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Michael Brenndoerfer

UMAP (Uniform Manifold Approximation and Projection) › Example

Michael BrenndoerferJuly 24, 20253 min read

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Part 6 of 9Explainability

Chapter 50UMAP (Uniform Manifold Approximation and Projection) › Example

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Let's work through a concrete example using a simple 2D dataset to demonstrate how UMAP works step by step. We'll use a dataset with two distinct clusters to make the process clear.

In[9]:

Code

# Create a simple 2D dataset with two clusters
np.random.seed(42)
cluster1 = np.random.normal([2, 2], 0.5, (50, 2))
cluster2 = np.random.normal([-2, -2], 0.5, (50, 2))
X = np.vstack([cluster1, cluster2])
y = np.hstack([np.zeros(50), np.ones(50)])

print(f"Dataset shape: {X.shape}")
print(f"Cluster 1 center: {cluster1.mean(axis=0)}")
print(f"Cluster 2 center: {cluster2.mean(axis=0)}")
print(
    f"Distance between cluster centers: {np.linalg.norm(cluster1.mean(axis=0) - cluster2.mean(axis=0)):.2f}"
)

Out[10]:

Visualization

Scatter plot showing two distinct clusters in 2D space, colored by cluster membership. — Original 2D dataset with two distinct clusters used to demonstrate UMAP's dimensionality reduction capabilities. The clusters are well-separated in the original space.

Now let's apply UMAP and examine the results:

In[11]:

Code

# Apply UMAP
reducer = UMAP(n_components=2, n_neighbors=15, min_dist=0.1, random_state=42)
X_umap = reducer.fit_transform(X)

print(f"UMAP embedding shape: {X_umap.shape}")
print(
    f"UMAP embedding range: X1=[{X_umap[:, 0].min():.2f}, {X_umap[:, 0].max():.2f}], X2=[{X_umap[:, 1].min():.2f}, {X_umap[:, 1].max():.2f}]"
)

Out[12]:

Visualization

Scatter plot of UMAP-embedded data showing preserved cluster separation, with points colored by their original cluster membership. — UMAP embedding of the original 2D cluster data demonstrating successful dimensionality reduction while preserving cluster structure. The two clusters remain well-separated in the UMAP projection, showing that UMAP maintains the essential topological relationships from the original space.

Let's also examine the local neighborhood structure that UMAP preserves:

In[13]:

Code

# Examine local neighborhoods
from sklearn.neighbors import NearestNeighbors

# Find nearest neighbors in original space
nbrs_original = NearestNeighbors(n_neighbors=5).fit(X)
distances_original, indices_original = nbrs_original.kneighbors(X)

# Find nearest neighbors in UMAP space
nbrs_umap = NearestNeighbors(n_neighbors=5).fit(X_umap)
distances_umap, indices_umap = nbrs_umap.kneighbors(X_umap)

# Check how well neighborhoods are preserved
neighborhood_preservation = []
for i in range(len(X)):
    original_neighbors = set(indices_original[i])
    umap_neighbors = set(indices_umap[i])
    intersection = len(original_neighbors.intersection(umap_neighbors))
    preservation = intersection / 5.0  # 5 neighbors
    neighborhood_preservation.append(preservation)

print(
    f"Average neighborhood preservation: {np.mean(neighborhood_preservation):.3f}"
)
print(
    f"Neighborhood preservation range: [{np.min(neighborhood_preservation):.3f}, {np.max(neighborhood_preservation):.3f}]"
)