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

Closed-Loop Data Transcription to an LDR via Minimaxing Rate Reduction