Mathematical Foundations of Generative AI Quiz 1: 26 October 2025 (September 2025 term)

Suppose you replace the log-loss in the original GAN with a general f-divergence minimization framework. Which of the following cannot be directly represented as an f-divergence?

Imagine in GAN, a simple setting where $p_{\text{data}} = \delta(x-2)$ and $p_g = \delta(x+4)$, i.e., both are Dirac delta functions located at different points. What would be the GAN value function $V(G)$?

Dirac delta function is defined as

$\delta(x) = \begin{cases} 0, & x \neq 0 \\ \infty, & x = 0 \end{cases}$

which satisfies

$\int_{-\infty}^{\infty} \delta(x)\, dx = 1.$