Why Manhattan Distance is More Reliable Than Euclidean Distance in Handling Outlier
Introduction
The K-Nearest Neighbors (KNN) algorithm is one of the simplest yet most powerful classification techniques in machine learning. It classifies a data point based on the majority label of its nearest neighbors. The algorithm’s effectiveness, however, heavily depends on the distance metric used to measure similarity between data points. While the Euclidean distance is the most commonly used metric, it is also highly sensitive to outliers. In real-world datasets, outliers are almost inevitable due to noise, errors, or extreme variations. These outliers can distort the distance calculations and reduce classification accuracy. Hence, choosing a robust distance metric that minimizes the influence of outliers becomes essential.
In such cases, the Manhattan distance, also known as the L1 norm or city block distance, is more robust and reliable than the Euclidean distance. This essay explores why Manhattan distance performs better in the presence of outliers and how it improves the overall performance of KNN models.
Understanding the Problem
The KNN algorithm works by finding the ‘k’ nearest data points to a given test point and assigning it the most frequent class among those neighbors. The closeness between points is measured using a distance metric—commonly Euclidean distance (L2 norm). Euclidean distance calculates the straight-line distance between two points in multidimensional space using the formula: DEuclidean=∑i=1n(xi−yi)2D_{Euclidean} = \sqrt{\sum_{i=1}^{n}(x_i – y_i)^2}DEuclidean=i=1∑n(xi−yi)2
While this metric is effective for clean datasets, it squares the differences between feature values. This squaring magnifies large deviations, making the distance highly sensitive to outliers. For example, if one feature contains an unusually large value, it disproportionately increases the overall distance, pulling the classification decision toward incorrect neighbors.
In contrast, Manhattan distance (L1 norm) calculates the sum of the absolute differences between the coordinates: DManhattan=∑i=1n∣xi−yi∣D_{Manhattan} = \sum_{i=1}^{n}|x_i – y_i|DManhattan=i=1∑n∣xi−yi∣
This approach avoids squaring the differences and therefore limits the impact of extreme values. It measures distance along coordinate axes—similar to how a car would travel through a city grid—which is why it is often called the city block metric.
Why Manhattan Distance Is More Robust to Outliers
1. Reduced Sensitivity to Extreme Values
In Manhattan distance, each feature contributes proportionally to the total distance. Unlike Euclidean distance, large deviations in a single feature do not dominate the overall measure. This property makes it far less affected by outliers. For example, if one feature in your dataset has a value that is ten times larger than others, Euclidean distance will exaggerate its influence due to squaring, while Manhattan distance will treat it linearly.
2. Stability in High-Dimensional Spaces
As the number of features increases, the effect of outliers in Euclidean space becomes even more pronounced. This phenomenon, known as the curse of dimensionality, causes distances to become less meaningful. Manhattan distance, however, remains more stable in higher dimensions because it scales linearly with feature differences, making it a better fit for high-dimensional datasets with noisy variables.
3. Better Alignment with Real-World Data
Real-world data often contain non-linear relationships, noise, and irregular distributions. In such environments, Manhattan distance reflects real differences more faithfully. For example, in financial, medical, or text data, where outliers can be large but rare, using Manhattan distance ensures that such anomalies don’t overpower the classification process.
4. Combination with Other Techniques
Manhattan distance also integrates well with feature scaling and normalization techniques. When data are standardized (using z-scores or min-max scaling), Manhattan distance further enhances robustness by ensuring all features contribute equally. It can also be paired with weighted KNN, where closer points get higher weights, ensuring that the influence of distant outliers remains minimal.
Conclusion
In summary, while K-Nearest Neighbors is a simple and intuitive algorithm, its performance depends critically on the choice of distance metric. The Euclidean distance, though popular, amplifies the effect of outliers because it squares deviations, making it unsuitable for noisy datasets. The Manhattan distance, on the other hand, is more robust, stable, and reliable in the presence of outliers. By measuring the absolute differences instead of squared ones, it provides a fairer and more balanced evaluation of proximity.
Therefore, when working on classification problems where data contain outliers, Manhattan distance should be preferred over Euclidean distance. It not only improves accuracy but also enhances the interpretability and resilience of your model. In practice, combining Manhattan distance with data normalization and careful feature engineering can lead to a KNN classifier that performs efficiently even in imperfect, real-world datasets—proving that sometimes, the simplest linear measure can be the most powerful defense against noisy data.