Content deleted Content added
Entropeneur (talk | contribs) →Simplex calibration transform: added reference to SGB distribution |
Entropeneur (talk | contribs) →Simplex calibration transform: added a subsection with a generalization |
||
Line 269:
* This result can also be obtained by factoring the density of the [[SGB distribution]],<ref name="sgb">{{cite web |last1=Graf |first1=Monique (2019)|title=The Simplicial Generalized Beta distribution - R-package SGB and applications |url=https://libra.unine.ch/server/api/core/bitstreams/dd593778-b1fd-4856-855b-7b21e005ee77/content |website=Libra |access-date=26 May 2025}}</ref> which is obtained by sending [[Dirichlet distribution|Dirichlet]] variates through <math>f_\text{cal}</math>.
While calibration transforms are most often trained as [[discriminative model|discriminative models]], the reinterpretation here as a probabilistic flow allows also the design of [[generative model|generative]] calibration models based on this transform.
====Generalized calibration transform====
The above calibration transform can be generalized to <math>f_\text{gcal}:\Delta^{n-1}\to\Delta^{n-1}</math>, with parameters <math>\mathbf c\in\R^n</math> and <math>\mathbf A</math> <math>n\text{-by-}n</math> invertible:
:<math>
\mathbf q = f_\text{gcal}(\mathbf p;\mathbf A,\mathbf c)
= \operatorname{softmax}(\mathbf A\log\mathbf p + \mathbf c)\,,\;\text{subject to}\;
\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 generalization ''is'' possible with non-diagonal matrices. The '''inverse''' is:
:<math>
\mathbf p = f_\text{gcal}^{-1}(\mathbf q;\mathbf A, \mathbf c)
= f_\text{gcal}(\mathbf q;\mathbf A^{-1}, -\mathbf A^{-1}\mathbf c)
</math>
The '''differential volume ratio''' is:
:<math>
R^\Delta_\text{gcal}(\mathbf p;\mathbf A,\mathbf c)
=\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 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 is the generalization of <math>a>0</math> in the <math>f_\text{cal}</math> parameter.
===Differential volume ratio for curved manifolds===
| |||