Fast Tucker Rank Reduction for Non-Negative Tensors Using Mean-Field Approximation
This work addresses a domain-specific problem in tensor decomposition for applications like data compression or analysis, but it is incremental as it builds on existing non-negative Tucker rank reduction techniques.
The paper tackles the problem of low-rank approximation for non-negative tensors by introducing a fast algorithm based on mean-field approximation, which avoids gradient methods and achieves competitive or better approximation results with improved speed compared to existing methods.
We present an efficient low-rank approximation algorithm for non-negative tensors. The algorithm is derived from our two findings: First, we show that rank-1 approximation for tensors can be viewed as a mean-field approximation by treating each tensor as a probability distribution. Second, we theoretically provide a sufficient condition for distribution parameters to reduce Tucker ranks of tensors; interestingly, this sufficient condition can be achieved by iterative application of the mean-field approximation. Since the mean-field approximation is always given as a closed formula, our findings lead to a fast low-rank approximation algorithm without using a gradient method. We empirically demonstrate that our algorithm is faster than the existing non-negative Tucker rank reduction methods and achieves competitive or better approximation of given tensors.