Improving the Threshold for Finding Rank-1 Matrices in a Subspace

We consider a basic computational task of finding $s$ planted rank-1 $m \times n$ matrices in a linear subspace $\mathcal{U} \subseteq \mathbb{R}^{m \times n}$ where $\dim(\mathcal{U}) = R \ge s$. The work of Johnston-Lovitz-Vijayaraghavan (FOCS 2023) gave a polynomial-time algorithm for this task and proved that it succeeds when ${R \le (1-o(1))mn/4}$, under minimal genericity assumptions on the input. Aiming to precisely characterize the performance of this algorithm, we improve the bound to ${R \le (1-o(1))mn/2}$ and also prove that the algorithm fails when ${R \ge (1+o(1))mn/\sqrt{2}}$. Numerical experiments indicate that the true breaking point is $R = (1+o(1))mn/\sqrt{2}$. Our work implies new algorithmic results for tensor decomposition, for instance, decomposing order-4 tensors with twice as many components as before.