Assumptions

1. Local Neighborhood Preservation

• Assumption: Points that are close in high-dimensional space should also be close in low-dimensional space.

• Implication: t-SNE is designed to preserve local structure, not global distances.

• If your data doesn’t have meaningful local clusters, t-SNE may produce misleading embeddings.

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2. Smooth Manifold Structure

• Assumption: High-dimensional data lies on or near a smooth, low-dimensional manifold.

• t-SNE tries to “unfold” this manifold in 2D or 3D.

• If the data is completely random or lacks structure, t-SNE embeddings will appear noisy.

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3. Pairwise Similarity Meaningful

• Assumption: The Euclidean distance (or chosen metric) in the high-dimensional space reflects meaningful similarity.

• t-SNE converts distances into probabilities, so if your distance metric is poor, t-SNE will produce misleading results.

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4. Noisy Distances Handled by Heavy-Tailed Distribution

• t-SNE assumes that small distances are meaningful, but larger distances may not be reliable.

• That’s why it uses a Student-t distribution in low-dimensional space:

  • Heavy tails allow distant points to be “pushed away” without distorting local neighborhoods.

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5. Perplexity Encodes Effective Neighborhood Size

• Assumption: There exists a meaningful “local neighborhood size” in the data.

• The perplexity parameter sets this expected neighborhood size.

• If the chosen perplexity is too small or too large relative to actual structure, embeddings may fail to reflect true clusters.

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6. Non-Determinism Assumption

• t-SNE assumes stochastic optimization is acceptable:

  • Different runs may produce different embeddings.

  • Multiple runs can reveal the robustness of clusters.

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Summary Table

Assumption	Explanation
Local structure matters	Nearby points should stay close in embedding
Data lies on a low-D manifold	t-SNE tries to unfold it in 2D/3D
Distances are meaningful	High-D metric reflects similarity
Noise tolerance	Heavy-tailed t-distribution separates distant points
Neighborhood size exists	Controlled via perplexity
Stochastic optimization	Non-deterministic embedding is acceptable

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Key Intuition

t-SNE is all about preserving local neighborhoods.

• If your data has meaningful clusters locally → t-SNE will show clear clusters.

• If the data is noisy or distances are meaningless → t-SNE can produce confusing embeddings.

import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import make_blobs
from sklearn.manifold import TSNE

# Generate synthetic 3-cluster dataset
np.random.seed(42)
X, y = make_blobs(n_samples=300, centers=3, n_features=5, cluster_std=1.0)

# Apply t-SNE with different perplexities
perplexities = [5, 30, 50]
tsne_embeddings = []

for p in perplexities:
    tsne = TSNE(n_components=2, perplexity=p, random_state=42)
    X_embedded = tsne.fit_transform(X)
    tsne_embeddings.append(X_embedded)

# Plot t-SNE embeddings
plt.figure(figsize=(18, 5))

for i, p in enumerate(perplexities):
    plt.subplot(1, 3, i+1)
    plt.scatter(tsne_embeddings[i][:, 0], tsne_embeddings[i][:, 1], c=y, cmap='viridis', alpha=0.7)
    plt.title(f"t-SNE Embedding (perplexity={p})")
    plt.xlabel("Dimension 1")
    plt.ylabel("Dimension 2")
    plt.grid(True)

plt.show()