Wasserstein Distance & Optimal Transport — Fully Explained

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What's the best way to measure the distance between two probability distributions? This video covers optimal transport theory from the ground up.

We start with the intuition of moving piles of sand, then formalize it with Monge's transport maps and Kantorovich's relaxation. Finally, we arrive at the Wasserstein distance — a true geometric metric on the space of distributions.

Topics covered:
• The sand-moving intuition
• Monge's deterministic transport maps
• Why Monge's problem sometimes has no solution
• Kantorovich's probabilistic transport plans
• The linear programming structure
• The Wasserstein distance and its properties

No prerequisites beyond basic calculus and probability.

A complete visual guide to optimal transport and the Wasserstein distance.

From moving sand piles to rigorous mathematics — learn how to measure the "effort" needed to transform one distribution into another. We cover Monge's original formulation, Kantorovich's elegant relaxation, and why the Wasserstein distance has become essential in machine learning and beyond.

Timestamps in comments.