Visualizing the Change-of-Variables Rule and the Jacobian-Inverse Term in Flow Models
CatAstro_Piyush · x · 2026-10-17
While re-reading flow model papers in a reading group, the author realized his grasp of the change-of-variables rule in probability was weaker than it should be, so he wrote a blog post with a collection of visuals to build intuition for the Jacobian-inverse determinant term.
Core formula: for Y = g(X) with invertible g, fY(y) = fX(g⁻¹(y)) × |det J g⁻¹(y)|. Normalizing flows rest on this: starting from samples of a simple Gaussian, composing invertible transformations yields samples from arbitrarily complex distributions (e.g., human face images). The author notes the flow literature treats the formula as axiomatic, yet it isn't immediately intuitive — so the post breaks it down term by term with geometric and probabilistic visual intuition.
More from Research
- Chartography leaderboard: Claude Sonnet 5.5 tops chart understanding at 0.902 pass@1 — NielsRogge · 2026-10-17
- Decentralized training works? REAX doubles baselines in first model competitions — airesearch12 · 2026-10-17
- Paper shows minimum variance portfolio rankings are only weakly consistent for the mean — PtrPomorski · 2026-10-17
- Dev's take: every new AI benchmark — StarCraft, GUIs, Minecraft, CAD — is secretly a coding benchmark — wateriscoding · 2026-10-17
- AI Beat His 2027 Prediction: Lean Proof of Fields Medal Sphere-Packing Result Arrived by March — TOEwithCurt · 2026-10-17
- Real-Time Hair Rendering: 100 Characters, 100K Strands Each, Just 1.7 MB — TinfoilTricorn · 2026-10-17