Chart Auto-Encoders for Manifold Structured Data
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Auto-encoding and generative models have made tremendous successes in image and signal representation learning and generation. These models, however, generally employ the full Euclidean space or a bounded subset (such as [0,1]^l) as the latent space, whose flat geometry is often too simplistic to meaningfully reflect the topological structure of the data. This paper aims at exploring a universal geometric structure of the latent space for better data representation. Inspired by differential geometry, we propose a Chart Auto-Encoder (CAE), which captures the manifold structure of the data with multiple charts and transition functions among them. CAE translates the mathematical definition of manifold through parameterizing the entire data set as a collection of overlapping charts, creating local latent representations. These representations are an enhancement of the single-charted latent space commonly employed in auto-encoding models, as they reflect the intrinsic structure of the manifold. Therefore, CAE achieves a more accurate approximation of data and generates realistic synthetic examples. We demonstrate the efficacy of CAEs through a series experiments with synthetic and real-life data which illustrate that CAEs can out-preform variational auto-encoders on reconstruction tasks while using much smaller latent spaces.
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