2024/02/07 by Kevin Kögler, Kögler, Kevin, Alexander P. Shevchenko +5
Computer Science · #Advanced Data Compression Techniques #FOS: Computer and information sciences #Information Theory (cs.IT) #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Neural Networks and Applications
paper · pdf · doi:10.48550/arxiv.2402.05013
openalex publication_date 2024/02/07 · openalex created_date 2024/02/09 · openalex updated_date 2026/07/28
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.