Range-Bounded Feature Scaling

In data science and machine learning, features often exist on vastly different scales. For instance, one feature might range from 0 to 1, while another spans from 1,000 to 1,000,000. When these raw features are fed directly into algorithms, those with larger numerical ranges tend to dominate the learning process, leading to biased or suboptimal models.

Range-Bounded Feature Scaling (commonly known as min-max scaling) is a powerful normalization technique that transforms numerical values into a standardized interval, typically [0, 1]. This ensures that every feature contributes proportionally to the model, regardless of its original magnitude.

The Transformation Formula:

For a given value x in a dataset with minimum value x_min and maximum value x_max, the scaled value x' is computed as:

$$x' = \frac{x - x_{min}}{x_{max} - x_{min}}$$

This linear transformation guarantees that:

• The smallest value in the dataset maps to exactly 0
• The largest value in the dataset maps to exactly 1
• All intermediate values are scaled proportionally based on their relative position within the original range

Handling Edge Cases:

When all values in the dataset are identical (i.e., x_min = x_max), the denominator becomes zero, making division undefined. In such cases, the function should return a list of zeros to indicate no meaningful variation exists in the data.

Your Task:

Implement a function that takes a list of numerical values and returns a new list where each value has been transformed to fall within the [0, 1] range using the formula above. Ensure proper handling of the edge case where all input values are the same.