The Change-of-Variables Formula Using Matrix Volume

By Adi Ben-Israel. 1999

The matrix volume is a generalization, to rectangular matrices, of the absolute value of the determinant. In particular, the matrix volume can be used in change-of-variables formulae, instead of the determinant (if the Jacobi matrix of the underlying transformation is rectangular). This result is applicable to integration on surfaces, illustrated here by several examples.

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Citations in the corpus

Bayesian Learning of Kernel Embeddings (2016, methodology): “Applying the change of variable x 7→ φz(x) and using the generalization of the change-of-variables formula to non-square Jacobian matrices as described in (Ben-Israel, 1999), we obtain a distribution for x conditionally on μθ and θ:”
Variational Autoencoders and Nonlinear ICA: A Unifying Framework (2019, methodology): “The parameter β of the β -VAE was chosen in the set [1 , 45] .”
Estimating individual treatment effect: generalization bounds and algorithms (2016, background): “[11] A.”
Refining Deep Generative Models via Discriminator Gradient Flow (2020, unknown): “If Jg has full column rank and g is an injective function, then we have the following change-of-variables equation (Ben-Israel, 1999; Gemici et al., 2016), ∫ X f(x)dx = ∫ Z (f ◦ g)(z) √ detJ>g Jg(z)dz.”
Bounding and Minimizing Counterfactual Error (2016, background): “See [3] for the case of a mapping Φ between spaces of different dimensions.”
Normalizing Flows on Riemannian Manifolds (2016, unknown): “The relation between these volumes is given by d~x = √ det G d~u, where G = JTφ Jφ is the metric induced by the embedding φ on the tangent space TxM, [18, 19, 20].”