Hyperplane Separation Width for Linear Classifiers

In the field of computational classification, Support Vector Machines (SVMs) stand as one of the most elegant and powerful algorithms for binary classification. At the heart of a linear SVM lies a fundamental geometric concept: the margin width, which quantifies the separation gap between two classes.

An SVM seeks to find an optimal decision hyperplane that separates two classes with maximum clearance. This hyperplane is defined by a weight vector w and a bias term b, expressed as the equation:

$$\mathbf{w} \cdot \mathbf{x} + b = 0$$

The margin is the region bounded by two parallel hyperplanes that pass through the support vectors (the data points closest to the decision boundary) of each class. These bounding hyperplanes are:

• Positive margin boundary: (\mathbf{w} \cdot \mathbf{x} + b = +1)
• Negative margin boundary: (\mathbf{w} \cdot \mathbf{x} + b = -1)

The margin width represents the total perpendicular distance between these two bounding hyperplanes. Mathematically, this width is computed as:

$$\text{Margin Width} = \frac{2}{|\mathbf{w}|}$$

where (|\mathbf{w}|) is the Euclidean (L2) norm of the weight vector:

$$|\mathbf{w}| = \sqrt{\sum_{i=1}^{d} w_i^2}$$

A larger margin indicates better class separation and typically leads to improved generalization on unseen data. The SVM optimization problem aims to maximize this margin while correctly classifying training samples.

Your Task: Write a Python function that computes the margin width given the weight vector w of a trained linear SVM classifier. Your function should:

• Accept a NumPy array representing the weight vector
• Return the margin width as a floating-point number
• Assume the weight vector is always non-zero (valid SVM output)