Frequently linear interpolation is used with a Gaussian or uniform prior which has unit variance and zero mean where the size of the vector can be defined in an arbitrary way e.g. 100 to generate initial random vectors for generator model in Generative Adversarial Neural (GAN).
Let's say we have 1000 images for training and batch size is 64. Then each epoch, need to generate a number of random vectors using prior distribution corresponding to each image given small batch. But the problem I see is that since there is no mapping between random vector and corresponding image, the same image can be generated using multiple initial random vectors. In this paper, it suggests overcoming this problem by using different spherical interpolation up to some extent.
So what will happens if initially generate random vectors corresponding to the number of training images and when train the model uses the same random vector which is generated initially?
In GANs the random seed used as input does not actually correspond to any real input image. What GANs actually do is learn a transformation function from a known noise distribution (e.g. Gaussian) to a complex unknown distribution, which is representated by i.i.d. samples (e.g. your training set). What the discriminator in a GAN does is to calculate a divergence (e.g. Wasserstein divergence, KL-divergence, etc.) between the generated data (e.g. transformed gaussian) and the real data (your training data). This is done in a stochastic fashion and therefore no link is neccessary between the real and the fake data. If you want to learn more about this on a hands on example, I can recommend that you train to train a Wasserstein GAN to transform one 1D gaussian distribution into another one. There you can visualize the discriminator and the gradient of the discriminator and really see the dynamics of such a system.
Anyways, what your paper is trying to tell you is after you have trained your GAN and want to see how it has mapped the generated data from the known noise space to the unknown image space. For this reason interpolation schemes have been invented like the spherical one you are quoting. They also show that the GAN has learned to map some parts of the latent space to key characteristics in images, like smiles. But this has nothing to do with the training of GANs.