Introduction

K-means clustering is one of the most widely used unsupervised learning algorithms. It partitions data into k groups by iteratively assigning each point to the nearest cluster center and then updating the cluster centers to be the mean of the assigned points.

While k-means works beautifully on data with roughly spherical clusters, its assumptions can fail dramatically when clusters are non-spherical. To make this clear, I built an interactive artifact that runs k-means side-by-side on two very different datasets:

• Left: Circular Gaussian blobs (k-means-friendly)
• Right: Moon-shaped, non-circular data (k-means-unfriendly)

You can step through the algorithm iteration-by-iteration or watch it run automatically, adjusting parameters to see how the results change.

K-means Clustering Comparison Demo

Compare k-means performance on circular vs. non-circular data side by side

k value:2345

Initialization:RandomK-means++

Auto RunNext StepReset

Circular Data

Iteration: 0

SSE: 0.0

Status:Running

Non-circular Data

Iteration: 0

SSE: 0.0

Status:Running

Key Learning Points

Left (Circular Data):

Spherical clusters that k-means handles well. Clean separation achieved in few steps.

Right (Non-circular Data):

Non-spherical clusters that k-means struggles with. Forces linear boundaries on curved structures.

How K-means Works

The k-means algorithm repeats two main steps until it converges:

1. Assignment step
   Each data point $x_i$ is assigned to the cluster whose center $\mu_j$ is closest in Euclidean distance:
   $$C_i = \arg\min_{j} \|x_i - \mu_j\|^2$$
2. Update step
   Each cluster center is updated to be the mean of all points assigned to it:
   $$\mu_j = \frac{1}{|C_j|} \sum_{x_i \in C_j} x_i$$

The algorithm starts with k initial centers (chosen randomly or with the smarter k-means++ method) and stops when the centers move very little between iterations or after a fixed number of steps.

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Measuring Progress: SSE

In our demo, we display the Sum of Squared Errors (SSE) at each iteration:

• A lower SSE means points are closer to their assigned center (tighter clusters).
• K-means always decreases SSE with each iteration until it converges.
• But low SSE does not always mean a good clustering — especially on non-spherical data.

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What You’ll See in the Demo

1. Initialization
   The cluster centers (× marks) appear in their initial positions.
2. Iteration
   Each step reassigns points to the nearest center, recoloring them, then moves the centers toward the mean of their assigned points.
3. Convergence
   On the left, clusters quickly form around the Gaussian blobs.
   On the right, k-means still forces linear boundaries, slicing curved clusters awkwardly.
4. Experimentation
   Adjust k and initialization methods to see:
   • How initial positions affect the final result
   • How k-means struggles with non-circular shapes

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Why the Difference?

K-means assumes:

• Clusters are spherical in shape
• Clusters have similar size
• Clusters are linearly separable

When these assumptions hold (e.g., Gaussian blobs), the algorithm works great.
When they don’t (e.g., moons, rings), k-means’ rigid boundaries can’t adapt to the actual structure of the data.

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With this interactive artifact, you can watch exactly how k-means evolves and see — in real time — both its strengths and limitations.