Remove condition-number dependence in sample complexity for nonconvex low-rank matrix recovery

Establish whether Scaled Gradient Descent for low-rank matrix recovery from Gaussian linear measurements can achieve the optimal sample complexity O((n1 + n2) r) independent of the condition number κ of the ground-truth matrix, thereby eliminating the κ^2 factor that arises in current analyses due to spectral initialization.

Background

The paper proves that Scaled Gradient Descent (ScaledGD) achieves linear convergence with sample complexity O((n1 + n2) r κ^2) under Gaussian measurements, improving iteration complexity over prior work but still retaining κ-dependence in the sampling requirement.

Convex methods for low-rank recovery can achieve sample complexity O((n1 + n2) r) without dependence on κ, revealing a gap between convex and currently analyzed nonconvex approaches.

The authors attribute the κ^2 factor in their analysis primarily to the spectral initialization step and note that alternate methods such as Stage-Alternating Minimization have removed κ-dependence in their sampling requirements, suggesting potential avenues to close this gap for ScaledGD.