Flow-based generative model: Difference between revisions

Revision as of 18:20, 15 June 2025

Revision as of 13:16, 18 June 2025

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If <math>f_\text{gcal}</math> is to be used as a calibration transform, a further constraint could be imposed that <math>\mathbf A</math> be [[positive definite matrix|positive definite]], so that <math>(\mathbf{Ax})'\mathbf x>0</math>, which avoids direction reversals. (This condition is the generalization of <math>a>0</math> in the <math>f_\text{cal}</math> parameter.)

IfFor <math>n=2</math>, <math>a>0</math> and <math>\mathbf A</math> positive definite, then <math>f_\text{cal}</math> and <math>f_\text{gcal}</math> are equivalent in the sense that in both cases, <math>\log\frac{p_1}{p_2}\mapsto\log\frac{q_1}{q_2}</math> is a straight line, the (positive) slope and offset of which are functions of the transform parameters. For <math>n>2</math> <math>f_\text{gcal}</math> ''does'' generalize <math>f_\text{cal}</math>.

It must however be noted that chaining mutliple <math>f_\text{gcal}</math> flow transformations does ''not'' give a further generalization, because:

:<math>

f_\text{gcal}(\cdot\,;\mathbf A_1,\mathbf c_1) \circ

f_\text{gcal}(\cdot\,;\mathbf A_2,\mathbf c_2)

= f_\text{gcal}(\cdot\,;\mathbf A_1\mathbf A_2,\mathbf c_1+\mathbf A_1\mathbf c_2)

</math>

In fact, the set of <math>f_\text{gcal}</math> transformations form a [[group mathematics|group]] under function composition. The set of <math>f_\text{cal}</math> transformations form a subgroup.

===Differential volume ratio for curved manifolds===