  This work proposes a new computational framework for learning a structured
generative model for real-world datasets. In particular, we propose to learn a
closed-loop transcription between a multi-class multi-dimensional data
distribution and a linear discriminative representation (LDR) in the feature
space that consists of multiple independent multi-dimensional linear subspaces.
In particular, we argue that the optimal encoding and decoding mappings sought
can be formulated as the equilibrium point of a two-player minimax game between
the encoder and decoder. A natural utility function for this game is the
so-called rate reduction, a simple information-theoretic measure for distances
between mixtures of subspace-like Gaussians in the feature space. Our
formulation draws inspiration from closed-loop error feedback from control
systems and avoids expensive evaluating and minimizing approximated distances
between arbitrary distributions in either the data space or the feature space.
To a large extent, this new formulation unifies the concepts and benefits of
Auto-Encoding and GAN and naturally extends them to the settings of learning a
both discriminative and generative representation for multi-class and
multi-dimensional real-world data. Our extensive experiments on many benchmark
imagery datasets demonstrate tremendous potential of this new closed-loop
formulation: under fair comparison, visual quality of the learned decoder and
classification performance of the encoder is competitive and often better than
existing methods based on GAN, VAE, or a combination of both. Unlike existing
generative models, the so learned features of the multiple classes are
structured: different classes are explicitly mapped onto corresponding
independent principal subspaces in the feature space. Source code can be found
at https://github.com/Delay-Xili/LDR.
