Flow-based generative model: Difference between revisions

Revision as of 10:32, 19 June 2025

Revision as of 10:41, 19 June 2025

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\mathbf{A1}=\lambda\mathbf1

</math>

where the condition that <math>\mathbf A</math> has <math>\mathbf1</math> as an [[eigenvector]] ensures invertibility by sidestepping the information loss due to the invariance: <math>\operatorname{softmax}(\mathbf x+\alpha\mathbf1)=\operatorname{softmax}(\mathbf x)</math>. Note in particular that <math>\mathbf A=\lambda\mathbf I_n</math> is the ''only'' allowed diagonal parametrization, in which case (for <math>\lambda>0</math>) we recover <math>f_\text{cal}(\mathbf p;\lambda^{-1},\mathbf c)</math>, while (for <math>n>2</math>) generalization ''is'' possible with non-diagonal matrices. The '''inverse''' is:

:<math>

\mathbf p = f_\text{gcal}^{-1}(\mathbf q;\mathbf A, \mathbf c)

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=\frac{\left|\operatorname{det}(\mathbf A)\right|}{|\lambda|}\prod_{i=1}^n\frac{q_i}{p_i}

</math>

If <math>f_\text{gcal}</math> is to be used as a calibration transform, a further constraint could be imposed, for example 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 theone possible generalization of <math>a>0</math> in the <math>f_\text{cal}</math> parameter.)

For <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:

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| arxiv = 1910.12656

| date = 28 October 2019

}}</ref> which generalizes <math>f_\text{gcal}</math>, by not placing any restriction on the matrix, <math>\mathbf A</math>, so that invertibility is not guaranteed. While Dirichlet calibration is trained as a discriminative model, <math>f_\text{gcal}</math> can also be trained as part of a generative calibration model.

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