Decomposing Overcomplete 3rd Order Tensors using Sum-of-Squares Algorithms
This addresses a bottleneck in tensor decomposition for applications in learning and complexity theory, representing a significant advance over prior methods limited to lower ranks.
The paper tackles the problem of decomposing overcomplete 3rd order tensors with super-linear rank, achieving a quasi-polynomial time algorithm for ranks up to n^{3/2}/polylog n and a polynomial time algorithm for certifying injective norms.
Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to $n^{\lfloor p/2 \rfloor}$ for a $p$-th order tensor in $\mathbb{R}^{n^p}$. Previously no efficient algorithm can decompose 3rd order tensors when the rank is super-linear in the dimension. Using ideas from sum-of-squares hierarchy, we give the first quasi-polynomial time algorithm that can decompose a random 3rd order tensor decomposition when the rank is as large as $n^{3/2}/\textrm{polylog} n$. We also give a polynomial time algorithm for certifying the injective norm of random low rank tensors. Our tensor decomposition algorithm exploits the relationship between injective norm and the tensor components. The proof relies on interesting tools for decoupling random variables to prove better matrix concentration bounds, which can be useful in other settings.