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Essays, notes, and technical writing.

Essays, notes, and technical writing on mathematics, machine learning, information geometry, and related topics.

Featured series: The Natural Geometry of Learning

The Second Ruler of Displacement

Why Wasserstein distance is the geometry of moving probability mass

Natural Geometry of Learning

Optimal Transport

Wasserstein Distance

Brenier Theorem

Benamou-Brenier

Otto's Calculus

This post introduces Wasserstein distance as the second “ruler” for probability distributions: a metric that measures the cost of moving probability mass through space. We will see why it arises from the Earth Mover’s problem, why the quadratic case has special geometric structure through Brenier’s theorem, and how the Benamou–Brenier formulation turns optimal transport into a dynamic geometry of probability flows.

Jun 13, 2026

Gaurav Khanal

The First Ruler of Distinguishability

Why Fisher information is the local geometry of statistical models

Natural Geometry of Learning

information geometry

fisher information

pullback metric

riemannian metric

This post introduces Fisher information as the first “ruler” for probability distributions: a local metric that measures how distinguishable nearby statistical models are from data. We will see why it arises naturally as a pullback metric, why Chentsov’s theorem makes it unique, and how it foreshadows the deeper connection between KL divergence, natural gradients, and learning.

Jun 5, 2026

Gaurav Khanal

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The Second Ruler of Displacement

The First Ruler of Distinguishability