Compression of Structured Data with Autoencoders: Provable Benefit of Nonlinearities and Depth

• URL: http://arxiv.org/abs/2402.05013v1
• Date: Wed, 7 Feb 2024 16:32:29 GMT
• Title: Compression of Structured Data with Autoencoders: Provable Benefit of Nonlinearities and Depth
• Authors: Kevin K\"ogler, Alexander Shevchenko, Hamed Hassani, Marco Mondelli
• Abstract summary: We prove that gradient descent converges to a solution that completely disregards the sparse structure of the input. We show how to improve upon Gaussian performance for the compression of sparse data by adding a denoising function to a shallow architecture. We validate our findings on image datasets, such as CIFAR-10 and MNIST.
• Score: 83.15263499262824
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Autoencoders are a prominent model in many empirical branches of machine learning and lossy data compression. However, basic theoretical questions remain unanswered even in a shallow two-layer setting. In particular, to what degree does a shallow autoencoder capture the structure of the underlying data distribution? For the prototypical case of the 1-bit compression of sparse Gaussian data, we prove that gradient descent converges to a solution that completely disregards the sparse structure of the input. Namely, the performance of the algorithm is the same as if it was compressing a Gaussian source - with no sparsity. For general data distributions, we give evidence of a phase transition phenomenon in the shape of the gradient descent minimizer, as a function of the data sparsity: below the critical sparsity level, the minimizer is a rotation taken uniformly at random (just like in the compression of non-sparse data); above the critical sparsity, the minimizer is the identity (up to a permutation). Finally, by exploiting a connection with approximate message passing algorithms, we show how to improve upon Gaussian performance for the compression of sparse data: adding a denoising function to a shallow architecture already reduces the loss provably, and a suitable multi-layer decoder leads to a further improvement. We validate our findings on image datasets, such as CIFAR-10 and MNIST.

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