{"id":"80da11db-28f0-429b-9f81-58af1a8f8a47","arxiv_id":"1908.03011","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"SINE, a conjugate-gradient-type shift-and-invert rational Krylov method, is an order-optimal regularizer for linear ill-posed problems and always stops no later than CGNE under the discrepancy principle.","lead":"An iterative method called SINE solves ill-posed linear inverse problems by minimizing the residual in a rational Krylov subspace, and is proven to be an order-optimal regularization scheme under the discrepancy principle. The paper shows SINE never needs more iterations than CGNE and can need far fewer, with an example that stops in 2 steps instead of 19.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.3's order-optimality proof stops at Eq. (27) and defers the remaining argument to a 'literal' transfer of the CGNE proof in [6]; that unverified rational-function continuation is the load-bearing gap.","rationale":"The reader's weakest assumption matches my own: the order-optimality claim rests on a deferred transfer of the CGNE proof to rational residual functions, and the manuscript does not supply that transfer. This is a genuine gap but not a demonstrated falsehood; the preceding derivation up to Eq. (27) is substantial and gives partial support. The rest of the paper, including the orthogonality lemmas, convergence theorem, and Theorem 5.1, is coherent and mostly self-contained. I therefore agree with the CONDITIONAL verdict rather than moving to rejection. One separate concrete misstatement in Section 6 strengthens the case for careful verification: for x+_2=t^3 ∈ X_{3/2,1}, Theorem 4.3 predicts a rate δ^(3/4), not δ^(3/2); the numerical illustration misstates its own prediction. This does not affect the central theorem directly, but it is an additional reason not to accept the proof transfer on faith.","tokens_in":16376,"tokens_out":21556,"duration_ms":217320,"concrete_test":"Complete the omitted part of Theorem 4.3 by translating Theorem 7.12 of [6] to rational residuals r_m=p_m/(1+λ/γ)^(m−1), starting from Eq. (27). Verify in particular that the proof yields π_m ≤ C(ρ/δ)^(2/(2μ+1)) and hence |r'_m(0)| ≤ C(ρ/δ)^(2/(2μ+1)), with constants independent of δ, and that no polynomial-only identity is used. If the translation cannot be carried out, Theorem 4.3 is unproven.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central order-optimality statement (Theorem 4.3) is not fully proved in the text. After deriving identity (27), the proof says 'the proof continues literally as the proof of Theorem 7.12 in [6]'. That deferred continuation is load-bearing: it must supply the bound |r'_m(0)| ≤ C(ρ/δ)^(2/(2μ+1)) that controls the noise term in Lemma 4.2. The CGNE proof is written for polynomial residual functions, whereas here r_m(λ)=p_m(λ)/(1+λ/γ)^(m−1); identity (27) contains the shifted factor (π_m−1/γ) in place of the polynomial-case π_m, and the orthogonality and approximation arguments for u_m and g_m are not written out. For fixed γ the shift is harmless as δ→0, but this is exactly the kind of step that must be verified before the order-optimality result is accepted. The manuscript gives no such verification, and no machine-checked proof is provided.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces SINE, a regularisation method for linear ill-posed operator equations, defined by minimizing the residual over shift-and-invert rational Krylov subspaces Q_m and stopping by Morozov's discrepancy principle. It presents the algorithm, proves orthogonality and rational residual representation, establishes convergence for exact data, claims order-optimal rates in Theorem 4.3, and shows in Theorem 5.1 that SINE's residual at each iteration is no larger than CGNE's, so SINE stops no later than CGNE. A simple multiplication-operator experiment illustrates the rates and the faster stopping behaviour.","tokens_in":16564,"tokens_out":10754,"duration_ms":106465,"significance":"If Theorem 4.3 is fully established, this is a worthwhile contribution: it extends order-optimal regularisation results from polynomial Krylov methods to shift-and-invert rational Krylov subspaces, with a fixed parameter gamma that is not fitted to the data. Theorem 5.1 gives a clean, falsifiable comparison with CGNE, and the algebraic lemmas in Section 2 are useful and coherent. The value of the paper is conditional on completing the rational-function continuation of the CGNE proof that is deferred in the central argument.","major_comments":[{"comment":"The order-optimality proof is not complete as written. The text states 'From here, the proof continues literally as the proof of Theorem 7.12 in [6],' but the deferred continuation is load-bearing. In the CGNE case, the remaining proof uses polynomial residual functions to obtain the bound |r'_m(0)| <= C(rho/delta)^(2/(2mu+1)) via the decay of [u_m,u_m] and the identity replacing Eq. (27). Here r_m is rational and Eq. (27) contains the shifted factor (pi_m - 1/gamma) instead of the polynomial-case pi_m, so the transfer is not verbatim; the sign and smallness of pi_m - 1/gamma, and the corresponding control of |r'_m(0)| from Eq. (22), must be verified explicitly. Without this, Theorem 4.3 is not established by the text alone.","section":"Section 4, proof of Theorem 4.3 after Eq. (27)"},{"comment":"Lemma 4.2 similarly rests on a 'literal copy of the proof of Lemma 7.11 in [6]'. That proof relies on estimates for lambda^mu r_m(lambda) and lambda^(1/2) g_m(lambda) on [0,epsilon] and on a final noise-term bound involving |r'_m(0)|. For rational r_m = p_m(lambda)/(1+lambda/gamma)^(m-1), these estimates are not identical to the polynomial case, and the text does not spell out why the same inequalities and constants hold. Since Lemma 4.2 feeds directly into Theorem 4.3, this deferral must be filled in before the central claim can be accepted.","section":"Section 4, Lemma 4.2, final paragraph"}],"minor_comments":[{"comment":"The text says the convergence for x2+ = t^3 is 'at least delta^(1/2) or delta^(3/2), respectively'; the theorem predicts delta^(2mu/(2mu+1)), which for mu = 3/2 is delta^(3/4), not delta^(3/2).","section":"Section 6, discussion of Figure 1"},{"comment":"The upper bound m_gamma(delta,y^delta) <= c(rho/delta)^(1/(2mu+1)) follows from Theorem 5.1 and Theorem 7.13 of [6], but the asserted sharpness of the exponent does not transfer automatically from CGNE to SINE. Since SINE may stop strictly earlier than CGNE, a separate lower-bound argument is needed; as written, the sharpness assertion is unsupported.","section":"Corollary 5.2"},{"comment":"Several typographical corruptions obscure the computation, for example 'Since 0 nu phi_m^2(0)' and 'By inserting in(...)'; these should be corrected to the intended formulas involving the derivative of phi_m and the maximum of lambda^nu phi_m^2(lambda).","section":"Section 3, proof of Theorem 3.1"}],"recommendation":"major_revision","confidential_remarks":"The central idea is sound and the presentation is mostly clear, but the gap in Theorem 4.3 and Lemma 4.2 is genuinely load-bearing. If the author supplies the missing rational-function continuation of the CGNE proof, I would support acceptance. I do not see circularity or data-fitting concerns: gamma is fixed a priori and the comparison with CGNE is used as a benchmark rather than as a tuning device."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives SINE, a shift-and-invert rational Krylov variant of CGNE, and proves it is order-optimal under the discrepancy principle for all μ>0. That is a real result. The subspace Q_m and the residual-minimization definition are new; previous rational Krylov regularizers (iterated Tikhonov, rational Arnoldi, implicit Euler) do not include this one. The analysis is mostly careful: Lemma 2.5 on the interlacing structure of the residual rational function, Lemma 2.6 on orthogonality, and the convergence proof in Section 3 are genuinely adapted to the rational case, not just copied. Theorem 5.1—that SINE stops no later than CGNE—follows immediately from the minimization property and is a nice selling point.\n\nThe soft spot is exactly where the reader's report puts it. At the end of Theorem 4.3's proof, after deriving identity (27), the text says 'from here, the proof continues literally as the proof of Theorem 7.12 in [6].' That continuation is load-bearing: it has to produce the bound on |r'_m(0)| and the decay of [u_m,u_m] for rational residual functions r_m(λ)=p_m(λ)/(1+λ/γ)^{m-1}. The polynomial proof in [6] does not contain the (π_m - 1/γ) shift that appears in (27), so 'literally' is an overstatement. I don't think the argument fails—the shift is harmless for fixed γ as δ→0—but it is not verified in the text. That is a genuine gap, and a referee should ask the author to write out the remaining lines rather than take the transfer on faith. Minor typos and the unreproducible Maple experiment are secondary.\n\nThe citation pattern is fine. There is no fitted parameter: γ and τ are arbitrary, and the rates are not post-hoc. The author also says up front that the proofs follow [6] closely, which is honest.\n\nBottom line: this deserves a serious referee. If the deferred step gets expanded (or verified to require no extra hypotheses), the paper is an important contribution to iterative regularization. I'd send it to review with a request for the missing proof details.","headline":"A genuinely new rational-Krylov iterative regularizer with a mostly careful analysis, but the order-optimality proof stops at a deferred 'literal' transfer from CGNE that the author needs to write out.","tokens_in":17066,"tokens_out":2707,"would_cite":true,"duration_ms":27692,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65J20","65F10","47A52"],"pacs":[],"model":"deepseek-v4-flash","headline":"Rational Krylov method SINE is order-optimal for all smoothness exponents","keywords":["ill-posed problems","regularisation","rational Krylov subspace","shift-and-invert","conjugate gradient normal equation","discrepancy principle","order-optimal","inverse problems"],"falsifier":"Set $T$ to the multiplication operator $Tf(t)=tf(t)$ on $L^2(0,1)$ and run SINE with small $\\gamma$ and known exact solutions $t$ and $t^3$ over noise levels $\\delta=10^{-2},\\dots,10^{-6}$; the measured error exponents must approach $1/2$ and $3/4$ if Theorem 4.3 holds. More directly, write out the omitted continuation of the CGNE proof: if the rational denominator $(1+\\lambda/\\gamma)^{m-1}$ breaks any of the inequalities involving $\\pi_m>1/\\gamma$ and the decay of $[u_m,u_m]$, the claimed bound is not established.","tokens_in":16131,"feed_emoji":"🧮","tokens_out":10271,"duration_ms":105432,"temperature":0.7,"pith_summary":"The paper proposes a new iterative regularisation method for ill-posed linear inverse problems: minimise the residual $\\|y^\\delta - Tx\\|$ over the shift-and-invert rational Krylov subspace $Q_m = \\operatorname{span}\\{T^*y^\\delta, (I+T^*T/\\gamma)^{-1}T^*y^\\delta, \\dots\\}$, then stop by Morozov's discrepancy principle. The central claim is that this method, called SINE, is order-optimal for every smoothness exponent $\\mu>0$: if the exact solution lies in the source set $X_{\\mu,\\rho}$, the error is bounded by $c\\,\\rho^{1/(2\\mu+1)}\\delta^{2\\mu/(2\\mu+1)}$. The paper also proves that the SINE residual at each step is no larger than the CGNE residual, so SINE never needs more iterations than CGNE to meet the discrepancy principle. A sympathetic reader would care because rational Krylov subspaces can accelerate convergence; if the claim is right, users can gain that speed while keeping the best worst-case accuracy known for CGNE.","feed_headline":"Rational Krylov method SINE is order-optimal for all smoothness exponents","feed_subtitle":"A shift-and-invert variant of CGNE provably matches its worst-case accuracy and never needs more iterations.","key_machinery":"SINE is defined by $x^\\delta_m=\\arg\\min_{x\\in Q_m}\\|y^\\delta-Tx\\|$ with $Q_m=\\operatorname{span}\\{T^*y^\\delta,(I+T^*T/\\gamma)^{-1}T^*y^\\delta,\\ldots,(I+T^*T/\\gamma)^{-(m-1)}T^*y^\\delta\\}$. The argument is carried by the rational residual function $r_m(\\lambda)=p_m(\\lambda)/(1+\\lambda/\\gamma)^{m-1}$, whose zeros are the interlacing positive Ritz values of the subspace; all estimates reduce to bounds on $|r'_m(0)|=\\sum_j 1/\\lambda_{j,m}+(m-1)/\\gamma$. Orthogonality is measured in the weighted inner product $[\\phi,\\psi]=\\int_0^{\\|T\\|^2}\\phi(\\lambda)\\psi(\\lambda)\\lambda\\,d\\|F_\\lambda y^\\delta\\|^2$. The proof of the main theorem reaches identity (27), $[u_m,u_m]=(\\pi_m-1/\\gamma)([r_{m-1},r_{m-1}/\\lambda]-[r_m,r_m/\\lambda])$ with $u_m=(r_{m-1}-r_m)/\\lambda$ and $\\pi_m=r'_{m-1}(0)-r'_m(0)>1/\\gamma$, and then claims the rest follows literally from the corresponding CGNE proof.","core_discovery":"The central result, Theorem 4.3, states: if $y\\in R(T)$, the noisy data satisfy $\\|y^\\delta-y\\|\\le\\delta$, SINE is stopped by the discrepancy principle with fixed $\\tau>1$, and $T^+y\\in X_{\\mu,\\rho}$, then $\\|T^+y - x^\\delta_{m(\\delta,y^\\delta)}\\| \\le c\\,\\rho^{1/(2\\mu+1)}\\delta^{2\\mu/(2\\mu+1)}$ for a constant $c$ independent of $\\delta$ and $\\rho$. This is exactly the definition of an order-optimal regularisation scheme for all $\\mu>0$. Theorem 5.1 adds a comparison with CGNE: for every iteration $m$, $\\min_{x\\in Q_m}\\|y^\\delta-Tx\\| \\le \\min_{x\\in K_m}\\|y^\\delta-Tx\\|$, so the discrepancy-principle stopping index for SINE is at most that of CGNE. The paper also shows convergence of the iterates for exact data and, as corollaries, transfers CGNE's iteration-count bounds to SINE.","pith_inferences":["A direct extension, only sketched in the paper, is that any rational Krylov subspace with negative real poles, chosen deterministically or randomly, should have residuals no larger than CGNE's; testing random pole choices on severely ill-posed problems would quantify how much early stopping improves.","The proof structure suggests the order-optimality result should transfer to other residual-minimising rational methods, such as nonstationary iterated Tikhonov with varying step sizes, although the missing CGNE transfer in Section 4 would have to be filled in first.","In finite-dimensional discretisations the worst-case rates are minimax, so SINE's practical payoff is likely lower iteration counts at the same worst-case accuracy; the paper's 2-versus-19 iteration example on a multiplication operator is a clean setting to test this on larger problems."],"forward_implications":["SINE is a fully order-optimal regularisation method with the discrepancy principle, matching CGNE's worst-case rate $\\delta^{2\\mu/(2\\mu+1)}$ for every source smoothness $\\mu>0$.","For every iterate, SINE's residual is at most CGNE's residual, so its stopping index $m_\\gamma(\\delta,y^\\delta)$ is never larger than $m_\\infty(\\delta,y^\\delta)$; all known iteration-count bounds for CGNE apply to SINE.","SINE is the fastest method among all regularisation schemes restricted to the shift-and-invert Krylov subspace $Q_m$: by construction it minimises the residual there, so no other method in that subspace can satisfy the discrepancy principle earlier.","Because $\\gamma\\to\\infty$ makes the rational Krylov subspace collapse to the polynomial Krylov subspace, SINE contains CGNE as a limit case and the two are governed by the same optimality framework.","The sharp bound $m_\\gamma(\\delta,y^\\delta)\\le c(\\rho/\\delta)^{1/(2\\mu+1)}$ carries over from CGNE, so the iteration count, not just the error, is order-optimal."],"supporting_citations":[{"why":"Supplies the CGNE theory, the discrepancy-principle convergence framework, and the proof template that Theorem 4.3 claims to follow.","marker":"[6]"},{"why":"Defines the Morozov discrepancy principle used as the stopping rule for SINE.","marker":"[23]"},{"why":"Establishes that CGNE is order-optimal, the property SINE is compared against and transfers.","marker":"[24]"},{"why":"Provide the sharp iteration-count bounds for CGNE that Corollary 5.2 extends to SINE.","marker":"[25, 26]"},{"why":"Supports identifying SINE iterates with rational functions and the functional-calculus representation used in the proofs.","marker":"[14]"}],"fun_headline_variants":["SINE rational Krylov is order-optimal for all smoothness","SINE rational Krylov needs no more iterations than CGNE","Shift-and-invert rational Krylov is order-optimal","Order-optimal rational Krylov regularisation with no extra iterations","SINE method provably order-optimal and never needs more iterations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, after identity (27), the remainder of the classical CGNE order-optimality proof can be copied verbatim to the rational residual functions $r_m(\\lambda)=p_m(\\lambda)/(1+\\lambda/\\gamma)^{m-1}$; the paper states this without writing the continuation, so the main theorem stands or falls on that unshown transfer.","fun_headline_variants_meta":{"raw":{"variants":["SINE rational Krylov is order-optimal for all smoothness","SINE rational Krylov needs no more iterations than CGNE","Shift-and-invert rational Krylov is order-optimal","Order-optimal rational Krylov regularisation with no extra iterations","SINE method provably order-optimal and never needs more iterations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001384,"raw_usage":{"total_tokens":5544,"prompt_tokens":827,"completion_tokens":4717,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":443,"completion_tokens_details":{"reasoning_tokens":4630}},"tokens_in":443,"tokens_out":4717,"duration_ms":32535,"temperature":1.0,"reasoning_tokens":4630,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:27:34.562257+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set $T$ to the multiplication operator $Tf(t)=tf(t)$ on $L^2(0,1)$ and run SINE with small $\\gamma$ and known exact solutions $t$ and $t^3$ over noise levels $\\delta=10^{-2},\\dots,10^{-6}$; the measured error exponents must approach $1/2$ and $3/4$ if Theorem 4.3 holds. More directly, write out the omitted continuation of the CGNE proof: if the rational denominator $(1+\\lambda/\\gamma)^{m-1}$ breaks any of the inequalities involving $\\pi_m>1/\\gamma$ and the decay of $[u_m,u_m]$, the claimed bound is not established.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the CGNE theory, the discrepancy-principle convergence framework, and the proof template that Theorem 4.3 claims to follow."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Morozov discrepancy principle used as the stopping rule for SINE."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that CGNE is order-optimal, the property SINE is compared against and transfers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supports identifying SINE iterates with rational functions and the functional-calculus representation used in the proofs."}],"review_version":1}