Add Weighted Euclidean Distance Metric

### Describe the workflow you want to enable

The workflow I want to enable is the ability for users to easily incorporate feature importance into distance-based algorithms like clustering (e.g., KMeans) and nearest neighbors (e.g., KNeighborsClassifier). Currently, scikit-learn allows users to define custom distance metrics, but there is no built-in support for weighted distance metrics, which are essential when certain features are more important than others.

Example Workflow:
A user has a dataset where some features are more relevant than others (e.g., in customer segmentation, age and income might be more important than the number of children).

The user wants to use a clustering algorithm like KMeans or a nearest neighbors algorithm like KNeighborsClassifier but needs to account for the varying importance of features.

The user specifies a vector of weights corresponding to the importance of each feature.

The algorithm uses the weighted Euclidean distance metric to compute distances, ensuring that more important features have a greater influence on the results.

### Describe your proposed solution

I propose adding a Weighted Euclidean Distance Metric to scikit-learn as a built-in distance metric. This will allow users to specify feature weights directly, making it easier to incorporate feature importance into distance-based algorithms.
 **Key Components of the Solution:**

1. New Class: 

- Add a WeightedEuclideanDistance class to the sklearn.metrics.pairwise module.
This class will accept a vector of weights during initialization.
- It will compute the weighted Euclidean distance between two points using the formula:
**d(x, y) = sqrt( summation from i = 1 to n of [ w_i * (x_i - y_i) squared ] )**
where ​wi are the user-defined weights.



### Describe alternatives you've considered, if relevant

_No response_

### Additional context

Why This Feature is Needed:

- Feature Importance: In many real-world datasets, not all features are equally important. For example, in medical diagnosis, certain biomarkers might be more relevant than others. A weighted distance metric allows users to account for this.
- 
- Ease of Use: While scikit-learn allows users to define custom distance metrics, this requires writing additional code and can be error-prone. A built-in weighted distance metric would simplify the process and make it more accessible to users.
- 
- Alignment with scikit-learn’s Design: Scikit-learn emphasizes ease of use and consistency. Adding a weighted distance metric aligns with this philosophy by providing a simple, consistent way to incorporate feature importance into distance-based algorithms.

Existing Alternatives:

- Custom Distance Functions: Users can define custom distance functions using the metric parameter. However, this requires additional coding and lacks the convenience of a built-in solution.
- 
- Feature Weighting via Preprocessing: Users can scale features by their importance before applying distance-based algorithms. However, this approach is less intuitive and can be cumbersome.

Potential Impact:

- This feature would benefit users in domains like bioinformatics, finance, and customer analytics, where feature importance is critical.
- 
- It would also make scikit-learn more competitive with other machine learning libraries that offer weighted distance metrics.

COMMENT:
From looking at the citation count of the two papers you linked to and a bit of googling it seems that weighted euclidean distance is too niche for the broad audience that scikit-learn targets. Users who need it can implement it themselves via the escape hatch we provide. So I am -1 on including it.

Labels: New Feature, Needs Decision - Include Feature