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Also known as: L1 Distance, Taxicab Distance
A metric equal to the sum of absolute differences of coordinates between two points in a grid-like space.
DESCRIPTION
Manhattan distance computes the L1 norm between points, summing absolute coordinate differences. It suits high-dimensional sparse data and grid-based movement. In machine learning, it influences clustering, nearest neighbors, and regularization (L1). Compared to Euclidean, it can be more robust to outliers and better reflect city-block constraints.
ALTERNATIVE DEFINITIONS
Wolfram MathWorld
The taxicab metric is the L1 metric defined by d(x,y) = sum_i |x_i − y_i|.
Provides a concise formula defining the L1 metric.
Encyclopedia of Mathematics
The L1 metric on R^n is defined by ρ(x,y) = Σ |x_i − y_i|, i = 1,…,n.
Formal definition in metric space terms.
scikit-learn
Supported metrics include ‘manhattan’, which computes the sum of absolute value differences across dimensions.
Practical implementation note within ML toolkit.
RELATED TERMS
Lp Norm K-Nearest Neighbors L1 Regularization Feature Scaling Cluster Analysis Similarity Search
COMPETING TERMS
Euclidean Distance Chebyshev Distance Minkowski Distance
KNOWLEDGE AREAS
TARGET AUDIENCE
Data Scientist AI Engineer Data Analyst
EXTERNAL SOURCES
- Wikipedia - Taxicab geometry
- scikit-learn - Usage in nearest neighbors
- Wolfram MathWorld - Formal metric definition
TOP QUESTIONS
When is Manhattan Distance preferred over Euclidean Distance?
In high-dimensional sparse spaces or grid-based movement, Manhattan distance can better reflect path costs and reduce sensitivity to large coordinate differences.
Category: Requirements
How does Manhattan Distance relate to L1 regularization?
Both use the L1 norm. L1 regularization encourages sparsity in parameters, analogous to summing absolute differences in distance.
Category: Example
What is a common mistake using Manhattan Distance in ML?
Failing to scale features. Different scales distort distance computations; standardize or normalize inputs to ensure meaningful comparisons.
Category: Mistakes
What is the risk of relying solely on Manhattan Distance?
It may misrepresent relationships in low dimensions where Euclidean geometry is more appropriate. Always validate distance choice against task performance.
Category: Risks
How is Manhattan Distance used in clustering?
It defines proximity between data points. Algorithms like k-medians leverage L1 distance to find representative centers robust to outliers.
Category: Example
Is Manhattan Distance metric a special case of Minkowski?
Yes. Manhattan distance is the Minkowski metric with p=1, while Euclidean is p=2.
Category: Example
How does Manhattan Distance impact k-NN classification?
It changes neighborhood shapes and sensitivity. For certain data distributions, L1 can improve performance compared to L2.
Category: Opinion
What is a hot topic around Manhattan Distance?
Efficient nearest neighbor search in high dimensions using L1-friendly indexes and embeddings.
Category: Topic
How to compute Manhattan Distance efficiently?
Vectorize absolute differences and sum across dimensions. Use optimized libraries and avoid Python loops for speed.
Category: Management