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arxiv: 1907.11705 · v1 · pith:EKHQBLWGnew · submitted 2019-07-27 · 💻 cs.DS · cs.IT· math.IT· math.OC

Low-Rank Matrix Completion: A Contemporary Survey

Luong Trung Nguyen , Junhan Kim , Byonghyo Shim This is my paper

Pith reviewed 2026-05-24 14:45 UTC · model grok-4.3

classification 💻 cs.DS cs.ITmath.ITmath.OC
keywords low-rank matrix completionsurveymatrix recoveryalgorithm classificationrecovery performancecomputational complexityCNN-based methodsgraph structure
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The pith

Low-rank matrix completion techniques are organized into two main categories to clarify their potentials and limitations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This survey arranges existing low-rank matrix completion methods into two primary categories and explains each in detail. It then addresses practical issues including the intrinsic matrix properties needed for recovery and the use of special structures in algorithm design. The paper also reviews CNN-based methods that use graph structure and compares recovery performance along with computational complexity across methods. A reader would care because the organization turns scattered theoretical results into an accessible guide for choosing and applying these techniques.

Core claim

The paper claims that classifying state-of-the-art low-rank matrix completion techniques into two main categories, then detailing each category along with required matrix properties and structure exploitation, supplies a structured view of the field's potentials and limitations. It incorporates CNN-based algorithms exploiting graph structure and supplies performance and complexity comparisons to aid understanding.

What carries the argument

The two-category classification of LRMC techniques, which structures scattered results to highlight essential trade-offs and design considerations.

If this is right

  • Practitioners gain clearer criteria for selecting a technique suited to a given matrix recovery task.
  • Design choices can explicitly incorporate special matrix structures to improve recovery.
  • CNN-based approaches become part of the standard toolkit when graph structure is present.
  • Performance and complexity tables allow direct comparison before implementation.
  • Assessment of intrinsic matrix properties becomes a standard first step before applying any method.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same two-category lens might be tested on hybrid methods that blend elements from both categories.
  • Fields using matrix recovery such as recommender systems could adopt the property checklist directly.
  • Future surveys could measure how many new techniques continue to fit the original two categories over time.
  • The structure might reveal gaps where no current method handles certain matrix properties well.

Load-bearing premise

That dividing the techniques into exactly two categories produces a structured and accessible view that captures the essential potentials and limitations.

What would settle it

A new low-rank matrix completion technique that fits neither category cleanly or requires a distinct third category would show the classification does not fully organize the field.

Figures

Figures reproduced from arXiv: 1907.11705 by Byonghyo Shim, Junhan Kim, Luong Trung Nguyen.

Figure 1
Figure 1. Figure 1: Recommendation system application of LRMC. Entries [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Localization via LRMC [4]. The Euclidean distance ma [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Image reconstruction via LRMC. Recovered images ach [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Outline of LRMC algorithms. 4) Image compression and restoration: When there is dirt or scribble in a two-dimensional image (see [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: LRMC with colored entries being observed. The dotted [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: An illustration of the worst case of LRMC. [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Coherence of matrices in (52) and (53): (a) maximum an [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Phase transition of LRMC algorithms. entries and the rank is large. Overall, NNM-based algorithms perform better than FNM-based algorithms. In particular, the NNM technique using SDPT3 solver outperforms the rest because the convex optimization technique always finds a global optimum while other techniques often converge to local optimum. In order to investigate the computational efficiency of LRMC algorit… view at source ↗
Figure 9
Figure 9. Figure 9: Running times of LRMC algorithms in noiseless scenar [PITH_FULL_IMAGE:figures/full_fig_p039_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: MSE performance of LRMC algorithms in noisy scenari [PITH_FULL_IMAGE:figures/full_fig_p040_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: MSE performance of LRMC algorithms in noisy scenari [PITH_FULL_IMAGE:figures/full_fig_p040_11.png] view at source ↗
read the original abstract

As a paradigm to recover unknown entries of a matrix from partial observations, low-rank matrix completion (LRMC) has generated a great deal of interest. Over the years, there have been lots of works on this topic but it might not be easy to grasp the essential knowledge from these studies. This is mainly because many of these works are highly theoretical or a proposal of new LRMC technique. In this paper, we give a contemporary survey on LRMC. In order to provide better view, insight, and understanding of potentials and limitations of LRMC, we present early scattered results in a structured and accessible way. Specifically, we classify the state-of-the-art LRMC techniques into two main categories and then explain each category in detail. We next discuss issues to be considered when one considers using LRMC techniques. These include intrinsic properties required for the matrix recovery and how to exploit a special structure in LRMC design. We also discuss the convolutional neural network (CNN) based LRMC algorithms exploiting the graph structure of a low-rank matrix. Further, we present the recovery performance and the computational complexity of the state-of-the-art LRMC techniques. Our hope is that this survey article will serve as a useful guide for practitioners and non-experts to catch the gist of LRMC.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper is a survey on low-rank matrix completion (LRMC) that organizes existing work by classifying state-of-the-art techniques into two main categories, explains each category in detail, discusses practical issues such as intrinsic properties required for recovery and exploiting special structures in LRMC design, covers CNN-based algorithms that exploit graph structure, and compares recovery performance and computational complexity of the techniques.

Significance. If the two-category classification is balanced and the performance/complexity comparisons are accurate and sourced, the survey could provide a useful structured guide for practitioners and non-experts; the organizational framing of scattered results is the primary contribution, with no new theorems or algorithms claimed.

minor comments (3)
  1. [Abstract] Abstract: the two main categories are referenced but not named, reducing immediate accessibility for readers seeking an overview.
  2. The manuscript should explicitly state the criteria used to partition techniques into the two categories (e.g., optimization vs. non-convex, or convex vs. non-convex) to allow readers to assess whether the taxonomy is exhaustive.
  3. Tables or sections presenting recovery performance and complexity should include explicit citations or references to the original sources for each reported number to support reproducibility of the comparison.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our survey and the recommendation for minor revision. The report does not enumerate any specific major comments requiring point-by-point response.

Circularity Check

0 steps flagged

Survey paper with no derivations or predictions

full rationale

This is a survey paper whose contribution is organizational: it classifies existing LRMC techniques into two main categories, explains each, and discusses practical considerations such as intrinsic properties and special structures. The abstract and full description confirm it presents early scattered results in a structured way without claiming new theorems, algorithms, empirical results, or any equations. No derivation chain exists, so no steps reduce by construction, self-citation, or fitted inputs. The classification itself is an expository choice, not a load-bearing prediction or self-referential claim.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a survey paper with no central mathematical claim, derivation, or empirical result; therefore no free parameters, axioms, or invented entities are introduced.

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