{"id":"f33a11a2-a721-476c-99f2-4d34a3239275","arxiv_id":"2411.14578","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Hermitian matrices, the optimal block subspace expansion that minimizes all principal angles to a target invariant subspace is explicitly characterized, with convergence bounds and computable Rayleigh-Ritz versions.","lead":"This paper extends the theory of optimal subspace expansion for eigenvalue problems from one eigenvector to several at once, giving explicit formulas for the best block expansion and convergence bounds for Hermitian matrices. The value for a generalist is a complete answer to a natural question in numerical linear algebra, plus a benchmark that computable eigensolvers can be compared against.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.8's convergence claim needs X∩V⊥={0}; without it V_t never approaches X, so the abstract's unqualified claim is false.","rationale":"The reader's weakest_assumption is exactly where the argument is most fragile. The condition X∩V⊥={0} is necessary: without it the tangent metric is undefined and convergence fails, so the abstract overstates. The theorem itself is correct under the stated hypothesis and the proof of Theorem 3.8 is sound (shifted polynomial φ(A), Theorem 4.6 with p=d, Proposition 4.7 gives the constant). The condition is generic for continuous random V, so the practical impact is limited, but the paper gives no probability estimate. I see no internal inconsistency in Theorems 3.2 or 3.8, and the numerical issues (single draw, no code) do not affect the mathematical claim. Hence the conditional verdict stands.","tokens_in":28109,"tokens_out":17554,"duration_ms":154129,"concrete_test":"Take A=diag(3,2,1), X=span{e1}, V=span{e2}, d=1. Compute the iterates of Algorithm 3.1: N_t=(1-P_{V_{t-1}})P_{S_t}(X). Since e1⊥V and A is diagonal, e1⊥V_t for all t, so θ(X,V_t)=π/2 and Eq. (7) is vacuous—this confirms failure when X∩V⊥≠{0}. Then repeat with V=span{e2+εe1} for ε=1 and ε=10^-8 and check that ∥tan Θ(X,V_t)∥/∥tan Θ(X,V_0)∥ ≤ ((λ2-λ3)/((λ1-λ3)+(λ1-λ2)))^t = (1/3)^t, verifying the bound in the generic regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim, Eq. (7), is only meaningful when X∩V⊥={0}. If a nonzero x∈X is orthogonal to V, then for Hermitian A, x^* A v = (Ax)^* v = λ x^* v = 0 for all v∈V, so x⊥A(V); by induction x⊥V_t for all t. Hence the principal angle between X and V_t is exactly π/2, tan Θ is undefined, and the iterates do not approximate X even though Algorithm 3.1 is well-defined. The paper states the condition in Theorem 3.8, so the theorem is internally consistent, but the abstract's 'approximate X arbitrarily well' is false without it. The 'generic' defense is reasonable for Gaussian V (probability 1 when r≥d), but no quantitative statement is given, and the numerics use a single draw, so the near-degenerate regime is untested. This makes the headline claim conditional on a non-degeneracy assumption that is necessary, not merely technical.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":28295,"tokens_out":15617,"duration_ms":130752,"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":[{"comment":"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.","section":"Abstract / Theorem 3.8"},{"comment":"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.","section":"Theorem 3.9 / Proposition 4.7"}],"minor_comments":[{"comment":"The phrase 'it's Moore–Penrose pseudo-inverse' should be 'its Moore–Penrose pseudo-inverse'.","section":"Section 2"},{"comment":"The name 'Raleigh-Ritz' is misspelled; it should be 'Rayleigh-Ritz'.","section":"Sections 3.3 and 5"},{"comment":"The caption 'Eliptical decay models' should read 'Elliptical decay models'.","section":"Figure 3"},{"comment":"The sentence 'Based on the previous ideas we consider the following' before Algorithm 3.2 is incomplete and should be finished or removed.","section":"Section 3.3"},{"comment":"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.","section":"Theorem 3.9, Eq. (8)"},{"comment":"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":"Sections 3.2 and 5"},{"comment":"In Proposition 6.5, 'Chebysehv polynomials' should be 'Chebyshev polynomials'.","section":"Section 6.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid and detailed contribution to subspace expansion methods. The main theorems are essentially correct in their stated form, but the abstract's overclaim and the missing hypothesis in Theorem 3.9 require revision. The numerical experiments are limited to diagonal matrices and single random draws, which is acceptable for an exploratory study. I recommend major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear —,\n\nThe 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.\n\nThe 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.\n\nTwo 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.\n\nWho 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.\n\nBest.","headline":"Necessary non-degeneracy condition missing from the abstract; otherwise the block optimal expansion and convergence bounds hold up.","tokens_in":28880,"tokens_out":2501,"would_cite":true,"duration_ms":23025,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65F15","15A18","65F10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["optimal subspace expansion","eigenvector approximation","block Krylov subspace","projection methods","computable subspace expansion","principal angles","Hermitian eigenvalue problem"],"falsifier":"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.","tokens_in":27898,"feed_emoji":"📐","tokens_out":12512,"duration_ms":110551,"temperature":0.7,"pith_summary":"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.","feed_headline":"A pseudoinverse formula finds the optimal block subspace expansion","feed_subtitle":"Project the target onto the block Krylov step, drop what V already knows, and the angles to the eigenvectors decay geometrically.","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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)."],"supporting_citations":[{"why":"introduced the d=1 optimal subspace expansion problem that this paper extends to blocks","marker":"[27]"},{"why":"recent d=1 optimal and computable expansions; the source of the computable-replacement strategy used in Algorithm 3.2","marker":"[10]"},{"why":"supplies the tangent-of-angles singular value identity used in the convergence proofs","marker":"[28]"},{"why":"provides the singular-gap amplification strategy and random-subspace control used in Theorem 4.6","marker":"[3]"},{"why":"Chebyshev-based convergence bounds for randomized block Krylov methods that Proposition 6.7 refines","marker":"[18]"},{"why":"Residual Arnoldi method used as a numerical baseline in the experiments","marker":"[15]"},{"why":"refined Rayleigh-Ritz projection method used in the computable Algorithm 3.2","marker":"[6]"},{"why":"comparison of Ritz and refined Ritz vectors cited in the analysis of the numerical results","marker":"[8]"}],"fun_headline_variants":["Pseudoinverse yields optimal block subspace expansion","Optimal eigenvector subspace step via pseudoinverse","Block Krylov meets pseudoinverse for optimal expansion","Closed-form optimal subspace expansion for eigenvalues","Geometric convergence from optimal block subspace updates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Pseudoinverse yields optimal block subspace expansion","Optimal eigenvector subspace step via pseudoinverse","Block Krylov meets pseudoinverse for optimal expansion","Closed-form optimal subspace expansion for eigenvalues","Geometric convergence from optimal block subspace updates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000343,"raw_usage":{"total_tokens":2002,"prompt_tokens":1176,"completion_tokens":826,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":792,"completion_tokens_details":{"reasoning_tokens":767}},"tokens_in":792,"tokens_out":826,"duration_ms":7552,"temperature":1.0,"reasoning_tokens":767,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:08:46.620773+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Ye, Optimal expansion of subspaces for eigenvector approximations","cited_arxiv_id":null,"evidence_quote":"introduced the d=1 optimal subspace expansion problem that this paper extends to blocks"},{"cited_title":"Jia, Theoretical and computable optimal subspace expansions for matrix eigenvalue problems","cited_arxiv_id":null,"evidence_quote":"recent d=1 optimal and computable expansions; the source of the computable-replacement strategy used in Algorithm 3.2"},{"cited_title":"Zhu, A.V","cited_arxiv_id":null,"evidence_quote":"supplies the tangent-of-angles singular value identity used in the convergence proofs"},{"cited_title":"Randomized Block Krylov Methods for Stronger and Faster Approximate Singular Value Decomposition","cited_arxiv_id":"1504.05477","evidence_quote":"Chebyshev-based convergence bounds for randomized block Krylov methods that Proposition 6.7 refines"},{"cited_title":"Lee and G","cited_arxiv_id":null,"evidence_quote":"Residual Arnoldi method used as a numerical baseline in the experiments"},{"cited_title":"Jia, Refined iterative algorithms based on Arnoldi’s process for unsymmetric eigenproblems, Linear Algebra Appl., 259 (1997), pp","cited_arxiv_id":null,"evidence_quote":"refined Rayleigh-Ritz projection method used in the computable Algorithm 3.2"},{"cited_title":"Jia, Some theoretical comparisons of refined Ritz vectors and Ritz vectors, Science in China Ser","cited_arxiv_id":null,"evidence_quote":"comparison of Ritz and refined Ritz vectors cited in the analysis of the numerical results"}],"review_version":1}