Mathematical Foundations of Generative AI Quiz 1: 13 July 2025 (May 2025 term)

Consider in GAN, a simple setting where $p_{\text{data}} = \delta(x-1)$ and $p_g = \delta(x+1)$, i.e., both are Dirac delta functions located at different points. What is the GAN value function $V(G)$ (adversarial loss)?

Dirac delta function is defined as

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

which satisfies

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

Where K is an integer.
What does this loop imply?