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Randomized Krylov-Schur eigensolver with deflation

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Pith's one-line read Randomized Krylov-Schur eigensolver computes a few eigenpairs of large nonsymmetric matrices 2–4x faster than deterministic restarted methods, with comparable accuracy.

desk verdict A genuinely useful randomized Krylov-Schur with sound algebra and real speedups, but the embedding assumption is unchecked and the headline deflation is never tested. read the letter →

arxiv 2508.05400 v1 pith:EJZE5MTF submitted 2025-08-07 math.NA cs.NA

Jean-Guillaume de Damas , Laura Grigori This is my paper · ORCID
classification math.NAcs.NA MSC 65F1565F2565F5015B52
keywords eigenvalueproblemrandomizedKrylov-SchursketchingdeflationnonsymmetricmatricesKrylovsubspacemethodsSchurformlinearalgebra
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces randomized Krylov-Schur (rKS), a restarted eigensolver for large sparse nonsymmetric matrices when only a small set of eigenpairs is wanted. The method works by sketching the Krylov basis into a low-dimensional space, performing orthonormalization and Rayleigh-Ritz there, and restarting with stable reordering of a Schur form instead of implicitly shifted QR. The authors prove that after each restart rKS produces the same Krylov spans as randomized Implicitly Restarted Arnoldi, so it inherits that method's polynomial-filtering convergence. They also prove a backward error bound for deflating converged eigenpairs, which lets the solver lock them and continue for the rest of the spectrum. The payoff is a simple implementation that runs 2 to 4 times faster in experiments than deterministic IRA and Krylov-Schur while finding the same eigenvalues.

What carries the argument

The engine of the method is the randomized $\varepsilon$-embedding—a matrix $\Omega\in\mathbb{R}^{d\times n}$ that approximately preserves norms on a $k$-dimensional subspace—used to define sketch-orthonormal bases, for which $(\Omega U_k)^T(\Omega U_k)=I_k$. Orthogonalization, Rayleigh-Ritz, residual estimation, and the deflation projector $P^\Omega_{U_q}=U_q(\Omega U_q)^\dagger\Omega$ all act on the sketch $S=\Omega U$ or in the small space $\mathbb{R}^d$, while the full-space vectors are updated by cheap triangular solves. This keeps per-iteration cost low, and the Schur form of the small Rayleigh quotient $T_k$ provides the stable reordering mechanism that replaces shifted QR for restart

What would settle it

Run rKS on a matrix with a clustered spectrum that forces many restarts, using the recommended sparse-sign sketch with d = 2m. If, before convergence, the condition number of the stored sketch S_k grows beyond a fixed factor, or the ratio between the true residual $\|A\tilde x - \tilde\lambda \tilde x\|/\|A\tilde x\|$ and the estimate $|b^T y|/|\tilde\lambda|$ exceeds $\sqrt{(1+\varepsilon)/(1-\varepsilon)} \approx 2.4$, then the ε-embedding assumption is violated for that run.

Watch

Extended reading notes

Core claim

The central object is the sketch-orthonormal Krylov-Schur decomposition $$AU_k = U_k T_k + u_{k+1} b_k^T,$$ where $(U_k,u_{k+1})$ is orthonormal after multiplication by a random $\varepsilon$-embedding $\Omega$ and $T_k$ is block upper-triangular (Definition 2.4). The paper shows that every Krylov decomposition is equivalent to a randomized Arnoldi factorization in the sense of spanning the same space (Theorem 2.2), and that one cycle of rKS—expand to length $k+p$, reorder the Schur form, truncate—produces the same subspace as one cycle of rIRA with the same shifts (Theorem 3.1). It then shows that a converged eigenpair can be locked by zeroing the corresponding entries of $b$; if those entr

Load-bearing premise

The argument assumes that the fixed random sketching matrix Ω remains an ε-embedding for the Krylov subspace at every restart; in the implementation this is only guaranteed with high probability, and if it fails the sketched basis becomes ill-conditioned, so the residual estimates and deflation error bounds no longer hold.

Editorial extensions

If this is right

  • rKS inherits the polynomial-filtering convergence of randomized IRA because each restart cycle yields the same Krylov subspace spans (Theorem 3.1).
  • A converged eigenpair can be locked when its $|b_i|$ entry is below $\eta$; the remaining computation runs on a perturbed operator whose restriction to the active Krylov space has error bounded by $\sqrt{q}\sqrt{(1+\varepsilon)/(1-\varepsilon)}\,\eta$ (Theorem 4.2).
  • In numerical experiments on synthetic and real sparse matrices, rKS is 2–4 times faster than deterministic IRA and Krylov-Schur, finds the same eigenvalues, and often needs the fewest restarts.
  • Because the sketch $S$ is stored and updated, the contraction and expansion steps operate in dimension $d\ll n$, so the method scales to matrices with millions of rows.
  • The method is less sensitive to the ratio between Krylov dimension and number of wanted eigenvalues, easing the choice of subspace size.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The authors leave implicit that the probabilistic guarantee on $\Omega$ suggests a practical safeguard: monitoring the condition number of the stored sketch $S_k$ during restarts and re-drawing or enlarging the embedding if it degrades.
  • Since the residual estimate (4.9) approximates $\|A\tilde x\|$ by $|\tilde\lambda|\,\|\tilde x\|$, it is weakest for smallest-modulus targets; the exact bound (4.10) avoids that approximation and would be the safer convergence test for smallest-modulus eigenproblems.
  • The convergence spikes seen in the residual plots hint that randomization can temporarily disturb the approximate eigenvalues; if confirmed more broadly, it may explain both the sometimes-fewer iterations and the need for a slightly relaxed deflation threshold.
  • The backward-error bound grows like $\sqrt{q}\,\eta$, so if the number of locked vectors $q$ is large, the threshold $\eta$ may need to shrink to keep the perturbed problem faithful.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper introduces a randomized Krylov-Schur eigensolver (rKS) for computing a small number of eigenpairs of large nonsymmetric matrices. It defines sketch-orthonormal Krylov(-Schur) decompositions, proves their equivalence to randomized Arnoldi factorizations, and shows that one rKS cycle is equivalent to the randomized Implicitly Restarted Arnoldi method, so that polynomial-filtering convergence transfers. It then derives residual estimators and a deflation procedure for converged eigenpairs, with backward-error bounds in Theorem 4.2, and reports numerical experiments on synthetic and SuiteSparse matrices showing 2--4x speedups over deterministic IRA/KS.

Significance. If the claims hold, the paper provides a simple, scalable randomized eigensolver with a plausible theoretical justification for convergence and deflation. The algebraic framework is clean and the sketch-orthonormal decomposition viewpoint is a natural extension of Stewart's Krylov-Schur framework. The main positive elements are the explicit factorization equivalences, the use of Schur reordering instead of shifted QR, and the experimental demonstration of consistent speedups on large sparse problems. However, the central guarantees are conditional on an epsilon-embedding hypothesis that the implemented algorithm does not verify, and the deflation backward-error theorem contains a real-diagonalization step that is not valid for real Schur forms with 2-by-2 blocks.

major comments (2)
  1. [Section 4.4, Theorem 4.2] The proof of Theorem 4.2 assumes an eigendecomposition T_qq W = W D with a real diagonal D (equivalently, real eigenvectors forming a basis). But T_qq is a block upper-triangular real Schur form with 1-by-1 and 2-by-2 diagonal blocks. A 2-by-2 block such as [[0,-1],[1,0]] has no real diagonalization, and even a triangular block with repeated real eigenvalues can be defective. Consequently the claimed existence of real unit vectors Z and a real diagonal D satisfying the bound in eq. (4.33) is not established for the general stated setting. This is load-bearing for the subspace-deflation contribution. The theorem should be restated either for Schur vectors (block-diagonal D), over the complex field, or under an explicit diagonalizability assumption.
  2. [Definition 1.1 / Remark 1, Section 5, and Theorem 4.2] All theoretical results are conditional on Omega being an epsilon-embedding for each Krylov subspace encountered, but the implementation uses one fixed sparse-sign sketch with d=2m, zeta=8 and no verification that the embedding property holds for the polynomial-filtered subspaces produced across restarts. The paper gives no oblivious-subspace-embedding guarantee for this parameter regime and no failure probability for the union of subspaces seen by the algorithm. This is not merely a formal gap: if at some restart the current subspace contains a direction nearly annihilated by Omega, then in Algorithm 2 line 8 beta_{j+1} can be tiny while ||u~_{j+1}|| is O(1), so u_{j+1} has huge norm. The sketched basis S remains orthonormal by construction, but U becomes ill-conditioned, and the residual estimator in eq. (4.9) and the deflation bound in eq. (4.32) can fail silently because they rely on
minor comments (6)
  1. [Algorithm 2, line 9] The expression 'U_{j+1} = (U_j u_{k+p+1} = ...)' contains an index typo; it should be 'u_{j+1}'.
  2. [Figure 7 caption] The caption says 'The execution times for each method are reported in Fig. 7'; this should refer to Fig. 6.
  3. [Section 4.2] The text refers to 'line 4 of Algorithm 2' for deflated sketch-orthogonalization. In Algorithm 2, line 4 is w_j = A u_j, while the projection is performed in line 6. Please update the reference.
  4. [Section 4.1, eq. (4.9)] Equation (4.9) is presented as a bound but uses the approximation ||Omega(A x~)|| ~= |lambda|. This is an estimate rather than a rigorous inequality unless ||A x~|| ~= |lambda| ||x~|| is justified. It would be clearer to state it as a heuristic and to base convergence certificates on the exact bound in eq. (4.10) or on true residuals.
  5. [Theorem 4.2, eq. (4.34)] The notation ||E|K||_{F,2} is nonstandard and ambiguous. Define the norm of the restriction of E to K explicitly.
  6. [Section 1.1] Corollary 1.0.1 should be numbered as Corollary 1.1; the current numbering is unusual.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the randomized Krylov-Schur derivations are self-contained; self-citations to prior rIRA work are not load-bearing.

full rationale

The paper's central results are derived from explicit algebraic relations and the stated ε-embedding hypothesis, not from fitted constants or circular definitions. Theorem 2.2 is proved by similarity and translation operations that preserve the span, followed by Householder reductions to Hessenberg form; the equivalence is established by construction rather than assumed. Theorem 3.1 proves span equality after a restart cycle using the fact that the two small matrices are similar via the orthonormal change of basis between sketch-orthonormal bases; the argument is self-contained once the rIRA and rKS update rules are fixed. The residual estimator in (4.9) is obtained from the algebraic identity A x̃ − λ̃ x̃ = u_{k+1} b_k^T y together with the ε-embedding inequalities; the '≈' step is explicitly flagged, and Remark 2 gives an exact alternative that does not rely on that approximation. Deflation bounds in Theorem 4.2 are direct consequences of the assumption |b_i| ≤ η under the stated ε-embedding for the Krylov subspace; no hidden parameter is fitted to the experimental outputs. The sketching parameters d = 2m and ζ = 8 are hand-chosen, not tuned to reproduce benchmark results, and η is a user-supplied tolerance. Citations to [dDG24] provide a uniqueness theorem and polynomial-filtering background, but the equivalence proof in this paper does not reduce to those citations: the existence direction is proved in detail, and the algorithm and its error bounds stand even without the uniqueness clause. The skeptic's concern that the fixed sparse-sign sketch may not remain an ε-embedding for every subspace encountered during restarts is a correctness and robustness gap, not a circularity of the derivation chain, since the paper explicitly states the embedding hypothesis and does not define its targets in terms of that hypothesis.

Assumptions & free parameters 4 free parameters · 8 assumptions · 0 invented entities

The central contribution is algorithmic; the only new objects are definitions (sketch-orthonormal Krylov(-Schur) decompositions) whose properties are proved. The main unstated burden is the epsilon-embedding assumption that applies to subspaces that evolve with restarts. Free parameters are hyperparameters of the algorithm and experiments, not fitted to data.

free parameters (4)
  • Sketch dimension ratio d/m = 2
    Chosen by hand in Section 5; d=2m gives eps about 1/sqrt(2) for sparse-sign embeddings.
  • Sparse embedding nonzeros per row zeta = 8
    Chosen by hand in Section 5 as the number of normalized random signs per row of the embedding.
  • Convergence tolerance eta = 1e-10
    Used as the threshold for residual estimates and deflation in all experiments (Sections 5.1, 5.2).
  • Krylov dimension m relative to k = 2k (or varying from 1.5k to 4k)
    Algorithmic choice; m=2k used in Section 5.1, varied in Section 5.2. Not fitted to data.
assumptions (8)
  • domain assumption Omega is an epsilon-embedding for the current Krylov subspace at each restart
    Invoked throughout Section 2 (Remark 1) and Section 4 to justify sketch-orthonormality, residual bounds, and deflation error bounds. The implementation uses an oblivious embedding, so this is assumed rather than verified per subspace.
  • standard math Oblivious subspace embeddings exist with d = O(k/eps^2)
    Standard JL-style result cited from [JL+84, MT20]; used to justify that sparse-sign matrices with d=2m approximately preserve norms.
  • standard math Randomized Implicit Q theorem from [dDG24]
    Stated in the authors' prior paper and used in the proof of Theorem 2.2 to assert essential uniqueness of the randomized Arnoldi factorization. Treating it as background literature, not derived here.
  • standard math Stewart's deterministic Krylov decomposition equivalence lemmas
    Used in Theorem 2.2 and Theorem 3.1, specifically the reduction of a Krylov decomposition to Arnoldi form via unitary transformations (cite [Ste02b, Theorem 2.2]).
  • domain assumption Krylov subspace Kk(A,v1) has full dimension k
    Section 2.1 says 'we assume that the dimension of any Krylov subspace Kk is exactly k and not less'. Needed to ensure bases are full rank.
  • domain assumption Retained Ritz values are distinct from discarded ones
    Used in Theorem 3.1 to ensure the Schur vectors associated to retained Ritz values span a unique subspace, so that reordering in rIRA and rKS yields the same span.
  • domain assumption No zero eigenvalues among locked vectors
    Section 4.2, before eq. (4.12): 'Assuming no zero eigenvalues among {lambda_1,...,lambda_k}, R is non-singular'. Needed for the existence of the similar triangular factor after sketch-orthonormalizing Schur vectors.
  • domain assumption Residual estimation uses ||Ax|| approx |lambda| ||x||
    Section 4.1 introduces the approximation to derive the cheap residual estimator (4.9), which is then used as the convergence test in all experiments.

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Pith. "Pith review of Randomized Krylov-Schur eigensolver with deflation." pith.science (2026). https://pith.science/paper/EJZE5MTF

@misc{pith2026250805400,
  author       = {Pith},
  title        = {Pith review of: Randomized Krylov-Schur eigensolver with deflation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EJZE5MTF}},
  note         = {Machine review of arXiv:2508.05400}
}
read the original abstract

This work introduces a novel algorithm to solve large-scale eigenvalue problems and seek a small set of eigenpairs. The method, called randomized Krylov-Schur (rKS), has a simple implementation and benefits from fast and efficient operations in low-dimensional spaces, such as sketch-orthogonalization processes and stable reordering of Schur factorizations. It also includes a practical deflation technique for converged eigenpairs, enabling the computation of the eigenspace associated with a given part of the spectrum. Numerical experiments are provided to demonstrate the scalability and accuracy of the method.

Figures

Figures reproduced from arXiv: 2508.05400 by the authors.

Figure 1
Figure 1. Different decompositions of A based on Krylov subspaces 3 The randomized Krylov-Schur method In this section, we introduce the randomized Krylov-Schur eigensolver, presented in Algorithm 1. We show that an iteration of rKS is equivalent to an iteration of rIRA, and describe a structure-preserving sketched re-orthogonalization process. 3.1 Deriving the algorithm The main idea behind the Krylov-Schur (KS) algorithm is… view at source ↗
Figure 2
Figure 2. Visual representation of the randomized Krylov-Schur from Algorithm 1 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Execution time of IRA, KS and rKS on synthetic matrices from Table 1. For each data point except [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Corresponding number of iterations from experiments of Fig. 3 for each method to converge and find [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: Maximum residual errors (left) alongside real parts of the eigenvalues computed (right), for the eight [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: Execution time to find k = 50 eigenpairs of different matrices summarized in [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: Number of iterations to find k = 50 eigenpairs of different matrices summarized in [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Randomized biorthogonalization through a two-sided Gram-Schmidt process

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    Randomized two-sided Gram-Schmidt builds bases Q and P with (ΩQ)^T ΩP = I, at about half the cost and often with better conditioning than the deterministic version.

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