ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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The score Lipschitz bound blows up for a point mass

Example

For $a \in R^{d}$ , $δ_{a}$ denotes the probability measure concentrated at $a$ . For $q > 0$ , write $ϕ_{q} (z) = (2 π q)^{- d / 2} \exp (- ∣ z ∣^{2} / (2 q))$ .

Let $d \geq 1$ , $T > 0$ , and $a \in R^{d}$ ; take $μ = δ_{a}$ . Let $g : [0, T] \to (0, \infty)$ be continuously differentiable with $0 < g_{\min} \leq g (t) \leq g_{\max}$ , and set $v (t) = \int_{0}^{t} g (r)^{2} d r$ . Then $p_{t} (x) = ϕ_{v (t)} (x - a)$ and

$s_{t} (x) = \nabla \log p_{t} (x) = - \frac{x - a}{v (t)}, D_{x} s_{t} (x) = - \frac{I_{d}}{v (t)} .$

The exact global Lipschitz constant of $s_{t}$ is $1 / v (t)$ . For every $0 < ϵ < T$ , its best uniform Lipschitz constant on $[ϵ, T]$ is $1 / v (ϵ)$ , which diverges as $ϵ ↓ 0$ .

Example references

Tang, Touzi, Zhang, and Zhou, Tweedie's Formulae and Diffusion Generative Models Beyond Gaussian, Section 1

Särkkä and Solin, Applied Stochastic Differential Equations, Chapter 5, §§5.1–5.2

Facts & Assumptions

Given: The point-mass initial law, Gaussian convolution, and score convention from the example.

[A1]

The full Axiom of Choice is inherited from the smoothing and score interfaces. The Axiom of Choice

[F1]

The Gaussian-smoothed density of a compact initial law is the mixture $p_{t} (x) = \int ϕ_{v (t)} (x - z) μ (d z)$ . Gaussian-smoothed data law and transition density

[F2]

The score and posterior mean satisfy $s_{t} (x) = - (x - m_{t, x}) / v (t)$ and its derivative is $- I_{d} / v (t) + C_{t, x} / v (t)^{2}$ . Gaussian-mixture score identities and a global Lipschitz bound

Proof

Given: The example's point mass and the score identities in [F2].

1.1A1F1F2given

Substituting $μ = δ_{a}$ into [F1] gives $p_{t} (x) = ϕ_{v (t)} (x - a)$ . The posterior measure $ν_{t, x}$ is then $δ_{a}$ , so its mean is $a$ and its covariance matrix is zero.

2.1F2step 1.1givenalgebra∎

Applying [F2] yields $s_{t} (x) = - (x - a) / v (t)$ and $D_{x} s_{t} = - I_{d} / v (t)$ . For any $x \neq y$ , $∣ s_{t} (x) - s_{t} (y) ∣ = ∣ x - y ∣ / v (t)$ , so the global Lipschitz constant is exactly $1 / v (t)$ . On $[ϵ, T]$ , $v$ is increasing, making the largest such constant $1 / v (ϵ)$ . Since $v (ϵ) \leq g_{\max}^{} ϵ \to 0$ and $v (ϵ) > 0$ , this bound tends to infinity.

Depends on

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Sources

Wenpin Tang, Nizar Touzi, Zikun Zhang, and Xun Yu Zhou, Tweedie's Formulae and Diffusion Generative Models Beyond Gaussian (standard reference, not scraped)
Simo Särkkä and Arno Solin, Applied Stochastic Differential Equations (standard reference, not scraped)