__
TR24-123 | 22nd July 2024 06:37
__

#### Faster & Deterministic FPT Algorithm for Worst-Case Tensor Decomposition

**Abstract:**
We present a deterministic $2^{k^{\mathcal{O}(1)}} \text{poly}(n,d)$ time algorithm for decomposing $d$-dimensional, width-$n$ tensors of rank at most $k$ over $\mathbb{R}$ and $\mathbb{C}$. This improves upon the previous randomized algorithm of Peleg, Shpilka, and Volk (ITCS '24) that takes $2^{k^{k^{\mathcal{O}(k)}}} \text{poly}(n,d)$ time and the deterministic $n^{k^k}$ time algorithms of Bhargava, Saraf, and Volkovich (STOC '21).

Our work resolves an open question asked by Peleg, Shpilka, and Volk (ITCS '24) on whether a deterministic FPT algorithm exists for worst-case tensor decomposition. We also make substantial progress on the fundamental problem of how the tractability of tensor decomposition varies as the tensor rank increases. Our result implies that we can achieve deterministic polynomial-time decomposition as long as the rank of the tensor is at most $\left({\log n}\right)^{1/C}$, where $C$ is some fixed constant independent of $n$ and $d$. Further, we note that there cannot exist a polynomial-time algorithm for $k = \Omega(\log n)$, unless ETH fails. Our algorithm works for all fields; however, the time complexity worsens to $2^{k^{k^{\mathcal{O}(1)}}}$ and requires randomization for finite fields of large characteristics. Both conditions are provably necessary unless there are improvements in the state of the art for system solving over the corresponding fields.

Our approach achieves this by designing a proper learning (reconstruction) algorithm for set-multilinear depth-3 arithmetic circuits. On a technical note, we design a "partial'' clustering algorithm for set-multilinear depth-3 arithmetic circuits that lets us isolate a cluster from any set-multilinear depth-3 circuit while preserving the structure of the circuit.