make_swiss_roll

# Generate Swiss Roll
X, _ = make_swiss_roll(n_samples=2000, noise=0.0)
X = X[:, [0, 2]] # Project to 2D for test? No, make_swiss_roll returns 3D.
# Actually make_swiss_roll returns (N, 3).
X, _ = make_swiss_roll(n_samples=2000, noise=0.01)

# PCA
pca_dim = effdim.compute(X, method='pca', threshold=0.95)
print(f"Global PCA Dimension: {pca_dim}")
# Likely 3, because the roll occupies 3D volume globally.

# kNN Intrinsic Dimension
knn_dim = effdim.compute(X, method='knn', k=5)
print(f"kNN Intrinsic Dimension: {knn_dim:.2f}")
# Should be close to 2.0

# Two-NN
twonn_dim = effdim.compute(X, method='twonn')
print(f"Two-NN Intrinsic Dimension: {twonn_dim:.2f}")
# Should be close to 2.0

When to use Geometric Estimators?

  1. Non-linear manifolds: Image datasets (digits, faces) often lie on low-dimensional non-linear manifolds.
  2. Manifold Learning: Checking if your autoencoder latent space has matched the intrinsic dimension of the data.
  3. Local Analysis: kNN can be computed per-point (though effdim currently returns the average).

Limitations

  • Computational Cost: Requires computing nearest neighbors, which can be slow for large \(N\). effdim uses scipy.spatial.cKDTree for efficiency.
  • Curse of Dimensionality: In extremely high dimensions, distance concentration can make geometric estimation unstable.