LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Reverse KL between isotropic Gaussian laws

Statement

Assume AC. Let $d \geq 1$ , $m, n \in R^{d}$ , and $v, w > 0$ . If $Q = N_{d} (m, v I_{d})$ and $P = N_{d} (n, w I_{d})$ , then $D_{K L} (Q ∥ P) = \frac{d}{2} (\frac{v}{w} - 1 + \log \frac{w}{v}) + \frac{∥ m - n ∥_{2}^{2}}{2 w},$ where the KL divergence is directional, its first argument is absolutely continuous with respect to its second, and $\log$ is the natural logarithm.

Statement references

Gaussian density formula: A positive-definite multivariate normal law has its Gaussian density.

Directional KL convention: Kullback–Leibler divergence.

Natural logarithm: The natural logarithm as the inverse of the exponential function.

Gaussian reverse-KL terms in diffusion objectives: Ho, Jain, and Abbeel, §3.2.

Gaussian reverse-process loss: Prince, Chapter 18, “The reverse process and the tractable conditional”.

Facts & Assumptions

Given: AC, $Q = N_{d} (m, v I_{d})$ and $P = N_{d} (n, w I_{d})$ with $d \geq 1$ and $v, w > 0$ .

[A1]

AC is assumed by the Gaussian-density supplier and the library KL definition. The Axiom of Choice.

[F1]

A positive-definite multivariate normal law has its displayed Lebesgue density. A positive-definite multivariate normal law has its Gaussian density.

[F2]

A multivariate normal law has its specified mean and covariance. Multivariate normal law, including singular covariance.

[F3]

For probability measures $Q ≪ P$ , the library divergence is the expectation under $Q$ of the log Radon–Nikodym density ratio. Kullback–Leibler divergence.

[F4]

Expectation of an integrable random variable is its integral with respect to its probability law. Expectation of a nonnegative or integrable random variable.

Proof

technique · expand the Gaussian log-density ratio and integrate its quadratic terms

Given: The hypotheses in the Statement.

1.1A1F1F3given

By [F1], both Lebesgue densities are strictly positive, so $Q ≪ P$ . Expanding their Gaussian density formulas gives the log-density ratio $\log \frac{d Q}{d P} (x) = \frac{d}{2} \log \frac{w}{v} - \frac{∥ x - m ∥_{2}^{2}}{2 v} + \frac{∥ x - n ∥_{2}^{2}}{2 w}$ . All terms are integrable under $Q$ because its covariance is finite.

2.1A1F2F3F4step 1.1algebra∎

Under $Q$ , [F2] gives $E_{Q} ∥ X - m ∥_{2}^{2} = d v$ , and expansion of $X - n = (X - m) + (m - n)$ gives $E_{Q} ∥ X - n ∥_{2}^{2} = d v + ∥ m - n ∥_{2}^{2}$ because $E_{Q} (X - m) = 0$ . Integrating the ratio in step 1.1 and applying the AC-dependent convention [F3] and expectation definition [F4] yields $D_{K L} (Q ∥ P) = \frac{d}{2} \log \frac{w}{v} - \frac{d}{2} + \frac{d v + ∥ m - n ∥_{2}^{2}}{2 w}$ , which is the stated formula.

Source notes

Expectation and covariance convention: Multivariate normal law, including singular covariance.

Natural logarithm: The natural logarithm as the inverse of the exponential function.

Depends on

Used by

Dependency tree · two levels

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Sources

Jonathan Ho, Ajay Jain, and Pieter Abbeel, Denoising Diffusion Probabilistic Models (2020) (standard reference, not scraped)
Simon J. D. Prince, Understanding Deep Learning, Chapter 18: Diffusion models (2026), pp. 363–387 (standard reference, not scraped)