⚡ 1. When Perceptron Fails

• The Perceptron algorithm won’t work well on datasets that are not linearly separable.
• If no hyperplane can separate the data, the perceptron will never converge.

✏️ 2. Perceptron Classification Rule

• The perceptron tries to find a hyperplane that makes no mistakes.

• Classification rule:

  y(wᵗx) → (w + kx)ᵗx
  

• If a point is misclassified, the hyperplane is adjusted; otherwise, it stays the same.

🔄 3. Adjusting the Hyperplane

• Start with a weight vector pointing to a point in your dataset.
• If you misclassify a point:
  • Combine (add) your current vector with the vector of the misclassified point.
  • Draw a new hyperplane based on this new vector.
  • Repeat until classification improves.

🧮 4. Ensuring Finite Adjustments

• To prevent infinite adjustments, the perceptron relies on this condition:

  k ≤ -(wᵗx) / (xᵗx)
  

• This ensures you won’t endlessly adjust the hyperplane for data that is actually linearly separable.

📜 5. Convergence Proof