# Absence of spurious solutions far from ground truth: A low-rank analysis with high-order losses

arXiv: https://arxiv.org/abs/2403.06056 · HTML: https://mlsanity.com/paper/2403.06056/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 47 references that are in the map.
- First posted on arXiv 10 Mar 2024.

## Builds on (most cited first, 19 of 19)

- [No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis](https://mlsanity.com/paper/1704.00708/) (2017)
- [Global Optimality of Local Search for Low Rank Matrix Recovery](https://mlsanity.com/paper/1605.07221/) (2016)
- [Small random initialization is akin to spectral learning: Optimization and generalization guarantees for overparameterized low-rank matrix reconstruction](https://mlsanity.com/paper/2106.15013/) (2021)
- [Understanding Incremental Learning of Gradient Descent: A Fine-grained Analysis of Matrix Sensing](https://mlsanity.com/paper/2301.11500/) (2023)
- [Global Optimality in Low-rank Matrix Optimization](https://mlsanity.com/paper/1702.07945/) (2017)
- [Nonconvex phase synchronization](https://mlsanity.com/paper/1601.06114/) (2016)
- [Sharp Restricted Isometry Bounds for the Inexistence of Spurious Local Minima in Nonconvex Matrix Recovery](https://mlsanity.com/paper/1901.01631/) (2019)
- [A Unified Computational and Statistical Framework for Nonconvex Low-Rank Matrix Estimation](https://mlsanity.com/paper/1610.05275/) (2016)
- [Global Convergence of Sub-gradient Method for Robust Matrix Recovery: Small Initialization, Noisy Measurements, and Over-parameterization](https://mlsanity.com/paper/2202.08788/) (2022)
- [Exact Guarantees on the Absence of Spurious Local Minima for Non-negative Rank-1 Robust Principal Component Analysis](https://mlsanity.com/paper/1812.11466/) (2018)
- [Improved Global Guarantees for the Nonconvex Burer--Monteiro Factorization via Rank Overparameterization](https://mlsanity.com/paper/2207.01789/) (2022)
- [General Low-rank Matrix Optimization: Geometric Analysis and Sharper Bounds](https://mlsanity.com/paper/2104.10356/) (2021)
- [How Many Samples is a Good Initial Point Worth in Low-rank Matrix Recovery?](https://mlsanity.com/paper/2006.06915/) (2020)
- [Local and Global Linear Convergence of General Low-rank Matrix Recovery Problems](https://mlsanity.com/paper/2104.13348/) (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)
- [Conic Relaxations for Power System State Estimation with Line Measurements](https://mlsanity.com/paper/1704.00133/) (2017)
- [Semidefinite Programming versus Burer-Monteiro Factorization for Matrix Sensing](https://mlsanity.com/paper/2208.07469/) (2022)
- [Over-parametrization via Lifting for Low-rank Matrix Sensing: Conversion of Spurious Solutions to Strict Saddle Points](https://mlsanity.com/paper/2302.07828/) (2023)

## Built on by (most cited first, 1 of 1)

- [Escaping Local Minima Provably in Non-convex Matrix Sensing: A Deterministic Framework via Simulated Lifting](https://mlsanity.com/paper/2602.05887/) (2026)

## Abstract (arXiv)

Matrix sensing problems exhibit pervasive non-convexity, plaguing optimization with a proliferation of suboptimal spurious solutions. Avoiding convergence to these critical points poses a major challenge. This work provides new theoretical insights that help demystify the intricacies of the non-convex landscape. In this work, we prove that under certain conditions, critical points sufficiently distant from the ground truth matrix exhibit favorable geometry by being strict saddle points rather than troublesome local minima. Moreover, we introduce the notion of higher-order losses for the matrix sensing problem and show that the incorporation of such losses into the objective function amplifies the negative curvature around those distant critical points. This implies that increasing the complexity of the objective function via high-order losses accelerates the escape from such critical points and acts as a desirable alternative to increasing the complexity of the optimization problem via over-parametrization. By elucidating key characteristics of the non-convex optimization landscape, this work makes progress towards a comprehensive framework for tackling broader machine learning objectives plagued by non-convexity.