Manhattan Distance

Imagine navigating a city laid out in a perfect grid: you can’t cut diagonally through buildings, so you move block by block. Manhattan Distance captures that same idea for data points—how far apart two feature vectors are when you add up the “straight-line” steps along each feature.

What it is (and the math behind it)

Manhattan Distance (also called L1 distance) between two vectors x and y is the sum of absolute differences across features:

d(x, y) = Σ |x_i - y_i|

So if you have two customers described by features like “number of purchases,” “days since last login,” and “support tickets,” Manhattan distance adds up how different they are on each of those dimensions, without squaring anything or taking square roots.

Why it matters in supervised learning (especially k-NN)

In k-Nearest Neighbors (k-NN), the distance metric decides who counts as a “neighbor,” which directly changes predictions. Manhattan distance tends to be more robust than Euclidean distance when:

• You have outliers: large deviations don’t get amplified by squaring.
• Your data is high-dimensional: L1 can preserve meaningful differences when distances start to look similar.
• You expect effects to be more “additive per feature” than “geometric.”

Practical examples and how you’ll see it in code

• Spam detection: comparing email feature counts (word frequencies) where a few extreme words shouldn’t dominate similarity.
• Credit risk: finding similar borrowers based on multiple numeric attributes with occasional unusual values.
• Churn prediction: matching customers by behavior counts (logins, purchases, complaints).

In scikit-learn, you’ll encounter it via k-NN’s metric="manhattan" (or equivalently p=1 for Minkowski distance).