Flow-based generative model: Difference between revisions

Revision as of 13:45, 11 June 2025

Revision as of 13:53, 11 June 2025

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* The positive orthant of <math>\R^n</math> is projected onto the '''simplex''' as: <math>\pi(\mathbf z)=\bigl(\sum_{i=1}^n z_i\bigr)^{-1}\mathbf z</math>

* Non-zero vectors in <math>\R^n</math> are projected onto the '''unitsphere''' as: <math>\pi(\mathbf z)=\bigl(\sum_{i=1}^n z^2_i\bigr)^{-\frac12}\mathbf z</math>

For every <math>\mathbf x\in\mathcal M</math>, we require of <math>\pi</math> that its <math>n\text{-by-}n</math> Jacobian, <math>\boldsymbol{\Pi_x}</math> has rank <math>m</math> (the manifold dimension), in which case <math>\boldsymbol{\Pi_x}</math> is an [[projection (linear algebra)|idempotent projection matrix]] onto the local tangent space (''orthogonal'' for the unitsphere: <math>\mathbf I_n-\mathbf{xx}'</math>; ''oblique'' for the simplex: <math>\mathbf I_n-\boldsymbol{x1}'</math>). The colums of <math>\boldsymbol{\Pi_x}</math> span the <math>m</math>-dimensional tangent space at <math>\mathbf x</math>. We use the notation, <math>\mathbf{T_x}</math> for any <math>n\text{-by-}m</math> matrix with orthonormal columns (<math>\mathbf T_{\mathbf x}'\mathbf{T_x}=\mathbf I_m</math>) that span the local tangent space. Also note: <math>\boldsymbol{\Pi_x}\mathbf{T_x}=\mathbf{T_x}</math>. We can now choose our local coordinate embedding function, <math>e_\mathbf x:\R^m\to\R^n</math>:

:<math>

e_\mathbf x(\tilde x) = \pi(\mathbf x + \mathbf{T_x\tilde x})\,,