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arxiv: 1907.11868 · v1 · pith:GHGQ5E4Znew · submitted 2019-07-27 · 💻 cs.IT · math.IT

A Greedy Algorithm for Matrix Recovery with Subspace Prior Information

Hamideh.S Fazael Ardakani , Sajad Daei , Farzan Haddadi This is my paper

Pith reviewed 2026-05-24 14:52 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords matrix recoverygreedy algorithmsubspace priorrank-restricted isometry propertylow-rank matrixnuclear norm minimizationlinear measurementscollaborative filtering
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The pith

A greedy algorithm recovers low-rank matrices under milder rank-restricted isometry conditions when accurate subspace priors are supplied.

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

The paper introduces a greedy recovery procedure that folds known information about the column and row spaces directly into each iteration. A reader would care because many matrix completion tasks already supply partial knowledge of these spaces, and the method is claimed to succeed with fewer measurements and lower run time than convex alternatives. Performance is bounded using the rank-restricted isometry property, with the proof showing that the prior relaxes the required constant. The same analysis also indicates higher empirical success rates than nuclear-norm minimization. The approach therefore converts readily available side information into a concrete reduction in sampling burden.

Core claim

We propose an efficient greedy algorithm that exploits prior information in the recovery procedure. The performance of the proposed algorithm is measured in terms of the rank restricted isometry property (R-RIP). Our proposed algorithm with prior subspace information converges under a more milder condition on the R-RIP in compared with the case that we do not use prior information. Additionally, our algorithm performs much better than nuclear norm minimization in terms of both computational complexity and success rate.

What carries the argument

A greedy iterative procedure that at each step projects onto the known column and row subspaces while selecting the next rank-one update.

If this is right

  • Exact recovery becomes possible from fewer linear measurements once the correct subspaces are known.
  • The per-iteration cost remains linear in the ambient dimensions rather than requiring semidefinite programming.
  • The success probability increases relative to nuclear-norm methods on the same measurement budget.
  • The convergence guarantee is stated directly in terms of an R-RIP constant that depends on the dimension of the supplied subspaces.

Where Pith is reading between the lines

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

  • If an approximate rather than exact subspace estimate is available, the same iteration might still converge, albeit under a somewhat stronger R-RIP requirement.
  • The technique could be combined with subspace estimation steps that learn the prior on the fly from auxiliary data.
  • In noisy settings the same prior-constrained greedy steps would likely reduce error compared with unconstrained pursuit, although that regime is not analyzed here.

Load-bearing premise

The subspace prior information about the column and row spaces is accurate and correctly supplied to the algorithm before recovery begins.

What would settle it

A low-rank matrix together with a measurement operator that satisfies the paper's milder R-RIP bound when the stated prior is used, yet for which the greedy iterates fail to recover the matrix exactly.

Figures

Figures reproduced from arXiv: 1907.11868 by Farzan Haddadi, Hamideh.S Fazael Ardakani, Sajad Daei.

Figure 1
Figure 1. Figure 1: Comparison of ADMiRA with RMSPI and GRMSPI in terms of success rate for (a) noiseless matrix recovery (b) noisy matrix recovery (c) matrix [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of ADMiRA with RMSPI and GRMSPI in terms of success rate for (a) noiseless matrix recovery (b) noisy matrix recovery (c) matrix [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of ADMiRA with RMSPI and GRMSPI in terms of success rate for (a) noiseless matrix recovery (b) noisy matrix recovery (c) matrix [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of ADMiRA with RMSPI and GRMSPI in terms of success rate for (a) noiseless matrix recovery (b) noisy matrix recovery (c) matrix [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

Matrix recovery is the problem of recovering a low-rank matrix from a few linear measurements. Recently, this problem has gained a lot of attention as it is employed in many applications such as Netflix prize problem, seismic data interpolation and collaborative filtering. In these applications, one might access to additional prior information about the column and row spaces of the matrix. These extra information can potentially enhance the matrix recovery performance. In this paper, we propose an efficient greedy algorithm that exploits prior information in the recovery procedure. The performance of the proposed algorithm is measured in terms of the rank restricted isometry property (R-RIP). Our proposed algorithm with prior subspace information converges under a more milder condition on the R-RIP in compared with the case that we do not use prior information. Additionally, our algorithm performs much better than nuclear norm minimization in terms of both computational complexity and success rate.

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 / 0 minor

Summary. The manuscript proposes a greedy algorithm for low-rank matrix recovery from linear measurements that incorporates prior information about the column and row subspaces of the unknown matrix. It claims that this algorithm converges under milder conditions on the rank-restricted isometry property (R-RIP) than the no-prior case and outperforms nuclear norm minimization in both computational complexity and empirical success rate.

Significance. If the central claims are substantiated, the work would provide a concrete mechanism to relax R-RIP requirements via subspace projections, which could improve both theoretical recovery guarantees and practical efficiency for applications such as collaborative filtering and seismic interpolation.

major comments (3)
  1. [Abstract] Abstract: the claim that the algorithm 'converges under a more milder condition on the R-RIP' is asserted without any derivation, proof sketch, or reference to a specific theorem; this is load-bearing for the central contribution.
  2. [Abstract] Abstract: the milder R-RIP guarantee is obtained by replacing standard greedy selection with projections onto the supplied subspaces, yet the text supplies no perturbation analysis or error term when the supplied bases differ from the true column/row spaces; under mismatch the effective RIP constant reverts to (or exceeds) the no-prior bound.
  3. [Abstract] Abstract: the claim of 'much better' performance than nuclear norm minimization in success rate is stated without any numerical evidence, simulation setup, or comparison table, leaving the empirical superiority unverified.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and detailed comments on our manuscript. We address each major comment point by point below, indicating where revisions to the abstract will be made for improved clarity and support of the claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the algorithm 'converges under a more milder condition on the R-RIP' is asserted without any derivation, proof sketch, or reference to a specific theorem; this is load-bearing for the central contribution.

    Authors: The milder R-RIP condition is derived in the main theoretical result (Section 4), where the convergence analysis explicitly shows how the subspace projections relax the required bound on the R-RIP constant relative to the no-prior case. We will revise the abstract to include a direct reference to this theorem. revision: yes

  2. Referee: [Abstract] Abstract: the milder R-RIP guarantee is obtained by replacing standard greedy selection with projections onto the supplied subspaces, yet the text supplies no perturbation analysis or error term when the supplied bases differ from the true column/row spaces; under mismatch the effective RIP constant reverts to (or exceeds) the no-prior bound.

    Authors: The presented analysis and guarantees assume exact knowledge of the column and row subspaces. We agree that the manuscript does not contain a perturbation analysis for approximate or mismatched priors; under mismatch the bound would indeed approach the standard (no-prior) case. This extension lies outside the scope of the current work, which focuses on the exact-prior setting. revision: no

  3. Referee: [Abstract] Abstract: the claim of 'much better' performance than nuclear norm minimization in success rate is stated without any numerical evidence, simulation setup, or comparison table, leaving the empirical superiority unverified.

    Authors: The empirical comparisons, including success-rate curves versus nuclear-norm minimization together with the associated simulation parameters, appear in Section 5. We will revise the abstract to reference these results and figures so that the performance claim is explicitly tied to the reported experiments. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation relies on independent R-RIP analysis of the proposed greedy steps

full rationale

The paper introduces a greedy algorithm that incorporates supplied column/row subspace bases to modify the selection and projection steps, then analyzes its convergence via the rank-restricted isometry property (R-RIP). The milder R-RIP bound is derived from the modified iteration that projects onto the given subspaces; this is a direct consequence of the algorithm definition and standard RIP arguments rather than a redefinition or fit. No self-citations are invoked as load-bearing uniqueness theorems, no parameters are fitted to data and relabeled as predictions, and no ansatz is smuggled via prior work. The analysis treats the supplied subspaces as exact (as stated in the weakest assumption), but this is an explicit modeling choice, not a circular reduction of the result to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not introduce or rely on any explicit free parameters, axioms, or invented entities beyond the standard rank-restricted isometry property already used in the compressive-sensing literature.

pith-pipeline@v0.9.0 · 5682 in / 1013 out tokens · 23806 ms · 2026-05-24T14:52:26.901894+00:00 · methodology

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