Exact-density normalizing flows — Examples

Given: Countable Choice and a Lebesgue probability density $p_U$ on ${\mathbb{R}}^2$.

The Axiom of Countable Choice, written ${\mathrm{A}\mathrm{C}}_{\omega }$, asserts that every sequence of nonempty sets has a choice function. The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

A $C^1$ map is totally differentiable at every point with continuous derivative matrix; a global $C^1$ diffeomorphism has a $C^1$ inverse, and a flow transports its base measure by ${\mu }_X(B)={\mu }_U(T^{-1}(B))$. A normalizing flow and its base density

Under Countable Choice, a $C^1$ diffeomorphism transforms density $p_U$ by $p_U\circ T^{-1}\mathrm{\mid }\det D(T^{-1})\mathrm{\mid }$. A diffeomorphic flow has the inverse-Jacobian density

The Jacobian matrix records the total derivative in standard coordinates, and the Jacobian determinant is its determinant. The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, The Jacobian determinant of a square-dimensional $C^1$ map is the determinant of its Jacobian matrix

The determinant of a lower triangular matrix is the product of its diagonal entries. The determinant of a triangular matrix is the product of its diagonal entries

Given: Countable Choice, the interval probability measure $\nu$, the continuous embedding $i(t)=(t,0)$, and its image measure $\mu$.

The Axiom of Countable Choice, written ${\mathrm{A}\mathrm{C}}_{\omega }$, asserts that every sequence of nonempty sets has a choice function. The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

A nonnegative measurable density $f$ defines a measure by $A\mapsto {\int }_{A}fd\lambda$. The measure with density $f$ relative to $\mu$

A flow from a Lebesgue density through a global $C^1$ diffeomorphism has density $p_U\circ T^{-1}\mathrm{\mid }\det D(T^{-1})\mathrm{\mid }$. A diffeomorphic flow has the inverse-Jacobian density

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

Every closed box $[a,b]$ in ${\mathbb{R}}^n$ is Borel and Lebesgue measurable; its measure is ${\prod }_{i<n}(b_i-a_i)$, so $[0,1]$ has one-dimensional Lebesgue measure $1$. A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included

A rectangle with a degenerate side is Lebesgue null under Countable Choice. A box with a degenerate side is Lebesgue null, and so is every coordinate hyperplane in ${\mathbb{R}}^n$

A nonnegative measurable density integrates to zero over a measurable null set. A nonnegative integral over a null set vanishes

A normalizing flow is a finite composition of global $C^1$ diffeomorphisms between open equal-dimensional Euclidean domains; its output measure is obtained by taking preimages under the flow. A normalizing flow and its base density