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Using Noise to Help Reach Global Minima: Turning Matrix Completion into Noisy Matrix Sensing

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
non-convex optimizationlow-rank matrix optimizationmatrix sensingimplicit biastensorover-parametrization
Abstract:

Matrix completion (MC) is an important yet challenging non-convex problem. In realistic settings, exact recovery of M∗M^* typically requires strong incoherence and an impractically large number of observed entries. Instead of enforcing exact recovery, we inject noise perturbation to construct a closely related surrogate that turns MC into a noisy matrix sensing problem with a more benign landscape. Although this surrogate permits a slight, controllable loss in accuracy, it can be solved effectively via over-parameterization (increasing model size), echoing modern machine learning practices where large models and stochasticity (e.g., SGD, dropout) make hard objectives tractable. Under the assumption that each entry of the matrix is observed independently and uniformly, we establish explicit accuracy–probability trade-offs as functions of the sampling rate pp and a user-chosen noise level. Empirically, our approach succeeds in low-observation regimes where classical exact-recovery pipelines are brittle. More broadly, our approach underscores a general paradigm in which noise perturbations are combined with large models to tackle modern ML tasks, and we use MC as a clean benchmark to formalize this perspective and unify noise, over-parameterization, and recoverability within a single framework.

Disclaimer
This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
If you have any questions, please contact: mingzhang23@m.fudan.edu.cn

Overview

Overall Novelty Assessment

The paper proposes a noise-perturbation approach to matrix completion, transforming the problem into a noisy matrix sensing task that can be solved via over-parameterization. According to the taxonomy, this work occupies a singleton leaf labeled 'Noise Perturbation for Matrix Completion,' with no sibling papers in that category. This placement suggests the paper explores a relatively sparse research direction within the broader field of low-rank matrix recovery, which encompasses seventeen papers across seven distinct branches. The taxonomy indicates that while over-parameterized recovery and robust recovery are well-populated areas, deliberate noise injection as a primary mechanism remains less explored.

The taxonomy reveals several neighboring branches that contextualize this work. The 'Over-Parameterized Matrix Recovery with Noise' branch contains nine papers across three sub-categories (optimization landscape analysis, algorithmic acceleration, validation approaches), focusing on rank over-specification in noisy settings but without deliberate noise injection. The 'Robust Recovery Under Measurement Corruption' branch addresses gross corruption rather than controlled perturbation. The 'Theoretical Foundations and General Frameworks' branch provides broader perspectives on over-parameterized low-rank estimation. The paper's approach bridges noise perturbation and over-parameterization, connecting to multiple branches while occupying a distinct methodological niche that emphasizes noise as a design choice rather than an obstacle.

Among twenty-two candidates examined, the contribution-level analysis shows varied novelty profiles. The perturbed matrix completion formulation (ε-MC) examined ten candidates with zero refutable matches, suggesting this specific noise-injection strategy is relatively unexplored in the limited search scope. The accuracy-probability trade-off guarantees similarly examined nine candidates without refutation. However, the lifted tensor framework extension shows two refutable candidates among three examined, indicating substantial prior work on tensor-based approaches to noisy matrix sensing. This pattern suggests the core noise-perturbation mechanism may be more novel than the tensor-based analysis techniques, though the limited search scope (twenty-two papers) means these findings reflect top semantic matches rather than exhaustive coverage.

Based on the limited literature search, the work appears to occupy a methodologically distinct position, particularly in its deliberate use of noise perturbation to enable over-parameterized recovery. The singleton taxonomy leaf and low refutation rates for the core formulation support this impression, though the tensor framework component shows more overlap with existing techniques. The analysis covers top semantic matches and does not claim exhaustive field coverage, leaving open the possibility of relevant work outside the examined candidates.

Taxonomy

Core-task Taxonomy Papers
17
3
Claimed Contributions
22
Contribution Candidate Papers Compared
2
Refutable Paper

Research Landscape Overview

Core task: Recovering low-rank matrices from incomplete observations via noise perturbation and over-parameterization. The field addresses the challenge of reconstructing structured matrices when only partial or corrupted measurements are available, leveraging both algorithmic and theoretical insights. The taxonomy reveals several complementary branches: some focus on over-parameterized formulations that exploit redundancy in factorized representations (e.g., Overparameterized Low Rank[11], Validation Overparameterized Recovery[3]), while others investigate how noise perturbation can regularize or guide recovery (e.g., Noise Regularizes Rank One[8], Geometric Noisy Recovery[4]). Additional branches examine robust recovery under measurement corruption (Sign RIP[10]), denoising under distribution shift (Denoising Distribution Shift[16]), and low-rank adaptation techniques such as tensor parameterizations (Tensor Train LoRA[12], SLDP LoRA[5]). Theoretical foundations (Blessing Nonconvexity[9], Sharp Nonconvex Recovery[1]) and general frameworks (Structured Models Review[15]) provide rigorous guarantees and unifying perspectives across these diverse settings. A particularly active line of work explores how noise and over-parameterization interact to enable efficient recovery. For instance, Noisy Matrix Sensing[0] sits within the noise perturbation branch, examining how additive noise influences matrix completion dynamics. This contrasts with approaches like Alternating Preconditioned Descent[2] or Efficient Overparameterized Sensing[17], which emphasize algorithmic design for over-parameterized regimes, and with works such as Double Descent Denoisers[6] that study generalization phenomena in denoising tasks. The original paper's emphasis on noise perturbation places it close to studies like Noise Regularizes Rank One[8] and Geometric Noisy Recovery[4], which similarly investigate how stochastic elements can regularize nonconvex optimization. Meanwhile, validation-based strategies (Validation Overparameterized Recovery[3]) and implicit bias analyses (Implicit Data Bias[14]) highlight alternative mechanisms for achieving low-rank solutions, underscoring ongoing questions about the interplay between model capacity, noise, and sample complexity.

Claimed Contributions

Perturbed matrix completion formulation (ε-MC)

The authors introduce a surrogate problem called ε-MC that converts matrix completion into noisy matrix sensing by perturbing the sensing operator. This formulation enables valid RSC/RSS constants and makes the problem amenable to over-parameterization techniques.

10 retrieved papers
Accuracy-probability trade-off guarantees

The authors prove that the global solution of ε-MC approximates the true matrix with explicit bounds that depend on sampling rate p and perturbation parameter ε, providing probabilistic guarantees even in low-observation regimes where exact recovery is impossible.

9 retrieved papers
Lifted tensor framework extension to noisy matrix sensing

The authors extend the lifted tensor framework from prior work to handle noisy matrix sensing problems, proving that over-parameterization can convert spurious solutions to strict saddles even under noise corruption.

3 retrieved papers
Can Refute

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Within the taxonomy built over the current TopK core-task papers, the original paper is assigned to a leaf with no direct siblings and no cousin branches under the same grandparent topic. In this retrieved landscape, it appears structurally isolated, which is one partial signal of novelty, but still constrained by search coverage and taxonomy granularity.

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Perturbed matrix completion formulation (ε-MC)

The authors introduce a surrogate problem called ε-MC that converts matrix completion into noisy matrix sensing by perturbing the sensing operator. This formulation enables valid RSC/RSS constants and makes the problem amenable to over-parameterization techniques.

Contribution

Accuracy-probability trade-off guarantees

The authors prove that the global solution of ε-MC approximates the true matrix with explicit bounds that depend on sampling rate p and perturbation parameter ε, providing probabilistic guarantees even in low-observation regimes where exact recovery is impossible.

Contribution

Lifted tensor framework extension to noisy matrix sensing

The authors extend the lifted tensor framework from prior work to handle noisy matrix sensing problems, proving that over-parameterization can convert spurious solutions to strict saddles even under noise corruption.

Table of Contents

Using Noise to Help Reach Global Minima: Turning Matrix Completion into Noisy Matrix Sensing | Novelty Validation