Optimization Landscape of Tucker Decomposition
This addresses the nonconvex optimization challenge in tensor analysis for data science and machine learning applications, offering theoretical guarantees for efficient algorithms.
The paper characterizes the optimization landscape of Tucker decomposition, showing that for tensors with an exact decomposition, all local minima are globally optimal, and provides a polynomial-time local search algorithm to find approximate solutions.
Tucker decomposition is a popular technique for many data analysis and machine learning applications. Finding a Tucker decomposition is a nonconvex optimization problem. As the scale of the problems increases, local search algorithms such as stochastic gradient descent have become popular in practice. In this paper, we characterize the optimization landscape of the Tucker decomposition problem. In particular, we show that if the tensor has an exact Tucker decomposition, for a standard nonconvex objective of Tucker decomposition, all local minima are also globally optimal. We also give a local search algorithm that can find an approximate local (and global) optimal solution in polynomial time.