Proof of Moving the Rescaled Head of the Diagonal Gaussian Measure by a Diffeomorphism: the Image Has an Explicit Positive Density Depending on the Head Only

Each result cited is universally quantified over the data in its own statement. Throughout, write $\rho=\rho_{\tilde{c}^{(n)}}$ and $\tilde{\gamma}=\tilde{\gamma}_{n}$, let $\lambda_{n}$ be Lebesgue measure on $\mathcal{B}(\mathbb{R}^{n})$, and write $u_{k}$ for the $k$th coordinate of $u\in\mathbb{R}^{n}$ and $F_{k}$ for the $k$th coordinate function of $F$. Continuity of maps between metric spaces is that of Continuous Map Between Metric Spaces. A composite $\beta\circ\alpha$ of continuous maps between metric spaces is continuous, by two applications of that definition: given a point $x$ and $\varepsilon>0$, choose $\eta>0$ for $\beta$ at $\alpha(x)$ and $\varepsilon$, and then $\delta>0$ for $\alpha$ at $x$ and $\eta$. Maps continuous for the Borel $\sigma$-algebras of their metric spaces are Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space, and composites of Borel maps are Borel by claim 4 there.

Claim 1.

Step 1 (Coordinates). For $u,u'\in\mathbb{R}^{n}$ and $k\in[n]$ we have $|u_{k}-u'_{k}|\le\lVert u-u'\rVert$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate, so the coordinate function $u\mapsto u_{k}$ is continuous from $(\mathbb{R}^{n},d_{E})$ to $\mathbb{R}$. Since $F$ is continuous by Change of Variables for the Lebesgue Integral under a Continuously Differentiable Bijection of Euclidean Space with Symmetric Positive Definite Jacobian Matrix, and the Density of a Push-Forward §regularity, each $F_{k}$ is continuous as a composite, and so is $u\mapsto F_{k}(u)-u_{k}$, by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set (a sum of $F_{k}$ and the multiple of the coordinate function by $-1$).

Step 2 (The inverse map satisfies the hypotheses). The map $F^{-1}$ is a bijection of $\mathbb{R}^{n}$ whose inverse is $F$, and the components of $F^{-1}$ and of $F$ are of class $C^{1}$ on $\mathbb{R}^{n}$ by hypothesis. Let $v\in\mathbb{R}^{n}$, put $u=F^{-1}(v)$, $A=DF(u)$ and $C=D(F^{-1})(v)$. Since $v=F(u)$, the hypothesis at $u$ says that $A$ is symmetric and positive definite (Symmetric, Positive Semidefinite, and Positive Definite Real Matrices) and that $C$ is the inverse matrix of $A$, that is, $AC=CA=I_{n}$. First, $C$ is symmetric: by claim 6 of Elementary Properties of the Transpose of a Real Matrix, $A^{\top}$ is invertible with $(A^{\top})^{-1}=(A^{-1})^{\top}$; as $A^{\top}=A$, this reads $C=C^{\top}$. Second, $C$ is positive definite: let $y\in\mathbb{R}^{n}$ be nonzero and put $z=Cy$. Then $Az=(AC)y=I_{n}y=y$, so $z\ne0$ (otherwise $y=A0=0$), and, the dot product being symmetric, $y\cdot(Cy)=(Az)\cdot z=z\cdot(Az)>0$ by positive definiteness of $A$. Third, the relations $CA=AC=I_{n}$ say that $A$ is an inverse of $C$, so $A$ is the inverse matrix of $C$ by the uniqueness recorded in Inverse Matrix and Invertible Real Square Matrix; and $A=DF(u)=D\bigl((F^{-1})^{-1}\bigr)\bigl(F^{-1}(v)\bigr)$. Hence for every $v$ the Jacobian matrix $D(F^{-1})(v)$ is symmetric and positive definite and the Jacobian matrix of the inverse map $F$ at $F^{-1}(v)$ is its inverse matrix: $F^{-1}$ satisfies the hypotheses imposed on $F$, and $T_{F^{-1}}$ is defined, with $(F^{-1})^{-1}=F$.

Step 3 (Shifted heads and tails). The map $\eta=F-\mathrm{id}$ is Borel, as recorded in the statement, so Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §shift, whose claims are in force by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §heads (for instance with $\mu=\gamma_{c}\in\mathcal{P}_{2}(X)$ by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian; that claim does not involve $\mu$), gives for every $x\in X$

$$r_{n}(T_{F}(x))=r_{n}(x)+F(r_{n}(x))-r_{n}(x)=F(r_{n}(x)),\qquad Q_{n}T_{F}(x)=Q_{n}x .$$

By Step 2, Steps 1 and 3 apply to $F^{-1}$; so the same holds with $F^{-1}$ in place of $F$.

Step 4 (A point is determined by its rescaled head and its tail). Let $x\in X$. By Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, $x=P_{n}x+Q_{n}x$ and $P_{n}x=p_{n}^{*}(p_{n}(x))=\sum_{k=1}^{n}x_{k}e_{k}=\sum_{k=1}^{n}a_{k}^{1/2}\,(r_{n}(x))_{k}\,e_{k}$, since $a_{k}^{1/2}a_{k}^{-1/2}x_{k}=x_{k}$. Hence if $x,y\in X$ satisfy $r_{n}(x)=r_{n}(y)$ and $Q_{n}x=Q_{n}y$, then $x=y$.

Step 5 (The inverse). Let $x\in X$, $y=T_{F}(x)$ and $z=T_{F^{-1}}(y)$. By Step 3, applied to $F^{-1}$ and then to $F$, $r_{n}(z)=F^{-1}(r_{n}(y))=F^{-1}(F(r_{n}(x)))=r_{n}(x)$ and $Q_{n}z=Q_{n}y=Q_{n}x$; so $z=x$ by Step 4. Interchanging the roles of $F$ and $F^{-1}$, which is legitimate by Step 2, gives $T_{F}(T_{F^{-1}}(y))=y$ for every $y\in X$. Hence $T_{F}$ is a bijection of $X$ and $T_{F^{-1}}$ is its inverse.

Step 6 (Continuity). For $k\in[n]$ let $\varphi_{k}(x)=F_{k}(r_{n}(x))-(r_{n}(x))_{k}$ for $x\in X$; it is continuous as the composite of $r_{n}$, continuous by Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space §head (for instance with $\mu=\gamma_{c}\in\mathcal{P}_{2}(X)$ by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian), with the continuous function of Step 1. By the definition of $\Lambda_{n}$ in Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space, $T_{F}(x)=x+\sum_{k=1}^{n}a_{k}^{1/2}\varphi_{k}(x)\,e_{k}$, so the triangle inequality and $|e_{k}|=1$ give, for $x,y\in X$,

$$|T_{F}(y)-T_{F}(x)|\le|y-x|+\sum_{k=1}^{n}a_{k}^{1/2}\,|\varphi_{k}(y)-\varphi_{k}(x)| .$$

Given $x\in X$ and $\varepsilon>0$, choose for each $k\in[n]$ a $\delta_{k}>0$ such that $|\varphi_{k}(y)-\varphi_{k}(x)|<\varepsilon/(2n\,a_{k}^{1/2})$ whenever $|y-x|<\delta_{k}$, and let $\delta$ be the least of $\varepsilon/2,\delta_{1},\dots,\delta_{n}$; then $|y-x|<\delta$ gives $|T_{F}(y)-T_{F}(x)|<\varepsilon/2+n\cdot\varepsilon/(2n)=\varepsilon$. So $T_{F}$ is continuous, hence Borel; by Step 2 the same argument applies to $T_{F^{-1}}$. With Steps 3 and 5 this proves claim 1.

Claim 2.

Step 7 ($G_{F}$ is Borel and positive). By The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §regularity, $\rho$ is smooth and positive on $\mathbb{R}^{n}$, hence continuous by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, $\mathbb{R}^{n}$ being open by claim 1 there. By Change of Variables for the Lebesgue Integral under a Continuously Differentiable Bijection of Euclidean Space with Symmetric Positive Definite Jacobian Matrix, and the Density of a Push-Forward §regularity, $F^{-1}$ and $u\mapsto\det DF(u)$ are continuous, and $\det DF(u)>0$ for every $u$, as recorded in the statement. So the functions $v\mapsto\rho(F^{-1}(v))$ and $v\mapsto\det DF(F^{-1}(v))$ are continuous composites, the latter and $\rho$ are nowhere zero, their reciprocals are continuous by claim 2 of Continuity of the Reciprocal of a Nonvanishing Real-Valued Function on a Metric Space, and $G_{F}$, the product of $\rho\circ F^{-1}$ with these two reciprocals, is continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, hence Borel (the Borel $\sigma$-algebra of $(\mathbb{R},d_{\mathbb{R}})$ being $\mathcal{B}(\mathbb{R})$ by claim 2 of Borel Measurability and Bounded Integration on a Metric Space). As a quotient of positive numbers, $G_{F}(v)>0$ for every $v$.

Step 8 ($G_{F}$ is a density of $F_{\#}\tilde{\gamma}$ with respect to $\tilde{\gamma}$). By Diagonal Gaussian Measures on Euclidean Space §measure, $\tilde{\gamma}(B)=\int_{\mathbb{R}^{n}}\mathbf{1}_{B}\,\rho\,d\lambda_{n}$ for $B\in\mathcal{B}(\mathbb{R}^{n})$, so the nonnegative Borel function $\rho$ is a density of $\tilde{\gamma}\in\mathcal{P}(\mathbb{R}^{n})$ with respect to $\lambda_{n}$ in the sense of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities. By Change of Variables for the Lebesgue Integral under a Continuously Differentiable Bijection of Euclidean Space with Symmetric Positive Definite Jacobian Matrix, and the Density of a Push-Forward §densities with $d=n$, the function

$$v\mapsto\frac{\rho(F^{-1}(v))}{\det DF(F^{-1}(v))}=G_{F}(v)\,\rho(v)$$

is a density of $F_{\#}\tilde{\gamma}$ with respect to $\lambda_{n}$; here $F_{\#}\tilde{\gamma}\in\mathcal{P}(\mathbb{R}^{n})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, $F$ being Borel. Hence for $B\in\mathcal{B}(\mathbb{R}^{n})$, using claim 3 of Image Measures, Measures with Densities, and Change of Variables with the density $h=\rho$ (whose measure is $\tilde{\gamma}$) and the nonnegative Borel function $\mathbf{1}_{B}G_{F}$,

$$(F_{\#}\tilde{\gamma})(B)=\int_{\mathbb{R}^{n}}\mathbf{1}_{B}\,G_{F}\,\rho\,d\lambda_{n}=\int_{\mathbb{R}^{n}}\mathbf{1}_{B}\,G_{F}\,d\tilde{\gamma},$$

so $G_{F}$, which is Borel and nonnegative by Step 7, is a density of $F_{\#}\tilde{\gamma}$ with respect to $\tilde{\gamma}$.

Step 9 (Tails have vanishing head coordinates). Let $N=\{w\in X:p_{n}(w)\ne0\}$, the complement of the preimage of the closed set $\{0\}$ under the Borel map $p_{n}$ (Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity); it is Borel by claim 1 of Borel Measurability and Bounded Integration on a Metric Space. For $x\in X$, the map $p_{n}$ is linear, its coordinates being inner products with $e_{1},\dots,e_{n}$, and $p_{n}(p_{n}^{*}(y))=y$ for $y\in\mathbb{R}^{n}$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity; hence $p_{n}(Q_{n}x)=p_{n}(x)-p_{n}(p_{n}^{*}(p_{n}(x)))=0$. So $Q_{n}^{-1}(N)=\emptyset$ and $\tau_{n}(N)=\gamma_{c}(\emptyset)=0$, by the definition of $\tau_{n}$ in A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §tails.

Step 10 (The map on the product). Let $(v,w)\in\mathbb{R}^{n}\times X$ with $w\notin N$, and $x=\Psi_{n}(v,w)=\sum_{j=1}^{n}a_{j}^{1/2}v_{j}e_{j}+w$. For $k\in[n]$, orthonormality of the basis and $w_{k}=0$ give $x_{k}=a_{k}^{1/2}v_{k}$, so $r_{n}(x)=v$. Therefore

$$T_{F}(\Psi_{n}(v,w))=x+\sum_{k=1}^{n}a_{k}^{1/2}\bigl(F_{k}(v)-v_{k}\bigr)e_{k}=\sum_{k=1}^{n}a_{k}^{1/2}F_{k}(v)\,e_{k}+w=\Psi_{n}(F(v),w).$$

Step 11 (Product measures). Let $F\times\mathrm{id}:\mathbb{R}^{n}\times X\to\mathbb{R}^{n}\times X$ be the map $(v,w)\mapsto(F(v),w)$. For the product metric, the distance between $(F(v),w)$ and $(F(v'),w')$ is the maximum of $\lVert F(v)-F(v')\rVert$ and $|w-w'|$, so $F\times\mathrm{id}$ is continuous because $F$ is: given $(v,w)$ and $\varepsilon>0$, take $\delta>0$ for $F$ at $v$ and $\varepsilon$; then every $(v',w')$ within distance $\min(\delta,\varepsilon)$ of $(v,w)$ has $\lVert v-v'\rVert<\delta$ and $|w-w'|<\varepsilon$, hence image within distance $\varepsilon$ of $(F(v),w)$. It is hence Borel, that is, measurable for $\mathcal{B}(\mathbb{R}^{n})\otimes\mathcal{B}(X)$ by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §tails. Its image measure $\kappa=(F\times\mathrm{id})_{\#}(\tilde{\gamma}\otimes\tau_{n})$, a measure on $\mathcal{B}(\mathbb{R}^{n})\otimes\mathcal{B}(X)$ by claim 1 of Image Measures, Measures with Densities, and Change of Variables, satisfies, for $A\in\mathcal{B}(\mathbb{R}^{n})$ and $C\in\mathcal{B}(X)$, $\kappa(A\times C)=(\tilde{\gamma}\otimes\tau_{n})(F^{-1}(A)\times C)=\tilde{\gamma}(F^{-1}(A))\,\tau_{n}(C)=(F_{\#}\tilde{\gamma})(A)\,\tau_{n}(C)$. The probability measures $F_{\#}\tilde{\gamma}$ and $\tau_{n}$ are finite, hence $\sigma$-finite, so the uniqueness assertion of Existence and Uniqueness of the Product Measure gives $\kappa=(F_{\#}\tilde{\gamma})\otimes\tau_{n}$.

Step 12 (The image of $\gamma_{c}$ is a Gaussian-tail extension). By Gaussian-Tail Extensions: the Gaussian Reference Measure, Marginals, Densities, Relative Entropy, Relative Score and the Noise Wasserstein Distance §reconstruction and The Gaussian-Tail Extension of a Probability Measure on the Rescaled Head §extension, $\gamma_{c}=(\Psi_{n})_{\#}(\tilde{\gamma}\otimes\tau_{n})$, with $\Psi_{n}$ Borel by The Gaussian-Tail Extension of a Probability Measure on the Rescaled Head §borel. Let $A\in\mathcal{B}(X)$, and let $S_{1}$ and $S_{2}$ be the preimages of $A$ under the Borel maps $T_{F}\circ\Psi_{n}$ and $\Psi_{n}\circ(F\times\mathrm{id})$; both belong to $\mathcal{B}(\mathbb{R}^{n})\otimes\mathcal{B}(X)$. By Step 10, a point of $S_{1}\setminus S_{2}$ or of $S_{2}\setminus S_{1}$ lies in $\mathbb{R}^{n}\times N$, whose $(\tilde{\gamma}\otimes\tau_{n})$-measure is $\tilde{\gamma}(\mathbb{R}^{n})\,\tau_{n}(N)=0$ by Step 9. By additivity and monotonicity of the measure, $(\tilde{\gamma}\otimes\tau_{n})(S_{1})=(\tilde{\gamma}\otimes\tau_{n})(S_{1}\cap S_{2})=(\tilde{\gamma}\otimes\tau_{n})(S_{2})$. Hence, by the definition of image measures (claim 1 of Image Measures, Measures with Densities, and Change of Variables), the formula for $\gamma_{c}$ above, $(T_{F}\circ\Psi_{n})^{-1}(A)=\Psi_{n}^{-1}(T_{F}^{-1}(A))$ and Step 11,

$$\bigl((T_{F})_{\#}\gamma_{c}\bigr)(A)=\gamma_{c}\bigl(T_{F}^{-1}(A)\bigr)=(\tilde{\gamma}\otimes\tau_{n})\bigl(\Psi_{n}^{-1}(T_{F}^{-1}(A))\bigr)=(\tilde{\gamma}\otimes\tau_{n})(S_{1})=(\tilde{\gamma}\otimes\tau_{n})(S_{2})=\kappa\bigl(\Psi_{n}^{-1}(A)\bigr)=\bigl((F_{\#}\tilde{\gamma})\otimes\tau_{n}\bigr)\bigl(\Psi_{n}^{-1}(A)\bigr),$$

and the right-hand side is $E_{n}(F_{\#}\tilde{\gamma})(A)$ by The Gaussian-Tail Extension of a Probability Measure on the Rescaled Head §extension. So $(T_{F})_{\#}\gamma_{c}=E_{n}(F_{\#}\tilde{\gamma})$.

Step 13 (Conclusion). By Step 8, $F_{\#}\tilde{\gamma}\in\mathcal{P}(\mathbb{R}^{n})$ has the density $G_{F}$ with respect to $\tilde{\gamma}$, so Gaussian-Tail Extensions: the Gaussian Reference Measure, Marginals, Densities, Relative Entropy, Relative Score and the Noise Wasserstein Distance §density shows that $G_{F}\circ r_{n}$ is a density of $E_{n}(F_{\#}\tilde{\gamma})=(T_{F})_{\#}\gamma_{c}$ with respect to $\gamma_{c}$. With Step 7 this proves claim 2.

Claim 3.

Step 14. Let $u\in\mathbb{R}^{n}$. Since $F^{-1}(F(u))=u$, the definition of $G_{F}$ gives $G_{F}(F(u))=\rho(u)/\bigl(\rho(F(u))\,\det DF(u)\bigr)$, a quotient of positive numbers. By The Natural Logarithm, $\log(st)=\log s+\log t$ for $s,t>0$; applied to $s'=G_{F}(F(u))$ and $t'=\rho(F(u))\det DF(u)$, whose product is $\rho(u)$, and then to the factors of $t'$, it gives

$$\log G_{F}(F(u))=\log\rho(u)-\log\rho(F(u))-\log\det DF(u).$$

By The Diagonal Gaussian Density on Euclidean Space and Its Notation §density, $\rho(y)=\exp\bigl(-\tfrac12|y|^{2}_{\tilde{c}^{(n)}}-Z_{\tilde{c}^{(n)}}\bigr)$ for $y\in\mathbb{R}^{n}$, and $\log(\exp(s))=s$ for real $s$ by The Natural Logarithm; so $\log\rho(y)=-\tfrac12|y|^{2}_{\tilde{c}^{(n)}}-Z_{\tilde{c}^{(n)}}$. Substituting $y=u$ and $y=F(u)$, the constants $Z_{\tilde{c}^{(n)}}$ cancel and

$$\log G_{F}(F(u))=\tfrac12\,|F(u)|^{2}_{\tilde{c}^{(n)}}-\tfrac12\,|u|^{2}_{\tilde{c}^{(n)}}-\log\det DF(u),$$

which is claim 3.