Compression of Structured Data with Autoencoders: Provable Benefit of Nonlinearities and Depth
<|reference_start|>Compression of Structured Data with Autoencoders: Provable Benefit of Nonlinearities and Depth: 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.<|reference_end|>
@article{kögler2024compression,
    title={Compression of Structured Data with Autoencoders: Provable Benefit of
  Nonlinearities and Depth},
    author={Kevin K"ogler, Alexander Shevchenko, Hamed Hassani, Marco Mondelli},
    journal={arXiv preprint arXiv:2402.05013},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.05013},
    primaryClass={cs.LG cs.IT math.IT stat.ML}
}