Near-Optimal Nonconvex Matrix Completion

• URL: http://arxiv.org/abs/2609.17048v1
• Date: Tue, 15 Sep 2026 11:57:44 GMT
• Title: Near-Optimal Nonconvex Matrix Completion
• Abstract summary: We study the problem of recovering a low-rank spectral error from a subset of its entries.<n>The method converges condition number methods for the problem.<n>The two methods achieve exact recovery with high probability from $O(nlog nlog(n)$(nrlogr)$(nrlogr)$ observations, respectively.
• Score: 6.287662267319984
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study nonconvex methods for matrix completion, the problem of recovering a low-rank matrix from a subset of its entries. Convex methods achieve sample complexity linear in the matrix dimension and the rank, up to logarithmic factors, whereas global guarantees for commonly used nonconvex methods require a higher polynomial dependence on the rank. We close this gap by analyzing Riemannian gradient descent (RGD) and Riemannian Gauss--Newton (RGN) methods. For an $n\times n$ matrix of rank $r$ with incoherence parameter $μ$ and condition number $κ$, the two methods achieve exact recovery with high probability from $O(μnr\log n\log(nκ))$ and $O(μnr\log n\log(2μrκ))$ observations, respectively. The methods use a multiscale residual initialization, while the analysis simultaneously controls the spectral error and incoherence. The resulting RGD iterates converge linearly, whereas RGN eventually converges Q-quadratically.

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