TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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The score-matching integration-by-parts identity

Statement

Assume the Axiom of Countable Choice used by the library's Lebesgue-measure and weak-derivative representative interfaces. Let $d \geq 1$ , let $λ_{d}$ be Lebesgue measure on $R^{d}$ , and let $p : R^{d} \to [0, \infty)$ be a continuously differentiable probability density, so $\int p d λ_{d} = 1$ . Write $P_{p} (A) : = \int_{A} p d λ_{d}, A \in L (R^{d}),$ which is a probability measure, and let $E_{p} [f] : = \int f d P_{p}$ for integrable $f$ . Define the data score by $s_{p} (x) : = \nabla \log p (x)$ when $p (x) > 0$ and set $s_{p} (x) : = 0$ when $p (x) = 0$ . For an energy model relative to $λ_{d}$ with $0 < Z_{θ} = \int e^{- E_{θ}} d λ_{d} < \infty$ , assume $E_{θ} \in C^{1} (R^{d})$ and define its model score field by $ψ_{θ} : = - \nabla E_{θ}$ . Assume each component of $ψ_{θ}$ is locally integrable and has locally integrable weak first derivatives: for each $i, j$ , a locally integrable representative $g_{i j}$ satisfies $\int_{R^{d}} ψ_{θ, i} \partial_{j} φ d λ_{d} = - \int_{R^{d}} g_{i j} φ d λ_{d} (φ \in C_{c}^{\infty} (R^{d})) .$ Here $C_{c}^{\infty} (R^{d})$ denotes the smooth compactly supported test functions. The weak derivatives are unique up to Lebesgue-null sets, and define $div ψ_{θ} : = \sum_{i = 1}^{d} g_{i i}$ ; the integrals below do not depend on the chosen representatives. The $C^{1}$ convention is that the first derivative exists everywhere and is continuous. For each admissible $θ$ , assume $E_{p} ∥ s_{p} ∥^{2} < \infty, E_{p} ∥ ψ_{θ} ∥^{2} < \infty, E_{p} ∣ div ψ_{θ} ∣ < \infty .$ Then the score-matching objective $J (θ) : = \frac{1}{2} E_{p} ∥ ψ_{θ} - s_{p} ∥^{2}$ satisfies $J (θ) = E_{p} [\frac{1}{2} ∥ ψ_{θ} ∥^{2} + div ψ_{θ}] + C_{p}, C_{p} : = \frac{1}{2} E_{p} ∥ s_{p} ∥^{2},$ where $C_{p}$ is independent of $θ$ . No pointwise boundary limit at infinity is assumed.

Statement references

Aapo Hyvärinen, Theorem 1 and Appendix A of the full JMLR paper states the score-matching identity under a pointwise boundary condition.

Kevin P. Murphy, Chapter 24, Sections 24.1–24.3.4 defines energy models, the model score and score matching.

Facts & Assumptions

Given: The hypotheses in the Statement, including Countable Choice for the library Lebesgue measure interface and the three stated finite $p$ -integrability conditions.

[A1]

$p$ is a nonnegative $C^{1}$ probability density; for each admissible parameter, $E_{θ} \in C^{1}$ , its score field is locally integrable with locally integrable weak first derivatives, and the score and divergence have the stated finite $p$ -moments.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every countable family of nonempty sets has a choice function; it is used through [F3], [F8], [F10], [F17], and [F19].

[F1]

Energy-based density and partition function: for $0 < Z_{θ} < \infty$ , the model density is $e^{- E_{θ}} / Z_{θ}$ , and $Z_{θ}$ is independent of $x$ .

[F2]

The measure with density $f$ relative to $μ$ : for a nonnegative measurable density $f$ , $f d μ$ is the measure $A \mapsto \int_{A} f d μ$ .

[F3]

Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume: under Countable Choice, $λ_{d}$ is a complete measure on the Lebesgue sigma-algebra of $R^{d}$ .

[F4]

Dominated convergence: almost-everywhere convergence under one integrable majorant implies convergence of the corresponding integrals.

[F5]

Probability measures and probability spaces: a measure of total mass one is a probability measure.

[F6]

Continuously differentiable maps, local inverses, and local diffeomorphisms: a map on an open Euclidean set is $C^{1}$ when it is differentiable everywhere and its derivative entries are continuous.

[F7]

Weak derivative of a locally integrable function: $g_{i j}$ is a weak derivative exactly when its signed test-function identity holds for every smooth compactly supported test function.

[F8]

Uniqueness of a weak derivative as an almost-everywhere class: under Countable Choice, any two locally integrable representatives satisfying the same weak-derivative identity agree almost everywhere.

[F9]

A continuous map has Borel preimages of Borel sets: a continuous map has Borel preimages of Borel sets.

[F10]

Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable: under Countable Choice, Borel subsets of $R^{d}$ are Lebesgue measurable.

[F11]

A smooth bump between concentric Euclidean balls: for $0 < r < R$ , a smooth function $χ : R^{d} \to [0, 1]$ can equal $1$ on ${\overline{B}}_{r} (0)$ and have support contained in $B_{R} (0)$ .

[F12]

Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line: closed bounded subsets of $R^{d}$ are compact.

[F13]

A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value: continuous real functions on nonempty compact metric spaces are bounded.

[F14]

Chain sum product and composition rules for Banach derivatives: total derivatives obey the bounded-bilinear product rule.

[F15]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: for $u > 0$ , $\log^{'} (u) = 1 / u$ .

[F16]

Monotonicity and nonnegative homogeneity of the nonnegative integral: the nonnegative integral is monotone and positively homogeneous.

[F17]

A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions: a $C^{1}$ diffeomorphism transforms nonnegative Lebesgue integrals by its absolute Jacobian determinant.

[F18]

The mollifier family generated by a unit-mass smooth bump: a unit-mass smooth compactly supported function generates the scaled family $ρ_{ϵ} (x) = ϵ^{- d} ρ_{0} (x / ϵ)$ .

[F19]

Convolution with a mollifier is smooth, and derivatives pass under the integral sign: convolution of a locally integrable function with a smooth compactly supported mollifier is smooth.

[F20]

Differentiation under the integral sign: an integrable parameter-independent majorant permits differentiation under the integral sign.

[F21]

Fermat's theorem: an interior differentiable local extremum has zero gradient: a differentiable function at an interior local extremum has zero gradient.

[F22]

The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case: the gradient coordinates of a scalar-valued function are its coordinate partial derivatives.

[F23]

Total differentiability gives a local $O (∥ h ∥_{2})$ increment bound and therefore continuity: total differentiability implies continuity.

[F24]

Chain sum product and composition rules for Banach derivatives: total derivatives obey the chain rule.

Proof

1.1givenA1A2F1F2F3F5F6F8F9F10F11F12F13F15F16F17F18F23F24

By [F1], the admissible model has density $e^{- E_{θ}} / Z_{θ}$ , with $Z_{θ}$ independent of $x$ . Since $p$ is $C^{1}$ , [F6, F23] make it continuous; [F9, F10] then make it Lebesgue measurable, so [F2] makes $P_{p}$ a measure, and [F5] makes it a probability measure. The local increment bound [F23] gives, at each $x \in U = {p > 0}$ , constants $C_{x}, δ_{x} > 0$ with $∣ p (y) - p (x) ∣ \leq C_{x} ∥ y - x ∥$ for $∥ y - x ∥ < δ_{x}$ ; taking $∥ y - x ∥ < \min {δ_{x}, p (x) / (2 (C_{x} + 1))}$ shows $p (y) > 0$ , so $U$ is open. On $U$ , [F6, F15, F24] make $s_{p} = \nabla \log p$ continuous. For each open $O \subseteq R^{d}$ , its score preimage is the union of the relative-open preimage in $U$ and $U^{c}$ when $0 \in O$ ; hence the zero extension is Borel and Lebesgue measurable by [F10]. By [F8], weak-derivative representatives agree off a Lebesgue-null set; [F2] makes every such set $P_{p}$ -null, so the divergence integral is representative-independent. Choose $χ$ from [F11] with $r = 1$ and $R = 2$ . By [F9, F10], this continuous bump is Lebesgue measurable. Its gradient vanishes outside $B_{2}$ ; [F12, F13] make $C : = ∥ \nabla χ ∥_{\infty}$ finite. The box $[- 1 / (2 d), 1 / (2 d))^{d}$ lies in $B_{1}$ , since its points have norm at most $\sqrt{d} / (2 d) \leq 1 / 2$ , while $supp χ \subset B_{2} \subset [- 3, 3)^{d}$ and $0 \leq χ \leq 1$ . Thus [F3, F16] give $d^{- d} \leq M : = \int χ d λ_{d} \leq 6^{d}$ . Set $ρ_{0} = χ / M$ ; positive homogeneity [F16] gives it unit mass. Let $ρ_{0, ε}$ be its scaled family from [F18]; [F18] gives support in ${\overline{B}}_{2 ε}$ and [F17] gives unit mass. For $R > 0$ , define $χ_{R} (x) = χ (x / R)$ . Then $χ_{R} = 1$ on ${\overline{B}}_{R}$ , vanishes outside $B_{2 R}$ , and [F24] gives $∥ \nabla χ_{R} ∥_{\infty} \leq C / R$ .

2.1A1F3F4F7F12F13F14F17F19F20F24step 1.1

Fix $R$ and put $φ = p χ_{R}$ . The product rule [F14] makes $φ$ a nonnegative $C^{1}$ function supported in ${\overline{B}}_{2 R}$ . By [F12, F13], $φ$ and its first derivatives are bounded there; the ball lies in a finite-volume box by [F3], so $φ$ is locally integrable. The convolution $φ_{ε} = φ * ρ_{0, ε}$ is smooth by [F19] and supported in ${\overline{B}}_{2 R + 2}$ for $ε \leq 1$ , since $supp φ \subseteq {\overline{B}}_{2 R}$ and $supp ρ_{0, ε} \subseteq {\overline{B}}_{2}$ . By [F17], applied to the nonnegative convolution integrand, $φ_{ε} (x) = \int φ (x - z) ρ_{0, ε} (z) d z$ . For each coordinate $j$ , [F24] gives the parameter derivative of $f (z, t) = φ (x + t e_{j} - z) ρ_{0, ε} (z)$ . Its sections are integrable because $φ$ is bounded and the kernel has mass one; its derivative is measurable and bounded by $∥ \partial_{j} φ ∥_{\infty} ρ_{0, ε} (z)$ , an integrable majorant. Thus [F20] gives $\partial_{j} φ_{ε} (x) = \int \partial_{j} φ (x - z) ρ_{0, ε} (z) d z$ . The unit mass and shrinking support give, for each $x$ , $∣ φ_{ε} (x) - φ (x) ∣ \leq \sup_{∥ z ∥ \leq 2 ε} ∣ φ (x - z) - φ (x) ∣ \to 0$ , with the same estimate for each first derivative. Since $φ$ and $\nabla φ$ are bounded by [F12, F13], the weak-identity integrands are dominated on this common compact support by constants times $∣ ψ_{θ, i} ∣$ and $∣ g_{i i} ∣$ . Local integrability and [F4] pass the identities from $φ_{ε}$ to $φ$ . Summing over $i$ and expanding $\nabla (p χ_{R})$ with [F14] yields $\int χ_{R} ψ_{θ} \cdot \nabla p d λ_{d} + \int p ψ_{θ} \cdot \nabla χ_{R} d λ_{d} = - \int p χ_{R} div ψ_{θ} d λ_{d}$ .

2.2givenA1F15F21F22F24step 1.1algebra

On ${p > 0}$ , [F15, F24] give $\nabla p = p s_{p}$ . At any point where $p = 0$ , nonnegativity makes it an interior local minimum, so [F21, F22] give $\nabla p = 0$ there too. Thus $\nabla p = p s_{p}$ everywhere. Componentwise $∣ ψ_{θ} \cdot s_{p} ∣ \leq \sum_{i} ∣ ψ_{θ, i} s_{p, i} ∣$ and $2 ∣ a b ∣ \leq a^{2} + b^{2}$ imply $\int ∣ ψ_{θ} \cdot \nabla p ∣ d λ_{d} \leq \frac{1}{2} E_{p} (∥ ψ_{θ} ∥^{2} + ∥ s_{p} ∥^{2}) < \infty$ and $\int p ∣ div ψ_{θ} ∣ d λ_{d} < \infty$ . Also $∥ ψ_{θ} ∥ \leq (1 + ∥ ψ_{θ} ∥^{2}) / 2$ , so the cutoff term from step 1.1 is bounded in absolute value by $\frac{C}{2 R} (1 + E_{p} ∥ ψ_{θ} ∥^{2}) \to 0$ .

3.1F4step 2.1step 2.2

Let $R = n$ for integers $n \to \infty$ . Then $χ_{n} \to 1$ pointwise and $∣ χ_{n} ∣ \leq 1$ . The first integrand in step 2.1 is dominated by $∣ ψ_{θ} \cdot \nabla p ∣$ and the third by $p ∣ div ψ_{θ} ∣$ , both integrable by step 2.2. Applying [F4] to both terms and using the vanishing cutoff term from step 2.2 gives $\int ψ_{θ} \cdot \nabla p d λ_{d} = - \int p div ψ_{θ} d λ_{d}$ .

4.1givenA1step 2.2step 3.1algebra∎

Expanding the square gives $J (θ) = \frac{1}{2} E_{p} ∥ ψ_{θ} ∥^{2} + \frac{1}{2} E_{p} ∥ s_{p} ∥^{2} - \int p ψ_{θ} \cdot s_{p} d λ_{d}$ . Since $\nabla p = p s_{p}$ by step 2.2, step 3.1 turns the last term into $+ E_{p} [div ψ_{θ}]$ ; the remaining data-score term is $C_{p}$ , independent of $θ$ .

Remarks

The cutoff and integration-by-parts argument is choice-free once the Lebesgue measure and weak-derivative classes are available. Countable Choice is used for those two interfaces; no full Axiom of Choice is used. The cutoff estimate supplies the multivariate limit directly and does not rely on pointwise vanishing of $p ψ_{θ}$ at infinity.

Depends on

Used by

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Sources

Aapo Hyvärinen, Estimation of Non-Normalized Statistical Models by Score Matching, Theorem 1 and Appendix A (standard reference, not scraped)
Kevin P. Murphy, Probabilistic Machine Learning: Advanced Topics, Chapter 24, Sections 24.1–24.3.4 (standard reference, not scraped)