Flow-based generative model: Difference between revisions

Revision as of 15:59, 10 June 2025

Revision as of 16:13, 10 June 2025

Line 221:

\end{bmatrix}

</math>.

To define <math>U</math>, the differential volume element at the transformation input in, <math>\mathbf p\in\Delta^{n-1}</math>-space, we start with a rectangle in <math>\tilde\mathbf p</math>-space, having (signed) differential side-lengths, <math>dp_1, \dots, dp_{n-1}</math> from which we form the square, diagonal matrix <math>\mathbf D</math>, the columns of which span the rectangle. TheAt differentialvery volumesmall elementscale, at the transformation input is the embeddedwe parallelotopeget <math>U=f(\mathbf D)=/\mathbf{ED}\!/</math>, and its volume iswith:

[[File:Simplex measure pullback.svg|frame|right|For the 1-simplex (blue) embedded in <math>\R^2</math>, when we pull back [[Lebesgue measure]] from [[tangent space]] (coinciding with the simplex), via the embedding <math>p_1\mapsto(p_1,1-p_1)</math>, with Jacobian <math>\mathbf E=\begin{bmatrix}1&-1\end{bmatrix}'</math>, a scaling factor of <math>\sqrt{\mathbf E'\mathbf E}=\sqrt2</math> results.]]

:<math>\operatorname{volume}(U) = \sqrt{\left|\operatorname{det}(\mathbf{DE}'\mathbf{ED})\right|}