Strictly subquadratic-rank tensor recovery

Determine whether some iterative method, using a number of components r<k=o(r^2), can recover a smoothed rank-r tensor decomposition from random initialization in polynomial time with high probability and relative Frobenius error at most ε.

Background

The paper studies a smoothed tensor decomposition model in which a rank-r tensor is formed from deterministic factor columns perturbed by independent Gaussian noise. The source problem asks for the smallest overparameterization k under which alternating least squares, gradient descent, or another iterative method converges from random initialization to a globally optimal or arbitrarily accurate decomposition.

The paper presents a conditional recovery result at k=Θ(r{5/3}(log r){5/2}), but only under additional bounded-scale, weak-interference, weight-balance, and smoothing/dimension assumptions. It explicitly identifies extending the result to arbitrary source-admissible base triples as unresolved.

References

Does some iterative method, with $r<k=o(r2)$ components, return in $\operatorname{poly}(n,r,\log(1/\epsilon))$ time a decomposition satisfying

\left|T-\sum_{i=1}k x_i\otimes y_i\otimes z_i\right|_F \le \epsilon|T|_F

with high probability over the smoothed instance?

— VALG: An Agentic System for ML Theory Research  (2608.13060 - Zhang et al., 13 Aug 2026) in Section 4, Subsection “How much overparametrization is needed for ALS in tensor decomposition?”, Subproblem 1: Upper Bound