Improve Your KNN Classifier: Best Distance Metrics to Use in High Dimensions

The K-Nearest Neighbors (KNN) algorithm is widely appreciated for its simplicity and effectiveness in classification and regression tasks. As a non-parametric method, KNN makes decisions based on the proximity of data points, assuming that similar data points reside near one another in feature space.

However, when working with high-dimensional datasets, KNN often encounters performance degradation due to the so-called curse of dimensionality. In such spaces, the concept of distance becomes increasingly ambiguous—most data points tend to appear nearly equidistant from each other. This diminishes the algorithm’s ability to identify meaningful neighbors, resulting in poor classification accuracy.

A practical solution to this challenge lies in tuning the distance metric used by the KNN algorithm. This post explores commonly used distance metrics, their mathematical properties, and how they perform in high-dimensional contexts.

1.Euclidean Distance (L2 Norm)

The Euclidean distance is perhaps the most commonly used metric in KNN implementations. It calculates the straight-line distance between two points in space, following the familiar formula derived from the Pythagorean theorem.

While Euclidean distance performs well in low-dimensional, continuous feature spaces, its reliability significantly diminishes as the number of dimensions increases. This is because in high-dimensional settings, the relative difference in distances between the nearest and farthest points becomes negligible. Consequently, the algorithm loses its ability to differentiate between truly “close” and “far” neighbors.

2. Manhattan Distance (L1 Norm or Cityblock Distance)

The Manhattan distance, also referred to as the L1 norm or cityblock distance, calculates the distance between two points by summing the absolute differences of their coordinates. It is analogous to navigating a grid-based city like Manhattan, where movement occurs along horizontal and vertical paths.

This metric tends to perform better in high-dimensional or sparse datasets, as it is more robust to outliers and extreme values than Euclidean distance. It is particularly suitable when individual features contribute independently and linearly to the output variable.

3. Minkowski Distance

The Minkowski distance is a generalization of both Euclidean and Manhattan distances. It introduces a parameter p, which determines the sensitivity of the metric:

• When p = 1, Minkowski distance becomes Manhattan distance.

• When p = 2, it becomes Euclidean distance.

• As p approaches infinity, it converges to Chebyshev distance.

This flexibility makes Minkowski distance an excellent candidate for tuning, especially in complex or high-dimensional datasets. For instance, using non-integer values such as p = 1.5 or p = 3 can help balance sensitivity to both small and large deviations across features, potentially improving model performance.

4. Chebyshev Distance (L∞ Norm)

Chebyshev distance considers only the maximum absolute difference across all dimensions. In practice, this means that it disregards smaller variations and focuses solely on the most significant discrepancy between two data points.

While this may be beneficial in specific applications—such as quality control or threshold-based systems—it is generally less effective for most classification problems, especially those involving noisy or distributed features. Nonetheless, it remains a useful option in scenarios where the largest deviation is the primary decision factor.

Comparison Summary

Among the various distance metrics available for KNN, Euclidean distance (L2 norm) is the most commonly used. However, it tends to perform poorly in high-dimensional spaces due to the diminishing contrast between near and far points, which undermines its ability to differentiate between neighbors effectively. In contrast, Manhattan distance (L1 norm), also known as cityblock distance, is more robust in such settings. It is particularly useful when dealing with sparse data or when features contribute independently and linearly, making it a suitable alternative in many high-dimensional classification tasks.

Minkowski distance offers the most flexibility, as it generalizes both Euclidean and Manhattan distances by adjusting the parameter p. This tunability allows practitioners to find a balance between sensitivity to individual feature variations and overall geometric distance. Custom values of p, such as 1.5 or 3, can often yield better performance than standard norms, especially in complex or non-linear datasets.

Lastly, Chebyshev distance (L∞ norm) focuses solely on the largest absolute difference among all features. While this metric is useful in scenarios where the maximum deviation is of primary concern—such as threshold-based decision rules—it is typically less effective in general classification tasks due to its insensitivity to cumulative feature differences.

Overall, the choice of distance metric should be guided by the nature of the dataset, the distribution of features, and the computational constraints of the problem at hand.

Is KNN Still Appropriate for High-Dimensional Data?

Despite its limitations, KNN can still perform well in high-dimensional contexts—provided certain considerations are taken into account. In addition to selecting an appropriate distance metric, it is advisable to apply data preprocessing techniques such as:

• Feature scaling (standardization or normalization), to prevent bias from dominant features.

• Dimensionality reduction (e.g., PCA, UMAP, or t-SNE), to reduce the effects of distance concentration and noise.

• Cross-validation, to empirically determine the optimal distance metric and value of p.

  
  

It is also important to consider computational cost. KNN requires calculating distances to all training points for each prediction, which becomes increasingly demanding as data volume and dimensionality grow.

Selecting an appropriate distance metric is a critical yet often overlooked aspect of optimizing KNN classifiers, especially in high-dimensional settings. While Euclidean distance is widely used by default, alternatives such as Manhattan, Minkowski (with tuned p values), and Chebyshev can offer significant improvements in model performance and interpretability.

By combining distance metric tuning with proper scaling and dimensionality reduction techniques, practitioners can significantly enhance the effectiveness of KNN and better navigate the challenges posed by high-dimensional data.