Statistical mechanics of low-rank tensor decomposition
This provides a theoretical framework for tensor decomposition, addressing a fundamental gap in machine learning and data analysis, though it is incremental in extending AMP methods to tensors.
The paper tackles the lack of theoretical understanding of low-rank tensor decomposition algorithms by deriving Bayesian approximate message passing (AMP) algorithms for recovering noisy tensors and using dynamic mean field theory to characterize their performance, revealing phase transitions and showing that AMP significantly outperforms alternating least squares (ALS) in noise.
Often, large, high dimensional datasets collected across multiple modalities can be organized as a higher order tensor. Low-rank tensor decomposition then arises as a powerful and widely used tool to discover simple low dimensional structures underlying such data. However, we currently lack a theoretical understanding of the algorithmic behavior of low-rank tensor decompositions. We derive Bayesian approximate message passing (AMP) algorithms for recovering arbitrarily shaped low-rank tensors buried within noise, and we employ dynamic mean field theory to precisely characterize their performance. Our theory reveals the existence of phase transitions between easy, hard and impossible inference regimes, and displays an excellent match with simulations. Moreover, it reveals several qualitative surprises compared to the behavior of symmetric, cubic tensor decomposition. Finally, we compare our AMP algorithm to the most commonly used algorithm, alternating least squares (ALS), and demonstrate that AMP significantly outperforms ALS in the presence of noise.