Block subspace expansions for eigenvalues and eigenvectors approximation

Demetrio Stojanoff; Francisco Arrieta Zuccalli; Pedro Massey

REVIEW 2 major objections 7 minor 28 references

Block subspace expansions for eigenvalues and eigenvectors approximation

T0 review · 2 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that the optimal block expansion of a search subspace is obtained by applying the pseudoinverse of the residual to the target subspace, and that the resulting iterates converge geometrically in the Hermitian case.

desk verdict Necessary non-degeneracy condition missing from the abstract; otherwise the block optimal expansion and convergence bounds hold up. read the letter →

arxiv 2411.14578 v1 pith:723ZARWL submitted 2024-11-21 math.NA cs.NAmath.FA

Francisco Arrieta Zuccalli , Pedro Massey , Demetrio Stojanoff This is my paper · ORCID

classification math.NAcs.NAmath.FA MSC 65F1515A1865F10

keywords optimalsubspaceexpansioneigenvectorapproximationblockKrylovprojectionmethodscomputableprincipalanglesHermitianeigenvalueproblem

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 solves, in a precise sense, the block version of a basic question in numerical linear algebra: given a search subspace $\mathcal V$ and a $d$-dimensional invariant subspace $\mathcal X$ of $A$, which vectors should one apply $A$ to so that $\mathcal V + A(\mathcal W)$ gets closest to $\mathcal X$? The answer is a subspace $\mathcal W_0\subset\mathcal V$ of dimension at most $d$, characterized by applying the Moore–Penrose pseudoinverse of the residual $R=(I-P_{\mathcal V})AV$ to $\mathcal X$. The optimal expanded space is $\mathcal V\oplus (I-P_{\mathcal V})P_{\mathcal S}(\mathcal X)$, where $\mathcal S=\mathcal V+A(\mathcal V)$, so the whole optimization reduces to projecting the target onto $\mathcal S$ and removing the part already in $\mathcal V$. When $A$ is Hermitian and $\mathcal X$ is the span of the $d$ largest eigenvectors with $\lambda_d>\lambda_{d+1}$, iterating this construction gives subspaces $\mathcal V_t$ whose principal angles to $\mathcal X$ decay at the explicit rate $((\lambda_{d+1}-\lambda_n)/((\lambda_d-\lambda_n)+(\lambda_d-\lambda_{d+1})))^t$, provided $\mathcal X\cap\mathcal V^\perp=\{0\}$. The paper also supplies computable Rayleigh–Ritz and refined Rayleigh–Ritz versions, with numerical comparisons against block Krylov methods.

What carries the argument

The load-bearing object is the residual matrix $R=(I-P_{\mathcal V})AV$, whose range is the part of $A(\mathcal V)$ lying outside $\mathcal V$, together with the subspace $\mathcal N=(I-P_{\mathcal V})P_{\mathcal S}(\mathcal X)$ obtained by projecting the target $\mathcal X$ onto $\mathcal S=\mathcal V+A(\mathcal V)$ and then removing the $\mathcal V$-component. The identity $\mathcal S=\mathcal V\oplus (I-P_{\mathcal V})A(\mathcal V)$ turns every expanded space $\mathcal V+A(\mathcal W)$ into an orthogonal sum, so the optimal choice becomes $\mathcal W=\mathcal V R^{\dagger}(\mathcal X)$, where $R^{\dagger}$ is the Moore–Penrose pseudoinverse. The convergence analysis reduces angles to singular values of $X_\perp^*V(X^*V)^{\dagger}$ and amplifies the spectral gap with the linear polynomial $x-(\lambda_{d+1}+\lambda_n)/2$ for the optimal iterates, and with Chebyshev polynomials for block Krylov spaces.

What would settle it

Take $A=\mathrm{diag}(1,0)$, $d=1$, $\mathcal X=\mathrm{span}\{e_1\}$, and $\mathcal V=\mathrm{span}\{e_2\}$: then $A(\mathcal V)=\{0\}$, every iterate stays in $\mathrm{span}\{e_2\}$, and $\tan\Theta(\mathcal X,\mathcal V_t)=\infty$, so the bound of Theorem 3.8 fails exactly when its hypothesis $\mathcal X\cap\mathcal V^\perp=\{0\}$ is violated. To test the optimality formula itself, enumerate all unit-norm $w\in\mathcal V$ in a small random instance and compare the minimum of the principal angle of $\mathcal V+A(\mathcal W)$ over $\dim\mathcal W\le d$ with the angle of $\mathcal V+A(\mathcal V R^\dagger(\mathcal X))$; any mismatch would disprove Theorem 3.2.

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Extended reading notes

Core claim

The central discovery is that the optimal block subspace expansion problem has a closed-form solution. Let $\mathcal S=\mathcal V+A(\mathcal V)$ and $\mathcal N=(I-P_{\mathcal V})P_{\mathcal S}(\mathcal X)$. Then $\mathcal V\oplus\mathcal N$ is the closest subspace to $\mathcal X$ among all spaces of the form $\mathcal V+A(\mathcal W)$ with $\mathcal W\subset\mathcal V$ and $\dim\mathcal W\le d$, in the componentwise sense of principal angles; a subspace $\mathcal W_0$ attaining this optimum can be chosen as $\mathcal W_0=\mathcal V R^{\dagger}(\mathcal X)$, uniquely when $\mathcal V\cap A^{-1}(\mathcal V)=\{0\}$ and $\dim\mathcal N=d$. This is Theorem 3.2 and Corollary 3.4. In the Hermitian case, Algorithm 3.1 produces $\mathcal V_t\subset\mathcal V_{t-1}+A(\mathcal V_{t-1})$ with $\Theta(\mathcal X,\mathcal V_t)=\Theta(\mathcal X,\mathcal V_{t-1}+A(\mathcal V_{t-1}))$, and if $\mathcal X\cap\mathcal V^\perp=\{0\}$ then every unitarily invariant norm of $\tan\Theta(\mathcal X,\mathcal V_t)$ satisfies the geometric bound of Theorem 3.8. Block Krylov spaces $\mathcal K_t=\mathcal V+A\mathcal V+\cdots+A^t\mathcal V$ satisfy a related Chebyshev bound that takes oversampling into account (Theorem 3.9).

Load-bearing premise

The load-bearing premise for the convergence theorem is that $\mathcal X\cap\mathcal V^\perp=\{0\}$, meaning no target eigenvector is orthogonal to the starting subspace; otherwise that direction is invisible to every iterate $\mathcal V_t$, the tangent angles stay infinite, and the claimed geometric decay cannot hold, and the paper gives no probability estimate for this 'generic' condition.

Editorial extensions

If this is right

At each step the optimal subspace $\mathcal V_t$ lies inside the block Krylov space $\mathcal K_t$ but has dimension $r+dt$ instead of $r(t+1)$; the angle to $\mathcal X$ equals that of the full one-step block Krylov expansion, giving the same angular quality in a much smaller space (Remark 3.5).
For Hermitian $A$ with $\lambda_d>\lambda_{d+1}$ and $\mathcal X\cap\mathcal V^\perp=\{0\}$, the tangent of the largest principal angle decays geometrically with rate $(\lambda_{d+1}-\lambda_n)/((\lambda_d-\lambda_n)+(\lambda_d-\lambda_{d+1}))$ in every unitarily invariant norm (Theorem 3.8).
The block Krylov sequence $\mathcal K_t$ converges with a rate that improves with the oversampling $\rho=r-d$ via a free parameter $p\in[d,r]$; this refines the bound one would get from $\mathcal V_t\subset\mathcal K_t$ alone (Theorem 3.9).
Replacing the unknown $\mathcal X$ in Algorithm 3.1 by a Rayleigh–Ritz or refined Rayleigh–Ritz approximation gives a computable algorithm that, in the numerical examples, stays competitive with block Krylov while storing substantially fewer vectors (Section 5).
Setting $d=1$ recovers the previously known optimal expansion of Ye and Jia, so the results form a genuine block extension of that line of work (Remark 3.3).

Reading between the lines

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

Because the optimal expansion requires the target $\mathcal X$, its practical role is as a benchmark; a natural extension would be to feed the same projection formula with a randomized sketch of $\mathcal X$ and prove a probabilistic error bound.
The hypothesis $\mathcal X\cap\mathcal V^\perp=\{0\}$ is called generic but carries no probability estimate; combining the paper's geometric rate with standard concentration bounds for random Gaussian $\mathcal V$ would turn Theorem 3.8 into a quantitative high-probability statement.
The improved Chebyshev separation estimate in Proposition 6.7 is stronger than the corresponding bound used in earlier randomized block Krylov analyses, so it could sharpen known convergence constants for dominant-subspace and singular-value approximation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear —,

The paper delivers a genuine block generalization of Ye's and Jia's optimal subspace expansion theory. Theorem 3.2 and Corollary 3.4 characterize the optimal W0 for d≥1 cleanly, recover the d=1 result as a special case, and the proof via PS(X) is actually simpler than the principal-angle optimization in the earlier papers. The Hermitian convergence bound in Theorem 3.8 follows from a correct choice of shift in Proposition 4.7, and Theorem 3.9's oversampling-aware bound for block Krylov spaces is new and sharpens existing Chebyshev estimates. The paper also states clearly that the optimal expansion is non-computable without the target, which is good discipline; the computable Algorithms 3.2 are presented honestly as heuristic counterparts.

The main soft spot is exactly what the stress-test note says: the abstract's claim that iterates approximate X arbitrarily well is false without the condition X∩V⊥={0}. If some x∈X is orthogonal to V, then for Hermitian A every iterate stays orthogonal to x, so the angle stays π/2 and tan Θ is undefined. Theorem 3.8 states the condition, so the theorem itself is internally consistent; the abstract just overstates it. Calling the condition generic is reasonable—Gaussian V gives it with probability 1 when r≥d—but the paper gives no probability estimate and the numerics rely on a single random draw per setting, so the near-degenerate regime is untested. That is a presentation gap, not a mathematical one.

Two smaller fixes: Theorem 3.9 silently needs λ_{p+1}>λ_n for the Chebyshev denominator to make sense, and the numerical section would be stronger shipping code and seeds. None of this touches the central algebraic characterization.

Who this is for: numerical linear algebra people working on subspace expansion, block Krylov methods, or eigenvector approximation. It deserves a serious referee; the theory is solid and the gaps are fixable at revision.

Best.

Referee Report

2 major / 7 minor

Summary. The paper studies the problem of expanding a current subspace V, of dimension r >= d, to V + A(W0) with W0 a subspace of V of dimension at most d, so as to approximate a d-dimensional A-invariant target subspace X in the sense of principal angles. It proves the existence of such optimal W0 and gives an explicit characterization in terms of the pseudo-inverse R†(X) (Theorem 3.2, Corollary 3.4). It then defines Algorithm 3.1, an iterative theoretical procedure that is step-wise optimal but depends on the unknown X, and shows that in the Hermitian case with X equal to the top-d eigenspace and λd > λd+1, the angle between X and the iterates converges linearly to zero under the non-degeneracy condition X∩V⊥={0} (Theorem 3.8). A related convergence bound for block Krylov subspaces is given in Theorem 3.9. Computable versions via Rayleigh-Ritz and refined Rayleigh-Ritz are proposed as Algorithm 3.2, and numerical experiments compare the algorithms on diagonal matrices with different eigenvalue decays.

Significance. The paper provides a natural generalization of the single-vector optimal expansion results of Ye and Jia to the block case d > 1, with a clean proof strategy based on the subspace PS(X) and pseudo-inverse formulas. The explicit characterization of all optimal expansions and the embedding of the optimal subspaces in block Krylov spaces are valuable structural insights. The proofs are detailed, and the auxiliary technical results (e.g., Theorem 6.2) are of independent interest. The paper is honest about the non-computability of the theoretical algorithm and presents the computable algorithm as a heuristic benchmark. Despite the issues noted below, the core mathematical claims in Theorems 3.2 and 3.8 are well supported.

major comments (2)

[Abstract / Theorem 3.8] The abstract states that the iterative sequence approximates X arbitrarily well when A is Hermitian and X is simple, but it omits the necessary hypothesis X∩V⊥={0} that appears in Theorem 3.8. This condition is not merely technical: if a nonzero x∈X is orthogonal to V0, then for Hermitian A, x is orthogonal to A(V0) and therefore to every iterate V_t, so the iterates cannot converge to X (indeed, the corresponding principal angle stays π/2 and its tangent is undefined). The theorem statement is correct, but the abstract's unqualified claim is false. Please include the condition, or an explicit 'generic' qualifier, in the abstract and in any summary of the convergence result.
[Theorem 3.9 / Proposition 4.7] The statement of Theorem 3.9 allows any d ≤ p ≤ r but does not require λ_{p+1} > λ_n. The bound in Eq. (8) contains (λ_{p+1} − λ_n) in the denominator, and the proof invokes Proposition 4.7 item 2, which uses the Chebyshev polynomial T_t((x−λ_n)/(λ_{p+1}−λ_n)). When λ_{p+1}=λ_n these expressions are undefined, so the theorem as stated is not meaningful for all allowed p. Add the hypothesis λ_{p+1} > λ_n to Theorem 3.9 and Proposition 4.7 item 2, or specify a limiting interpretation for the degenerate case.

minor comments (7)

[Section 2] The phrase 'it's Moore–Penrose pseudo-inverse' should be 'its Moore–Penrose pseudo-inverse'.
[Sections 3.3 and 5] The name 'Raleigh-Ritz' is misspelled; it should be 'Rayleigh-Ritz'.
[Figure 3] The caption 'Eliptical decay models' should read 'Elliptical decay models'.
[Section 3.3] The sentence 'Based on the previous ideas we consider the following' before Algorithm 3.2 is incomplete and should be finished or removed.
[Theorem 3.9, Eq. (8)] The typesetting of the exponent 3^{−t·min{...}} in Eq. (8) is ambiguous; please format the expression so that the argument of min and the exponentiation are clear.
[Sections 3.2 and 5] The paper calls the hypotheses X∩V⊥={0} and span{x1,...,xr}∩V⊥={0} 'generic' but gives no probability estimate. Since the numerical section uses a single Gaussian draw per configuration, a brief statement or reference on the probability of the non-degeneracy condition would strengthen the presentation.
[Section 6.4] In Proposition 6.5, 'Chebysehv polynomials' should be 'Chebyshev polynomials'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: optimal expansions are explicitly non-computable and the convergence bounds are derived from spectral data, not fitted or self-cited.

full rationale

The central theoretical construction (Theorem 3.2, Corollary 3.4, Algorithm 3.1) chooses W0 from the projection of the target subspace X onto S = V + A(V), and the paper states plainly that this makes the expansion optimal but not computable ('although the expansions V + A(W0) are optimal, they are not computable'). This is not a hidden prediction: it is an explicit optimization characterization, and no fitted quantity is renamed as a result. The quantitative convergence claims (Theorems 3.8 and 3.9) are derived from the eigenvalues λi of A through polynomial filter estimates (Proposition 4.7), with constants such as (λd+1 − λn)/((λd − λn) + (λd − λd+1)) computed from A, not fitted to the numerical examples. The stated assumptions X∩V⊥ = {0} and span{x1,...,xr}∩V⊥ = {0} are genuine non-degeneracy conditions needed for the tangent of principal angles to be finite; they are correctness conditions, not circularity. References to Jia [10], Ye [27] and Zhu-Knyazev [28] are external prior results used as tools, and no load-bearing self-citation chain appears. The numerical section benchmarks the stated algorithms and upper bounds; it does not supply the theorems.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central derivation uses standard linear algebra facts (principal angles, interlacing, pseudoinverse inequalities, Chebyshev bounds) plus the explicitly stated Hermitian spectral gap and genericity assumptions. There are no fitted free parameters, and no new entities are postulated. One implicit spectral assumption (λp+1 > λn) is not stated.

assumptions (7)

standard math Interlacing inequalities and monotonicity of principal angles under subspace inclusion (Proposition 4.1)
Used throughout Section 4 to compare angles of nested subspaces, e.g., in the proof of Theorem 3.8.
standard math Zhu-Knyazev identity: for orthonormal bases, tangents of principal angles are singular values of X_⊥^* V (X^* V)^†, and the inequality in Theorem 4.5/6.2
Used to turn subspace angle comparisons into matrix singular value bounds in Theorem 4.6 and Section 6.2.
standard math Chebyshev polynomial growth estimates (Propositions 6.6 and 6.7)
Used to prove the block Krylov bound in Theorem 3.9 via Proposition 4.7, item 2.
domain assumption Target subspace X is the span of eigenvectors for the d largest eigenvalues with λd > λd+1
Stated in Notation 3.7 and required for the convergence ratio in Theorem 3.8 to be less than 1.
domain assumption Non-degeneracy: X ∩ V⊥ = {0} in Theorem 3.8 and span{x1..xr} ∩ V⊥ = {0} in Theorem 3.9
These ensure tan(Θ(X,V0)) is finite and the Hp construction works; the authors note they hold generically but do not quantify the probability.
domain assumption Unstated condition λp+1 > λn for the Chebyshev rescaling in Proposition 4.7 and Theorem 3.9
The polynomial φt(x) = Tt((x - λn)/(λp+1 - λn)) is undefined when λp+1 = λn; the theorem statements do not exclude this case.
domain assumption Exact arithmetic in the theoretical algorithms; no roundoff analysis is included
All convergence bounds are for the exact theoretical iterates, not for floating-point implementations.

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Pith. "Pith review of Block subspace expansions for eigenvalues and eigenvectors approximation." pith.science (2026). https://pith.science/paper/723ZARWL

@misc{pith2026241114578,
  author       = {Pith},
  title        = {Pith review of: Block subspace expansions for eigenvalues and eigenvectors approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/723ZARWL}},
  note         = {Machine review of arXiv:2411.14578}
}

abstract

Let $A\in\mathbb C^{n\times n}$ and let $\mathcal X\subset \mathbb C^n$ be an $A$-invariant subspace with $\dim \mathcal X=d\geq 1$, corresponding to exterior eigenvalues of $A$. Given an initial subspace $\mathcal V\subset \mathbb C^n$ with $\dim \mathcal V=r\geq d$, we search for expansions of $\mathcal V$ of the form $\mathcal V+A(\mathcal W_0)$, where $\mathcal W_0\subset \mathcal V$ is such that $\dim \mathcal W_0\leq d$ and such that the expanded subspace is closer to $\mathcal X$ than the initial $\mathcal V$. We show that there exist (theoretical) optimal choices of such $\mathcal W_0$, in the sense that $\theta_i(\mathcal X,\mathcal V+A(\mathcal W_0))\leq \theta_i(\mathcal V+A(\mathcal W))$ for every $\mathcal W\subset \mathcal V$ with $\dim \mathcal W\leq d$, where $\theta_i(\mathcal X,\mathcal T)$ denotes the $i$-th principal angle between $\mathcal X$ and $\mathcal T$, for $1\leq i\leq d\leq \dim \mathcal T$. We relate these optimal expansions to block Krylov subspaces generated by $A$ and $\mathcal V$. We also show that the corresponding iterative sequence of subspaces constructed in this way approximate $\mathcal X$ arbitrarily well, when $A$ is Hermitian and $\mathcal X$ is simple. We further introduce computable versions of this construction and compute several numerical examples that show the performance of the computable algorithms and test our convergence analysis.

Figures

Figures reproduced from arXiv: 2411.14578 by the authors.

**Figure 2.** Linear decay models Analysis of the numerical results. In all the computed examples, the block Krylov method produces the best approximation of the target space X after 60 iterations. This is in accordance with the fact that Vt ⊂ Kt , for 0 ≤ t ≤ q, and that dim V ≥ dim X (see the comments after Theorem 3.8). On the other hand, the theoretical optimal subspace expansion computed by Algorithm 3.1 typically provides b… view at source ↗

**Figure 3.** Eliptical decay models 0 15 30 45 60 10 0 10 2 10 4 10 6 10 8 10 10 ( , q) ( , q) UB on ( , q) UB on ( , q) RR with refRR with (a) r = 30, p = 20 and G = 0 0 20 40 60 80 10 0 10 3 10 6 10 9 10 12 10 15 ( , q) RR with RR with R RA (b) r = 30, p = 20 and G = 0 0 15 30 45 60 0 1000 2000 3000 4000 5000 dim( t), r = 60 dim( t), r = 30 dim( t), r = 60 dim( t), r = 30 (c) Dimensions of various subspaces [PITH_FULL_IMAGE:f… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Polynomial (baseline) decay model with moderate oversampling and various dimensions. [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Bhatia, Matrix analysis

R. Bhatia, Matrix analysis. Graduate Texts in Mathematics, 169. Springer-Verlag, New York, 1997

work page 1997

[2] [2]

Drineas, I.F

P. Drineas, I.F. Ipsen, E.M. Kontopoulou, M. Magdon-Ismail, Structural convergence results for approximation of dominant subspaces from block Krylov spaces. SIAM J. Matrix Anal. Appl. 39 (2018), no. 2, 567-586

work page 2018

[3] [3]

Gu, Subspace Iteration Randomization and Singular Value Problems

M. Gu, Subspace Iteration Randomization and Singular Value Problems. SIAM Journal on Scientific Computing, vol. 37, n.3, January 2015, pp. A1139-73., 10.1137/130938700

doi:10.1137/130938700 2015

[4] [4]

Hochstenbach, A Jacobi-Davidson type SVD method

M. Hochstenbach, A Jacobi-Davidson type SVD method. Copper Mountain Conference (2000). SIAM J. Sci. Comput. 23 (2001), no. 2, 606-628

work page 2000

[5] [5]

Jia, The convergence of generalized Lanczos methods for large unsymmetric eigenproblems, SIAM Journal on Matrix Analysis and Applications 16 (1995), 843–862

Z. Jia, The convergence of generalized Lanczos methods for large unsymmetric eigenproblems, SIAM Journal on Matrix Analysis and Applications 16 (1995), 843–862

work page 1995

[6] [6]

Jia, Refined iterative algorithms based on Arnoldi’s process for unsymmetric eigenproblems, Linear Algebra Appl., 259 (1997), pp

Z. Jia, Refined iterative algorithms based on Arnoldi’s process for unsymmetric eigenproblems, Linear Algebra Appl., 259 (1997), pp. 1-23

work page 1997

[7] [7]

Jia, Generalized block Lanczos methods for large unsymmetric eigenproblems, Numerische Mathematik 80 (1998), 171–189

Z. Jia, Generalized block Lanczos methods for large unsymmetric eigenproblems, Numerische Mathematik 80 (1998), 171–189

work page 1998

[8] [8]

Jia, Some theoretical comparisons of refined Ritz vectors and Ritz vectors, Science in China Ser

Z. Jia, Some theoretical comparisons of refined Ritz vectors and Ritz vectors, Science in China Ser. A Math., 47 (2004), Supp. 222-233

work page 2004