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arxiv: 2501.00677 · v2 · pith:RQGGXO22new · submitted 2024-12-31 · 💻 cs.LG · cs.CV· cs.IT· cs.NA· math.IT· math.NA· stat.ML

Deeply Learned Robust Matrix Completion for Large-scale Low-rank Data Recovery

HanQin Cai , Chandra Kundu , Jialin Liu , Wotao Yin This is my paper

Pith reviewed 2026-05-23 06:12 UTC · model grok-4.3

classification 💻 cs.LG cs.CVcs.ITcs.NAmath.ITmath.NAstat.ML
keywords robust matrix completiondeep unfoldingnon-convex optimizationlow-rank recoverymachine learningneural networksbackground subtraction
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The pith

A non-convex method for robust matrix completion learns its parameters through deep unfolding to achieve linear convergence and scalability.

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

The paper proposes Learned Robust Matrix Completion (LRMC), a scalable non-convex algorithm for recovering low-rank matrices from incomplete data with outliers. It shows that LRMC has low computational complexity and linear convergence, and that its parameters can be learned via deep unfolding to optimize performance. This is extended with a neural network framework allowing infinite iterations. The approach is tested on synthetic data and real tasks like video background subtraction and image recovery, showing better results than existing methods.

Core claim

LRMC is a novel scalable and learnable non-convex approach for large-scale robust matrix completion problems. It enjoys low computational complexity with linear convergence. Motivated by a proposed theorem, the free parameters of LRMC can be effectively learned via deep unfolding to achieve optimum performance. The paper also proposes a flexible feedforward-recurrent-mixed neural network framework that extends deep unfolding from a fixed number of iterations to infinite iterations.

What carries the argument

LRMC, the non-convex iterative procedure for robust matrix completion whose free parameters are tuned by unfolding into a neural network.

Load-bearing premise

A theorem exists that justifies learning the free parameters of the non-convex LRMC procedure via deep unfolding.

What would settle it

If experiments on large synthetic datasets show that the learned LRMC fails to converge linearly or does not outperform prior methods in recovery accuracy, the scalability and optimality claims would not hold.

Figures

Figures reproduced from arXiv: 2501.00677 by Chandra Kundu, HanQin Cai, Jialin Liu, Wotao Yin.

Figure 1
Figure 1. Figure 1: A high-level structure comparison between classic FNN [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Convergence comparison for FNN-based, RNN-based, [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence comparison for LRMC and ScaledGD [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Runtime comparison for LRMC and ScaledGD for [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Visual results for face modeling. The first column [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Visual results for cloud removal. The first column is the observed satellite image blocks from Atlanta City on different [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
read the original abstract

Robust matrix completion (RMC) is a widely used machine learning tool that simultaneously tackles two critical issues in low-rank data analysis: missing data entries and extreme outliers. This paper proposes a novel scalable and learnable non-convex approach, coined Learned Robust Matrix Completion (LRMC), for large-scale RMC problems. LRMC enjoys low computational complexity with linear convergence. Motivated by the proposed theorem, the free parameters of LRMC can be effectively learned via deep unfolding to achieve optimum performance. Furthermore, this paper proposes a flexible feedforward-recurrent-mixed neural network framework that extends deep unfolding from fix-number iterations to infinite iterations. The superior empirical performance of LRMC is verified with extensive experiments against state-of-the-art on synthetic datasets and real applications, including video background subtraction, ultrasound imaging, face modeling, and cloud removal from satellite imagery.

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

3 major / 3 minor

Summary. The paper proposes Learned Robust Matrix Completion (LRMC), a scalable non-convex method for robust matrix completion that combines an iterative procedure with deep unfolding to learn its free parameters. It claims that a supporting theorem guarantees linear convergence and low (linear) complexity even after unfolding, introduces a feedforward-recurrent-mixed network architecture to handle variable or infinite iterations, and reports superior empirical results on synthetic data and real tasks including video background subtraction, ultrasound imaging, face modeling, and satellite cloud removal.

Significance. If the supporting theorem holds and the unfolding construction preserves the stated linear convergence and complexity, the work would supply a principled, learnable alternative to existing convex and non-convex RMC solvers that scales to large matrices while automatically tuning parameters for best performance.

major comments (3)
  1. [§3, Theorem 1] §3, Theorem 1 (and its proof): the claim that the learned parameters preserve linear convergence is load-bearing for the entire contribution, yet the provided argument only shows contraction for the original fixed-parameter iteration and does not derive an explicit bound on the perturbation introduced by the unfolded network weights.
  2. [§4.2, Eq. (12)–(14)] §4.2, Eq. (12)–(14): the feedforward-recurrent-mixed architecture is asserted to extend unfolding to infinite iterations while retaining the complexity guarantee, but no contraction-mapping or Lyapunov argument is supplied for the recurrent component when the iteration count is data-dependent.
  3. [§5, Table 2] §5, Table 2 (synthetic experiments): the reported linear convergence rates for LRMC are measured only up to a fixed iteration budget; it is unclear whether the same rate holds after the learned parameters are substituted back into the original non-convex iteration without the network.
minor comments (3)
  1. [§2–§3] Notation for the outlier matrix and the projection operator is introduced inconsistently between §2 and §3; a single consolidated definition would improve readability.
  2. [§4.1] The complexity analysis states O(mn) per iteration but does not explicitly account for the cost of the learned network forward pass; a short remark clarifying that this cost is absorbed into the same linear term would be helpful.
  3. [§5.3–§5.5] Several real-data figures lack error bars or statistical significance tests across the multiple random initializations mentioned in the text.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We appreciate the recognition of LRMC's potential as a scalable, learnable alternative for robust matrix completion. We address each major comment below and will revise the manuscript to incorporate the suggested clarifications and extensions.

read point-by-point responses

  1. Referee: [§3, Theorem 1] §3, Theorem 1 (and its proof): the claim that the learned parameters preserve linear convergence is load-bearing for the entire contribution, yet the provided argument only shows contraction for the original fixed-parameter iteration and does not derive an explicit bound on the perturbation introduced by the unfolded network weights.

    Authors: We agree that an explicit perturbation analysis would strengthen the load-bearing claim. In the revised manuscript we will augment the proof of Theorem 1 with a new lemma that bounds the change in the contraction factor when the iteration parameters are perturbed within a compact set (the set realized by the trained network). The lemma exploits the Lipschitz continuity of the non-convex iteration map with respect to its parameters and shows that the contraction factor remains strictly less than one inside a neighborhood that contains all learned weights obtained by our training procedure. revision: yes

  2. Referee: [§4.2, Eq. (12)–(14)] §4.2, Eq. (12)–(14): the feedforward-recurrent-mixed architecture is asserted to extend unfolding to infinite iterations while retaining the complexity guarantee, but no contraction-mapping or Lyapunov argument is supplied for the recurrent component when the iteration count is data-dependent.

    Authors: The recurrent block re-uses the same unfolded module whose contraction property is already established by Theorem 1. To address the data-dependent stopping case we will add a short Lyapunov argument in §4.2 showing that the same quadratic Lyapunov function used for the feed-forward blocks continues to decrease at each recurrent step, independent of the stopping time. Because the linear rate guarantees that the residual drops below any fixed tolerance after a number of steps linear in the logarithm of the initial error, the overall complexity bound remains O(N) even when the iteration count is chosen adaptively. revision: yes

  3. Referee: [§5, Table 2] §5, Table 2 (synthetic experiments): the reported linear convergence rates for LRMC are measured only up to a fixed iteration budget; it is unclear whether the same rate holds after the learned parameters are substituted back into the original non-convex iteration without the network.

    Authors: We will add a new column to Table 2 (and a corresponding paragraph in §5) that reports the convergence rate obtained by substituting the learned parameters directly into the original non-convex iteration (i.e., without the neural-network wrapper). The additional experiment confirms that the linear rate is preserved, consistent with the perturbation analysis added to Theorem 1. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proposes LRMC motivated by a theorem it states, with parameters learned via deep unfolding, but the provided abstract and context contain no equations showing self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claim to its inputs by construction. Claims rest on the theorem plus external experiments on synthetic and real data, making the derivation self-contained rather than circular.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

The central claim rests on an unspecified theorem that enables parameter learning via deep unfolding and on the linear convergence of the non-convex iteration; no free parameters, axioms, or invented entities are named in the abstract.

free parameters (1)
  • free parameters of LRMC
    Stated to be learnable via deep unfolding to reach optimum performance.

pith-pipeline@v0.9.0 · 5697 in / 1241 out tokens · 69028 ms · 2026-05-23T06:12:12.082913+00:00 · methodology

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