# How Many Samples is a Good Initial Point Worth in Low-rank Matrix Recovery?

arXiv: https://arxiv.org/abs/2006.06915 · HTML: https://mlsanity.com/paper/2006.06915/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jun 2020.

## Built on by (most cited first, 10 of 10)

- [On the computational and statistical complexity of over-parameterized matrix sensing](https://mlsanity.com/paper/2102.02756/) (2021)
- [Improved Global Guarantees for the Nonconvex Burer--Monteiro Factorization via Rank Overparameterization](https://mlsanity.com/paper/2207.01789/) (2022)
- [Fundamental limits for rank-one matrix estimation with groupwise heteroskedasticity](https://mlsanity.com/paper/2106.11950/) (2021)
- [Noisy Low-rank Matrix Optimization: Geometry of Local Minima and Convergence Rate](https://mlsanity.com/paper/2203.03899/) (2022)
- [Sharp Restricted Isometry Property Bounds for Low-rank Matrix Recovery Problems with Corrupted Measurements](https://mlsanity.com/paper/2105.08232/) (2021)
- [Sharp Global Guarantees for Nonconvex Low-rank Recovery in the Noisy Overparameterized Regime](https://mlsanity.com/paper/2104.10790/) (2021)
- [Semidefinite Programming versus Burer-Monteiro Factorization for Matrix Sensing](https://mlsanity.com/paper/2208.07469/) (2022)
- [Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective](https://mlsanity.com/paper/2209.15130/) (2022)
- [Algorithmic Regularization in Tensor Optimization: Towards a Lifted Approach in Matrix Sensing](https://mlsanity.com/paper/2310.15549/) (2023)
- [Absence of spurious solutions far from ground truth: A low-rank analysis with high-order losses](https://mlsanity.com/paper/2403.06056/) (2024)

## Abstract (arXiv)

Given a sufficiently large amount of labeled data, the non-convex low-rank matrix recovery problem contains no spurious local minima, so a local optimization algorithm is guaranteed to converge to a global minimum starting from any initial guess. However, the actual amount of data needed by this theoretical guarantee is very pessimistic, as it must prevent spurious local minima from existing anywhere, including at adversarial locations. In contrast, prior work based on good initial guesses have more realistic data requirements, because they allow spurious local minima to exist outside of a neighborhood of the solution. In this paper, we quantify the relationship between the quality of the initial guess and the corresponding reduction in data requirements. Using the restricted isometry constant as a surrogate for sample complexity, we compute a sharp threshold number of samples needed to prevent each specific point on the optimization landscape from becoming a spurious local minimum. Optimizing the threshold over regions of the landscape, we see that for initial points around the ground truth, a linear improvement in the quality of the initial guess amounts to a constant factor improvement in the sample complexity.