  Generative models rely on the key idea that data can be represented in terms
of latent variables which are uncorrelated by definition. Lack of correlation
is important because it suggests that the latent space manifold is simpler to
understand and manipulate. Generative models are widely used in deep learning,
e.g., variational autoencoders (VAEs) and generative adversarial networks
(GANs). Here we propose a method to build a set of linearly independent vectors
in the latent space of a GANs, which we call quasi-eigenvectors. These
quasi-eigenvectors have two key properties: i) They span all the latent space,
ii) A set of these quasi-eigenvectors map to each of the labeled features
one-on-one. We show that in the case of the MNIST, while the number of
dimensions in latent space is large by construction, 98% of the data in real
space map to a sub-domain of latent space of dimensionality equal to the number
of labels. We then show how the quasi-eigenvalues can be used for Latent
Spectral Decomposition (LSD), which has applications in denoising images and
for performing matrix operations in latent space that map to feature
transformations in real space. We show how this method provides insight into
the latent space topology. The key point is that the set of quasi-eigenvectors
form a basis set in latent space and each direction corresponds to a feature in
real space.
