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Generative models rely on the key idea that data can be represented in termsof latent variables which are uncorrelated by definition. Lack of correlationis important because it suggests that the latent space manifold is simpler tounderstand 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 vectorsin the latent space of a GANs, which we call quasi-eigenvectors. Thesequasi-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 featuresone-on-one. We show that in the case of the MNIST, while the number ofdimensions in latent space is large by construction, 98% of the data in realspace map to a sub-domain of latent space of dimensionality equal to the numberof labels. We then show how the quasi-eigenvalues can be used for LatentSpectral Decomposition (LSD), which has applications in denoising images andfor performing matrix operations in latent space that map to featuretransformations in real space. We show how this method provides insight intothe latent space topology. The key point is that the set of quasi-eigenvectorsform a basis set in latent space and each direction corresponds to a feature inreal space.

Using Deep LSD to build operators in GANs latent space with meaning in real space