Lecture 1.5 — Gaussian Primer

(a)  every Gaussian is a linear image of the standard oneOn the left, 700 draws of g ∼ N(0, I₂) with the circles ‖x‖ = 1, 2. On the right, the same 700 points after x = μ + Ag with A = U(θ) diag(√λ₁, √λ₂) — the circles become the ellipses of Σ = AA⊤. Slide λ₁, λ₂, θ and drag the mean; hover any point to see it on both sides at once: the cloud is not resampled, only mapped. The chip checks the fact numerically: the sample covariance of the mapped points sits on top of AA⊤.

λ₁2.20
λ₂0.50
θ0.60
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(b)  scaling, shifting, summing — still GaussianThe bars are honest histograms of 5000 seeded draws; the gold curve is the density the closure fact predicts, not a fit. Scale & shift: from X ∼ N(0.5, 1), the variable aX+b lands on N(a·0.5+b, a²) — a stretches, b slides, the shape stays Gaussian, and a negative a flips it. Sum: with independent Y ∼ N(−0.3, τ²), the sum lands on N(0.2, 1+τ²): means add, variances add — never the standard deviations. The chip compares predicted mean and variance against the empirical ones.

rule
a1.4
b-0.6
τ0.80
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(a)  rotational invariance, one direction at a timeThe density of g ∼ N(0, I) depends only on ‖x‖, so the gold circles never move: press rotate and the 700 points visibly turn while the cloud’s distribution stays exactly what it was — Ug ∼ g. Drag the direction w around the rim: the histogram below is the projection g⊤w, and at every angle it hugs the same N(0,1) curve, with empirical mean ≈ 0 and variance ≈ 1. Hover a point to see its decomposition g = (g⊤w)w + g⊥ — the two parts are independent Gaussians.

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(b)  the score, and the integer cousinSince log p(x) = const − ‖x‖²/2, the score field is ∇log p(x) = −x: every arrow points straight at the origin with length ‖x‖. Drag the probe and read it off. This is the field score estimation targets and diffusion sampling runs along (Lecture 5). On the right, the discrete Gaussian on ℤ assigns k probability proportional to exp(−k²/2s²); slide s and the bars follow the continuous bell drawn through them — narrow s concentrates on 0, wide s flattens toward the lattice.

discrete s1.40
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