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Plot T Sne Perplexity

============================================================================= t-SNE: The effect of various perplexity values on the shape

An illustration of t-SNE on the two concentric circles and the S-curve datasets for different perplexity values.

We observe a tendency towards clearer shapes as the perplexity value increases.

The size, the distance and the shape of clusters may vary upon initialization, perplexity values and does not always convey a meaning.

As shown below, t-SNE for higher perplexities finds meaningful topology of two concentric circles, however the size and the distance of the circles varies slightly from the original. Contrary to the two circles dataset, the shapes visually diverge from S-curve topology on the S-curve dataset even for larger perplexity values.

For further details, “How to Use t-SNE Effectively” https://distill.pub/2016/misread-tsne/ provides a good discussion of the effects of various parameters, as well as interactive plots to explore those effects.

Imports for Exploring t-SNE Perplexity Effects

Perplexity controls the effective number of neighbors in t-SNE’s probability model: For each point, TSNE constructs a Gaussian distribution over all other points, with the bandwidth (sigma) chosen so that the distribution’s Shannon entropy equals log(perplexity). Low perplexity (e.g., 5) makes each point attend only to its closest few neighbors, producing tight but potentially artificial micro-clusters; high perplexity (e.g., 100) broadens attention to more distant points, encouraging the algorithm to preserve larger-scale structure at the cost of local detail. The init="random" setting means the optimization starts from a random configuration, so the final layout depends on both perplexity and the random seed – unlike PCA-initialized t-SNE, random initialization can produce qualitatively different results across runs.

Three synthetic datasets reveal different perplexity sensitivities: The two concentric circles (make_circles) require t-SNE to discover that the inner and outer rings are separate structures – low perplexity may fail to separate them because each point’s neighborhood spans both circles, while higher perplexity provides enough global context to pull them apart. The S-curve (make_s_curve) is a continuous manifold where perplexity affects whether t-SNE preserves the smooth color gradient or fragments it into disconnected blobs. The uniform 2D grid tests whether t-SNE can reproduce a regular structure that has no clusters at all – low perplexity tends to create spurious clumping even in uniform data, a well-known artifact that users must be aware of when interpreting t-SNE visualizations. The max_iter parameter (250-400) is deliberately kept low for speed, but production use typically requires 1000+ iterations for convergence.

# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause

from time import time

import matplotlib.pyplot as plt
import numpy as np
from matplotlib.ticker import NullFormatter

from sklearn import datasets, manifold

n_samples = 150
n_components = 2
(fig, subplots) = plt.subplots(3, 5, figsize=(15, 8))
perplexities = [5, 30, 50, 100]

X, y = datasets.make_circles(
    n_samples=n_samples, factor=0.5, noise=0.05, random_state=0
)

red = y == 0
green = y == 1

ax = subplots[0][0]
ax.scatter(X[red, 0], X[red, 1], c="r")
ax.scatter(X[green, 0], X[green, 1], c="g")
ax.xaxis.set_major_formatter(NullFormatter())
ax.yaxis.set_major_formatter(NullFormatter())
plt.axis("tight")

for i, perplexity in enumerate(perplexities):
    ax = subplots[0][i + 1]

    t0 = time()
    tsne = manifold.TSNE(
        n_components=n_components,
        init="random",
        random_state=0,
        perplexity=perplexity,
        max_iter=300,
    )
    Y = tsne.fit_transform(X)
    t1 = time()
    print("circles, perplexity=%d in %.2g sec" % (perplexity, t1 - t0))
    ax.set_title("Perplexity=%d" % perplexity)
    ax.scatter(Y[red, 0], Y[red, 1], c="r")
    ax.scatter(Y[green, 0], Y[green, 1], c="g")
    ax.xaxis.set_major_formatter(NullFormatter())
    ax.yaxis.set_major_formatter(NullFormatter())
    ax.axis("tight")

# Another example using s-curve
X, color = datasets.make_s_curve(n_samples, random_state=0)

ax = subplots[1][0]
ax.scatter(X[:, 0], X[:, 2], c=color)
ax.xaxis.set_major_formatter(NullFormatter())
ax.yaxis.set_major_formatter(NullFormatter())

for i, perplexity in enumerate(perplexities):
    ax = subplots[1][i + 1]

    t0 = time()
    tsne = manifold.TSNE(
        n_components=n_components,
        init="random",
        random_state=0,
        perplexity=perplexity,
        learning_rate="auto",
        max_iter=300,
    )
    Y = tsne.fit_transform(X)
    t1 = time()
    print("S-curve, perplexity=%d in %.2g sec" % (perplexity, t1 - t0))

    ax.set_title("Perplexity=%d" % perplexity)
    ax.scatter(Y[:, 0], Y[:, 1], c=color)
    ax.xaxis.set_major_formatter(NullFormatter())
    ax.yaxis.set_major_formatter(NullFormatter())
    ax.axis("tight")


# Another example using a 2D uniform grid
x = np.linspace(0, 1, int(np.sqrt(n_samples)))
xx, yy = np.meshgrid(x, x)
X = np.hstack(
    [
        xx.ravel().reshape(-1, 1),
        yy.ravel().reshape(-1, 1),
    ]
)
color = xx.ravel()
ax = subplots[2][0]
ax.scatter(X[:, 0], X[:, 1], c=color)
ax.xaxis.set_major_formatter(NullFormatter())
ax.yaxis.set_major_formatter(NullFormatter())

for i, perplexity in enumerate(perplexities):
    ax = subplots[2][i + 1]

    t0 = time()
    tsne = manifold.TSNE(
        n_components=n_components,
        init="random",
        random_state=0,
        perplexity=perplexity,
        max_iter=400,
    )
    Y = tsne.fit_transform(X)
    t1 = time()
    print("uniform grid, perplexity=%d in %.2g sec" % (perplexity, t1 - t0))

    ax.set_title("Perplexity=%d" % perplexity)
    ax.scatter(Y[:, 0], Y[:, 1], c=color)
    ax.xaxis.set_major_formatter(NullFormatter())
    ax.yaxis.set_major_formatter(NullFormatter())
    ax.axis("tight")


plt.show()