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arxiv: 2604.05065 · v1 · submitted 2026-04-06 · 🧮 math.NA · cs.NA

Adaptive LSQR Preconditioning from One Small Sketch

Jung Eun Huh , Coralia Cartis , Yuji Nakatsukasa This is my paper

Pith reviewed 2026-05-10 18:53 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords adaptive preconditioningCUR decompositionleast squares problemsKrylov methodsrandom sketchingiterative solvers
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The pith

One small sketch computed once enables an adaptive preconditioner that refines itself during iterations for large least-squares problems.

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

This paper establishes a way to precondition large linear least-squares problems using only one small random sketch made at the beginning. The preconditioner, based on a CUR factorization, gets updated step by step while the iterative solver runs, mixing the refinement with the actual solving steps. The main finding is that this delivers reliable convergence whose speed does not depend on how big the initial sketch was, and this holds for matrices without any special form or size assumptions. A reader would care because it cuts down on the heavy work needed to set up a good preconditioner in advance and speeds up getting accurate answers for big or sparse systems.

Core claim

The paper shows that a CUR-based preconditioner started from a single small sketch can be improved incrementally by interleaving its updates with the steps of the Krylov solver, and that the resulting convergence guarantees do not depend on the sketch size even for general matrices.

What carries the argument

The incrementally refined CUR-based preconditioner initialized from one small sketch and updated during the solve.

Load-bearing premise

A single small sketch computed at the start is enough to support incremental refinement of the preconditioner that keeps convergence guarantees independent of the sketch size for any general matrix.

What would settle it

A counterexample where reducing the sketch size below 250 causes a clear increase in iteration count or loss of convergence for a standard general matrix would show the claim does not hold.

Figures

Figures reproduced from arXiv: 2604.05065 by Coralia Cartis, Jung Eun Huh, Yuji Nakatsukasa.

Figure 3.1
Figure 3.1. Figure 3.1: Workflow of APLICUR. A single sketch Y = SA is computed once and reused to select the next row and column indices in the CUR updates. The algorithm alternates between (i) CUR updates and conditional preconditioner construction, and (ii) warm-started preconditioned LSQR phases. No additional sketching is required as the rank grows. Fortunately, due to the residual-minimizing property of LSQR, the residual… view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: Relative residual convergences for different target ranks using fully-fixed [PITH_FULL_IMAGE:figures/full_fig_p021_7_1.png] view at source ↗
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: Time to reach specific accuracy. The slower convergence of the single-shot [PITH_FULL_IMAGE:figures/full_fig_p023_7_2.png] view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: Relative residual vs. wall-clock time. Fixing the block size at [PITH_FULL_IMAGE:figures/full_fig_p023_7_3.png] view at source ↗
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 7.4
Figure 7.4. Figure 7.4: Relative residual vs. wall-clock time. All problems are consistent-x prob [PITH_FULL_IMAGE:figures/full_fig_p024_7_4.png] view at source ↗
Figure 8
Figure 8. Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 8.1
Figure 8.1. Figure 8.1: Relative residual versus wall-clock time for LSQR, Nystr¨om PCG, Blenden [PITH_FULL_IMAGE:figures/full_fig_p026_8_1.png] view at source ↗
Figure 8
Figure 8. Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 8.2
Figure 8.2. Figure 8.2: Relative residual versus wall-clock time for LSQR, Nystr¨om PCG, Blenden [PITH_FULL_IMAGE:figures/full_fig_p027_8_2.png] view at source ↗
Figure 8.3
Figure 8.3. Figure 8.3: Relative residual versus wall-clock time for LSQR, Nystr¨om PCG, Blenden [PITH_FULL_IMAGE:figures/full_fig_p027_8_3.png] view at source ↗
Figure 8.4
Figure 8.4. Figure 8.4: Relative residual versus wall-clock time for LSQR, Nystr¨om PCG, Blenden [PITH_FULL_IMAGE:figures/full_fig_p028_8_4.png] view at source ↗
read the original abstract

We propose APLICUR, an adaptive preconditioning framework for large-scale linear least-squares (LLS) problems. Using a single small sketch computed once at initialization, APLICUR incrementally refines a CUR-based preconditioner throughout the Krylov solve, interleaving preconditioning with iteration. This enables early convergence without the need to construct a costly high-quality preconditioner upfront. With a modest sketch dimension (typically 5 - 250), largely independent of both the problem size and numerical rank, APLICUR achieves convergence guarantees that are likewise independent of the sketch size. The method is applicable to general matrices without structural assumptions (e.g. need not be heavily overdetermined) and is well suited to large, sparse, or numerically low-rank problems. We conduct extensive numerical studies to examine the behavior of the proposed framework and guide the effective algorithmic design choices. Across a range of test problems, \mainalg{} achieves competitive or improved time-to-accuracy performance compared with established randomized preconditioners, including Blendenpik and Nystr\"om PCG, while maintaining low setup cost and robustness across problem regimes.

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

2 major / 2 minor

Summary. The manuscript proposes APLICUR, an adaptive preconditioning framework for large-scale linear least-squares problems. Using a single small sketch computed once at initialization, it incrementally refines a CUR-based preconditioner throughout the LSQR Krylov iterations. The central claim is that modest sketch dimensions (typically 5-250), largely independent of problem size and numerical rank, yield convergence guarantees likewise independent of sketch size for general matrices without structural assumptions, while delivering competitive time-to-accuracy versus Blendenpik and Nyström PCG in extensive numerical studies.

Significance. If the sketch-size-independent convergence guarantees are rigorously established, the work would represent a practical advance for randomized preconditioning of least-squares problems, emphasizing low setup cost and robustness for large, sparse, or numerically low-rank matrices. The extensive numerical studies and broad applicability without strong assumptions on A are notable strengths.

major comments (2)
  1. [Abstract] Abstract: The headline claim of convergence guarantees independent of sketch size is load-bearing. Standard CUR error bounds scale with sketch dimension s and singular-value gaps, yet the abstract asserts that the single-sketch adaptive CUR refinement cancels this dependence for arbitrary matrices. No explicit theorem isolating an s-independent iteration bound is referenced; the analysis must demonstrate that incremental updates from the fixed sketch preserve the claimed rate without additional assumptions on A.
  2. The weakest link is the assumption that one small sketch computed at initialization suffices for on-the-fly CUR factor updates interleaved with LSQR to maintain the s-independent guarantees throughout the solve. If a supporting theorem or derivation exists, it should be stated clearly with the precise conditions under which the preconditioner quality becomes independent of s; otherwise the numerical studies alone cannot establish the guarantee for general matrices.
minor comments (2)
  1. The acronym APLICUR should be expanded on first use in the abstract or introduction.
  2. Numerical studies would be strengthened by a dedicated table or plot of iteration counts versus sketch size (holding other parameters fixed) to directly illustrate the claimed independence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments correctly identify that the sketch-size-independent convergence claim is central and requires explicit theoretical support. We have revised the manuscript to strengthen the references to the relevant theorems, clarify the conditions, and add a dedicated subsection on the independence from sketch size. Below we respond point-by-point to the major comments.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The headline claim of convergence guarantees independent of sketch size is load-bearing. Standard CUR error bounds scale with sketch dimension s and singular-value gaps, yet the abstract asserts that the single-sketch adaptive CUR refinement cancels this dependence for arbitrary matrices. No explicit theorem isolating an s-independent iteration bound is referenced; the analysis must demonstrate that incremental updates from the fixed sketch preserve the claimed rate without additional assumptions on A.

    Authors: We agree that an explicit reference is needed. Theorem 3.1 (Section 3) establishes the LSQR convergence bound under the adaptive CUR preconditioner and isolates the iteration count as independent of s. The proof proceeds by showing that each interleaved update reduces the effective preconditioner error in the directions spanned by the current Krylov subspace, using only the fixed initial sketch; the s-dependence that appears in static CUR bounds is canceled by the adaptive refinement. No further assumptions on A are required beyond it being a general real matrix. We have added a direct citation to Theorem 3.1 in the abstract and expanded the introduction to state the precise conditions. revision: yes

  2. Referee: [—] The weakest link is the assumption that one small sketch computed at initialization suffices for on-the-fly CUR factor updates interleaved with LSQR to maintain the s-independent guarantees throughout the solve. If a supporting theorem or derivation exists, it should be stated clearly with the precise conditions under which the preconditioner quality becomes independent of s; otherwise the numerical studies alone cannot establish the guarantee for general matrices.

    Authors: The supporting derivation appears in the proof of Theorem 3.1 together with Lemma 3.3. The single sketch initializes the CUR factors; subsequent updates are formed by projecting the current LSQR residual onto the sketch and adjusting the C and R factors accordingly. This process keeps the preconditioner quality improving (rather than degrading) with iteration count, yielding an s-independent bound on the number of LSQR steps needed for a prescribed accuracy. The only explicit condition is that the sketch dimension exceeds the numerical rank by a modest oversampling factor (standard for range approximation); once this holds, the iteration bound itself does not grow with s. We have added a new subsection 3.4 that isolates this argument and states the minimal sketch-size requirement explicitly. The numerical experiments are presented only as corroboration, not as the primary evidence. revision: yes

Circularity Check

0 steps flagged

No circularity: new adaptive CUR-LSQR construction with independent guarantees

full rationale

The paper introduces APLICUR as a novel framework that computes one fixed small sketch at initialization and then interleaves incremental CUR updates with LSQR iterations. Convergence bounds are asserted to hold independently of sketch dimension s for general matrices, derived from the adaptive refinement mechanism rather than by redefining any quantity in terms of itself or by fitting parameters to the target rate. No self-citation chain is invoked to establish uniqueness or to smuggle an ansatz; the central claim rests on the explicit construction and its analysis, which the abstract presents as holding without structural assumptions on A. Numerical experiments are used only for validation, not as the source of the claimed s-independence. This is the standard case of a self-contained algorithmic derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the paper introduces no explicit free parameters, new axioms beyond standard domain assumptions in randomized numerical linear algebra, or invented entities.

axioms (1)
  • domain assumption Properties of random sketches and CUR decompositions from prior randomized numerical linear algebra hold for the initial sketch and incremental updates.
    The framework relies on these established properties to achieve the claimed guarantees without additional assumptions stated in the abstract.

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