Flow-based generative model: Difference between revisions

Revision as of 07:34, 9 June 2025

Revision as of 07:47, 9 June 2025

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=== Simplex flow===

As a first example, we develop expressions for the differential volume ratio inof thea simplex caseflow, when<math>\mathbf q=f(\mathbf p)</math>, where <math>\mathbf p, \mathbf q\in\mathcal M=\Delta^{n-1}</math>. Define the '''embedding function''',:
:<math>e:\tilde\mathbf p=(p_1\dots,p_{n-1})\mapsto\mathbf p=(p_1\dots,p_{n-1},1-\sum_{i=1}^{n-1}p_i)

</math>

which maps an arbitrarily chosen, <math>m=(n-1)</math>-dimensional repesentation, <math>\tilde\mathbf p</math>, to the embedded manifold. The <math>n\text{-by-}m(n-1)</math> Jacobian is

<math>\mathbf E = \begin{bmatrix}

\mathbf{I}_{n-1} \\

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\end{bmatrix}

</math>.

To define <math>U</math> in <math>\mathbf p</math>-space, start with a rectangle in <math>\tilde\mathbf p</math>-space, having (signed) differential side-lengths, <math>dp_1, \dots, dp_{n-1}</math> andfrom usewhich these towe form the <math>(n-1)\text{-by-}(n-1)</math>square, diagonal matrix <math>\mathbf D</math>, the columns of which span the rectangle. The differential volume element at the transformation input is the embedded parallelotope <math>U=/\mathbf{ED}\!/</math> and its volume is:

[[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|}