{"id":"fd537e66-f492-4a55-bfd7-8ef78a4724ad","arxiv_id":"2608.11091","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Two sketched GMRES variants in Tucker format, using randomized HOSVD and multilinear Nyström compression, solve tensor linear systems with reduced memory and time, and low-rank compression regularizes inverse problems.","lead":"This paper proposes two randomized low-rank Tucker solvers, RHOSVD-Tucker sGMRES and MLN-Tucker sGMRES, that replace full GMRES orthogonalization with short recurrences to solve large tensor-structured linear systems faster and with less memory. A numerical study on Poisson, convection-diffusion, and 3D deblurring problems reports gains over existing Tucker GMRES variants, with the low-rank constraint acting as an implicit regularizer in inverse problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sketched-residual stopping rule is the load-bearing risk: RHOSVD-Tucker sGMRES stops with true residual 5.09e-5 versus target 5e-5 in Table 2, and no bound is given for the Khatri-Rao sketch's norm preservation.","rationale":"I agree with the reader that the most insecure condition is the reliability of the sketched residual for the structured sketch. What makes this concern load-bearing is not only the missing theory but a direct counterexample in the paper: in the convection-diffusion benchmark, the RHOSVD-Tucker sGMRES run stops on the sketched residual criterion and the true residual is above tolerance. This means the method's output contract in Algorithm 2 is violated for a reported configuration. The MLN variants happen to meet the tolerance in the same table, so the failure is not universal, but it shows the correction factor η cannot be a fixed constant across the method family. A concrete pair of runs differing only in the sketch distribution (structured vs dense Gaussian) would settle whether the failure is due to the Khatri-Rao structure or merely to the sketch size/η tuning. This does not change the reader's conditional verdict: the paper needs either a proof of residual-norm preservation for the structured sketch or a reliable a posteriori true-residual check, plus a systematic η sensitivity study. The availability of code and data is a plus and makes the proposed test feasible.","tokens_in":25279,"tokens_out":13471,"duration_ms":127275,"concrete_test":"Using the released code [14], rerun the §4.2 convection-diffusion test with Algorithm 2, keeping all parameters identical (η=0.3, k_trunc=2, s=240, same seed), but replace the Khatri-Rao structured sketch S by a dense Gaussian matrix of the same row dimension s. If the dense-sketch run reaches a true relative residual below 5e-5 while the structured run stops at 5.09e-5, the structured sketch's residual distortion is the cause; if the dense run also stops prematurely, the issue is sketch size or η rather than the Khatri-Rao structure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Algorithm 2 (lines 17-19) and Algorithm 6 (lines 27-29) terminate when the sketched relative residual ∥M y_k - b_s∥/β_s drops below η·tol. This is valid only if the Khatri-Rao structured Gaussian sketch S = ⊙ S_k is a subspace embedding for the Krylov residual subspace of dimension up to k+1. No OSE-type theorem for this structured sketch is provided, and the sketch dimension is set by tensor-rank heuristics (RHOSVD) or fixed a priori (MLN), not by the subspace dimension or a target distortion ε. The paper acknowledges the risk in §3.2, and §4.2 provides a concrete failure: Table 2 reports RHOSVD-Tucker sGMRES stopping in 97 iterations with true relative residual 5.09e-5, above the target 5e-5, because the sketched residual underestimated the true one. The η=0.3 correction is hand-tuned, and no experiment shows it generalizes across problems, ranks, or random draws. Since the central claim is that both methods solve tensor systems faster and with less memory, a method whose stopping rule can silently return an iterate above tolerance does not support that claim as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two randomized GMRES-type solvers for linear systems with Tucker-structured tensors. RHOSVD-Tucker sGMRES combines a truncated-window partial orthogonalization with randomized HOSVD-based summation and rank selection; MLN-Tucker sGMRES instead uses a fixed-rank multilinear Nyström approximation, so that only compact sketched representations of the Krylov basis are retained. Both methods replace the full Arnoldi orthogonalization with a sketched least-squares problem whose stopping criterion uses the sketched residual. Numerical experiments on a preconditioned 3D Poisson equation, a 3D convection-diffusion problem, and a 3D image deblurring inverse problem compare the methods with RHOSVD-Tucker GMRES and, in the symmetric case, TK-SS-SD. The paper claims that the proposed methods outperform standard low-rank Tucker solvers in time and memory and that the low-rank Tucker constraint acts as an implicit regularizer for inverse problems when combined with adaptive projected Tikhonov penalization and GCV parameter selection.","tokens_in":25576,"tokens_out":10026,"duration_ms":87003,"significance":"The paper addresses a real computational bottleneck: rank growth and full orthogonalization make standard Krylov methods expensive for Tucker-format equations. The algorithmic idea of reusing the MLN sketch, and in particular the identity diag(G_Ψ) = (Ψ_1 ⊙ ⋯ ⊙ Ψ_d)^T vec(Σ c_j X^{(j)}) proved in Section 3.3, is clean and useful, and the streaming memory-efficient variant is a meaningful contribution. The manuscript also ships code, data, and results with a Zenodo DOI, which is commendable. However, the central algorithmic claims currently rest on single runs with many hand-tuned parameters and without a supporting theory for the structured sketch; the paper itself documents a failure of the sketched-residual stopping criterion in Table 2. These issues are load-bearing because the algorithms' output guarantee and the claimed speed/memory advantages depend on the reliability of that stopping rule. If the stopping rule and the structured-sketch embedding question are resolved, the paper would be a solid contribution to randomized tensor solvers.","major_comments":[{"comment":"The stopping rule is based on the sketched relative residual ∥M y_k − b_s∥_2/β_s, with termination when it falls below η·tol, but no bound relates this quantity to the true relative residual ∥L(X_k) − B∥_F/∥B∥_F. The paper acknowledges the risk in Section 3.2, and Section 4.2 documents the failure: Table 2 reports RHOSVD-Tucker sGMRES stopping after 97 iterations with true residual 5.09·10^{-5}, above the target 5·10^{-5} stated in the table (and ten times above the 5·10^{-6} target stated in the prose of Section 4.2). The correction η = 0.3 is hand-tuned, and no experiment shows that it transfers across problems, tolerances, or random draws. This is load-bearing because the output guarantee of Algorithms 2 and 6 ('such that ∥L(eX)−B∥_F ≤ tol') is exactly what the stopping rule must certify. Please either provide a certified bound for the structured sketch or modify the algorithms to verify the true residual, for example by occasional exact residual computations, and re-report the experiments accordingly.","section":"Section 3.2, Algorithm 2 lines 17–19; Algorithm 6 lines 27–29; Table 2"},{"comment":"The reduction to the sketched least-squares problem requires S to be an oblivious subspace embedding for the subspace containing the residual vectors. In this paper S = S_1 ⊙ ⋯ ⊙ S_d with Gaussian S_k (Algorithm 2 lines 3–4; Section 3.3, Eqs. (14) and (21)), but no theorem is given showing that this Khatri–Rao-structured sketch satisfies the (ε, δ, k)-OSE property for the Krylov subspace. The sketch size is chosen by tensor-rank heuristics (RHOSVD) or fixed a priori (MLN), not by the subspace dimension k and the desired distortion ε. The cited analysis in [22] concerns structured sketches for the MLN approximation itself and, as far as the manuscript shows, does not establish a residual-norm subspace embedding for the sGMRES least-squares problem. This gap directly underlies the failure described in the previous comment and should be addressed, even if only with a restricted result for tensors of bounded multilinear rank.","section":"Section 3.2, Definition 4 and Eq. (11)"},{"comment":"All reported numerical claims are based on single runs with hand-picked hyperparameters (ktrunc, η, r, p, r_sol, p_sol, τ_outer), and no standard errors or multiple random draws are provided. The comparison is also dominated by baselines from the same research group: RHOSVD-Tucker GMRES is Ref. [72] with two coauthors, and TK-SS-SD is Ref. [40] with one coauthor. Since the paper provides no convergence theory, the central claim that the proposed methods 'outperform standard low-rank Tucker solvers' rests entirely on these runs. Please add multiple trials or sensitivity studies for the main parameters (especially η and ktrunc), and include at least one independent baseline or clearly specify the settings used for the baselines.","section":"Section 4, Tables 1–2 and Figures 1, 4, 5"},{"comment":"The claim that 'the low-rank Tucker constraint acts as an implicit regularizer' and that this is the first demonstration of the idea is stronger than what the experiments show. Figure 5 compares regularized versus unregularized variants of the same Tucker solvers, and Figure 4 varies the right-hand-side compression tolerance, but there is no comparison with, for example, a standard hybrid GMRES on the vectorized problem or a non-tensor regularized solver, and no formal argument (e.g., noise propagation or regularization property of the rank-truncation operator) is provided. The observed semiconvergence behavior is suggestive, but the abstract's claim of 'stable reconstructions' should either be supported by a theorem or stated more cautiously.","section":"Section 4.3, Figure 5, and Section 5"}],"minor_comments":[{"comment":"The target tolerance is inconsistent: the prose states a relative residual target of 5×10^{-6}, while Table 2's caption states 5×10^{-5}. This should be reconciled, since it changes the interpretation of the reported premature convergence.","section":"Section 4.2 and Table 2"},{"comment":"Algorithm 6 line 3 generates matrices Ψ_k ∈ R^{n_k × r}, whereas Section 3.3 defines the left sketching matrices as n_k × (r + p) and later uses ̃r = r + p in the memory comparison; the relationship between the generated matrices, the oversampling parameter p, and the sketch size ̃r should be clarified.","section":"Section 3.3 and Algorithm 6"},{"comment":"The notation for U^{(j)}_{Ψ_k} and the barred variant used for the first r rows is nearly indistinguishable in print, and the two symbols are used interchangeably in later equations; please use clearly distinct symbols.","section":"Section 3.3, Eq. (15)"},{"comment":"The cross-references are incorrect: the text says 'Figure 4(a) displays the RRE over 100 iterations', but the RRE histories with and without regularization are shown in Figure 5(a)–(b), while Figure 4 concerns the effect of the RHS compression tolerance.","section":"Section 4.3"},{"comment":"The symbol η is overloaded: it denotes the sketched-residual correction factor in Algorithms 2 and 6, but in Eq. (24) it is also used for the exponential-sum parameter η = π/√q. Please rename one of the two.","section":"Section 3.4"},{"comment":"The text refers to 'tol = 0.05' when reporting the rank-(80,57,3) Tucker tensor and the RRE histories, while the algorithm parameter for right-hand-side compression is τ_outer; please align the notation and state explicitly that τ_outer = tol.","section":"Section 4.3"}],"recommendation":"major_revision","confidential_remarks":"The core algorithmic ideas are reasonable and the code release is a strength, but the manuscript's central claims are currently undermined by the unproven and demonstrably unreliable sketched-residual stopping rule and by the absence of any subspace-embedding analysis for the Khatri–Rao-structured sketch. These issues are fixable in revision, so I do not recommend rejection. The editor may also wish to note that the main baselines come from the same research group, which weakens the 'outperform standard low-rank Tucker solvers' claim unless independent baselines or a parameter sensitivity study are added."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: this is a coherent, useful paper that deserves a serious referee, but it should not be accepted as-is. The two new solvers—RHOSVD-Tucker sGMRES and MLN-Tucker sGMRES—are sensible combinations of known randomized tools, and the package includes code and data, which is more than most submissions in this area manage. The cleanest novelty is the reuse of the Nyström sketch diagonal as the sGMRES sketch in the MLN variant, and the downsampled-sketches memory option; both are clearly described and the diag(GPsi) equivalence in Section 3.3 is proven correctly. The experiments on the Poisson, convection-diffusion, and deblurring problems are reproducible from the Zenodo archive and support the main performance claims on those benchmarks.\n\nThe soft spot is the stopping rule, and the paper knows it. Algorithms 2 and 6 terminate when the sketched residual drops below eta*tol, but no subspace-embedding bound is given for the Khatri-Rao structured Gaussian sketch, and the sketch dimension is set by rank heuristics or a priori, not by the subspace dimension. The acknowledged premature convergence in the convection-diffusion test (true residual 5.09e-5 vs target 5e-5 with eta=0.3) is a concrete instance of the risk. It is a minor miss numerically, but it is load-bearing for the claim that the solvers \"solve\" the system to tolerance. The hand-tuning of eta is also not explored across problems or random draws. I'd like to see either a theoretical argument for why this structured sketch preserves norms on the relevant Krylov residual subspace, or a more conservative stopping rule (e.g., a safety factor and a posteriori verification of the true residual).\n\nOther smaller issues: the \"first\" claims are overbroad—\"first work demonstrating that compressed tensor decompositions serve as an effective regularizer\"—given their own TT-sGMRES paper and the existing literature on low-rank regularization. The experiments are single runs with hand-picked hyperparameters and no error bars; adding multiple seeds and external baselines (not just the authors' own RHOSVD-Tucker GMRES and their close collaborators' TK-SS-SD) would materially strengthen the comparison.\n\nNone of this undermines the core construction. The algorithms are coherent, the code is there, and the tested benchmarks do show gains. I would send it to a competent referee with a request for revision, not desk-reject it. It just needs its claims trimmed and its stopping rule either justified or made conservative.\n\nLet me know if you want to discuss.","headline":"Solid, citable solvers with a load-bearing stopping-rule gap; deserves revision, not rejection.","tokens_in":26137,"tokens_out":3934,"would_cite":true,"duration_ms":49217,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A69","65F10","65F55","68W20","65J22","65N22","65F22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proposes two randomized Tucker-format GMRES variants that replace full Arnoldi orthogonalization with short recurrences, and it argues that the low-rank Tucker constraint itself acts as an implicit regularizer in inverse problems.","keywords":["tensor equations","randomized numerical linear algebra","Tucker decomposition","sketched GMRES","Krylov subspace method","inverse problems","preconditioning","compression"],"falsifier":"Run either proposed solver on the paper's unpreconditioned 3D convection-diffusion benchmark with the orthogonalization window $k_{\\mathrm{trunc}}=1$ and the sketch-correction factor $\\eta=0.3$; the reported figure already shows true-residual stagnation for RHOSVD-Tucker sGMRES. If increasing the sketch size $s$ while keeping the window fixed still cannot drive the true residual below the prescribed tolerance, then the structured sketch is not a reliable stand-in for the true residual in exactly the nonsymmetric regime the method targets.","tokens_in":25080,"feed_emoji":"🧮","tokens_out":11698,"duration_ms":101192,"temperature":0.7,"pith_summary":"This paper is about solving linear equations whose unknown is a multidimensional array (a tensor), with operators that are sums of separable terms, as arise in high-dimensional PDEs and imaging inverse problems. Standard Krylov solvers become impractical here because each new basis vector must be stored and orthogonalized against all previous ones, and in low-rank tensor formats the ranks of these basis vectors inflate as iteration proceeds. The paper proposes two randomized solvers in the sketched-GMRES family—one using randomized HOSVD with adaptive rank selection, one using a streamable Nyström approximation with fixed rank—that orthogonalize only against a short window of recent vectors and keep ranks bounded by randomized Tucker compression. The claim is that both beat existing low-rank Tucker solvers in symmetric and nonsymmetric benchmarks, and that for inverse problems the low-rank constraint acts as an implicit regularizer, so adding an adaptive projected Tikhonov term with automatic parameter selection gives stable reconstructions without knowing the noise level.","feed_headline":"Two randomized solvers make tensor GMRES faster and lighter","feed_subtitle":"Short recurrences replace full orthogonalization, and low-rank truncation doubles as regularization in inverse problems.","key_machinery":"The argument runs on the Tucker format, which stores a $d$-dimensional tensor as a small core tensor plus one factor matrix per mode, and on a Khatri-Rao structured Gaussian sketch $\\mathbf S = \\mathbf S_1 \\odot \\cdots \\odot \\mathbf S_d$, defined by folding independent Gaussian matrices $\\mathbf S_j$ with $\\odot$ denoting the column-wise Kronecker (Khatri-Rao) product. This sketch maps the vectorization of any basis tensor to a short vector without forming dense intermediates, turning GMRES's least-squares problem into the sketched problem $\\min_{\\mathbf y}\\|\\mathbf M\\mathbf y-\\mathbf b_s\\|_2$ and permitting a short-recurrence window of width $k_{\\mathrm{trunc}}$ in place of full Arnoldi orthogonalization. In the MLN variant the same sketch is produced at no extra cost as the superdiagonal of the Nyström-sketched core, using the identity $[\\mathrm{diag}(\\mathbf G_\\Psi)]_i = (\\boldsymbol\\psi_{1,i}\\otimes\\cdots\\otimes\\boldsymbol\\psi_{d,i})^T\\mathrm{vec}(\\mathbf X)$, where $\\otimes$ is the Kronecker product. The role of this machinery is to keep the multilinear ranks of the basis tensors bounded while making the residual computation and stopping test cheap, which is what converts GMRES from an iteration-count-limited method into one whose per-step cost is stable.","core_discovery":"The central discovery, stated on the paper's own terms, is that the two bottlenecks of Tucker-format GMRES—rank growth of the Krylov basis and the quadratic cost of full Arnoldi orthogonalization—can be attacked simultaneously within the sketched-GMRES framework. RHOSVD-Tucker sGMRES applies randomized HOSVD with an adaptive rank heuristic inside every tensor summation, so basis tensors are rounded without forming dense intermediates, and it orthogonalizes only against the previous $k_{\\mathrm{trunc}}$ basis tensors. MLN-Tucker sGMRES uses the multilinear Nyström approximation with a fixed rank, sketches each basis tensor in a single pass, discards the tensor itself, and reconstructs the final solution from the accumulated sketches; the sketch needed for the residual test is obtained for free as the superdiagonal of the Nyström-sketched core. The paper's experiments show these variants reaching the target residual faster than the full-orthogonalization baseline on a preconditioned 3D Poisson equation and an unpreconditioned 3D convection-diffusion equation, and on a 3D image-deblurring inverse problem the paper argues that the low-rank Tucker constraint suppresses high-frequency noise and, together with GCV-selected projected Tikhonov regularization, prevents the semi-convergence that makes unregularized iterates useless.","pith_inferences":["Beyond the paper, the two solvers' rank strategies are complementary, suggesting a hybrid that starts with adaptive RHOSVD ranks and switches to a fixed-rank streaming MLN phase once a stable rank is observed.","Beyond the paper, the claim that Tucker truncation regularizes inverse problems could be quantified by computing the equivalent filter factors of the rank-truncated iteration and comparing them with classical regularization parameter choices.","Beyond the paper, the memory-efficient MLN downsampling introduces a user-set reconstruction resolution $r_{\\mathrm{sol}}$; a natural extension is an adaptive rule that grows $r_{\\mathrm{sol}}$ until two successive reconstructions agree.","Beyond the paper, for operators with slow singular-value decay the fixed-rank MLN approximation will lose accuracy, so updating the target rank adaptively while preserving the streaming structure is a testable extension the paper does not develop."],"forward_implications":["Per-iteration cost no longer grows with the total iteration count, because only a window of $k_{\\mathrm{trunc}}$ recent basis tensors is orthogonalized instead of all previous ones.","The MLN variant never needs to store the full Krylov basis: each tensor is sketched once and discarded, and the final solution is assembled from the accumulated sketches, cutting memory from linear-in-iterations growth to a bounded window plus compact sketch storage.","In inverse problems, restricting all iterates to a manifold of bounded multilinear rank suppresses high-frequency noise, and the GCV-selected Tikhonov parameter removes the need for oracle-based stopping rules.","The same short-recurrence construction works with preconditioning through a flexible-GMRES wrapper, so rank-truncated preconditioner actions can be used without breaking the iteration.","The numerical comparison indicates the methods are competitive with symmetric short-recurrence solvers on a preconditioned Poisson problem, not only with other GMRES variants."],"supporting_citations":[{"why":"It supplies the sketched GMRES framework, replacing the full least-squares problem with minimization of the sketched residual norm.","marker":"[58]"},{"why":"It supplies the Khatri-Rao randomized tensor summation that RHOSVD-Tucker sGMRES inherits and whose baseline RHOSVD-Tucker GMRES is the main comparison.","marker":"[72]"},{"why":"It supplies the multilinear Nyström approximation that defines MLN-Tucker sGMRES and its fixed-rank streaming compression.","marker":"[16]"},{"why":"It establishes the streaming low-rank Tucker approximation paradigm that the MLN single-pass accumulation builds on.","marker":"[73]"},{"why":"It supplies the randomized HOSVD routine used for per-iteration rank selection in RHOSVD-Tucker sGMRES.","marker":"[1]"},{"why":"It shows how sketched GMRES adapts to tensor formats with short recurrences and memory-efficient reconstruction, the direct precedent for the memory-saving variant.","marker":"[15]"},{"why":"It provides the symmetric Tucker subspace-gradient baseline that the Poisson benchmark compares against.","marker":"[40]"},{"why":"It provides the hybrid projection and Tikhonov regularization framework used inside the projected least-squares problem.","marker":"[29]"},{"why":"It supplies generalized cross-validation, the rule that selects the Tikhonov parameter automatically in the inverse-problem experiments.","marker":"[30]"},{"why":"It frames the semi-convergence of iterative methods for discrete inverse problems, which motivates the regularization component of the paper's claim.","marker":"[34]"}],"fun_headline_variants":["Sketched GMRES cuts tensor cost with randomized rounding","Faster tensor GMRES via short recurrences and sketching","Two randomized Tucker GMRES variants beat full orthogonalization","Tensor linear solves sped up by low-rank sketching","Randomized Tucker GMRES: cheaper, streaming, self-regularizing"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the Khatri-Rao structured Gaussian sketch preserves residual norms well enough that minimizing the sketched least-squares problem and stopping on the sketched residual behave like the true residual, a property the paper assumes without proof and that the convection-diffusion experiment shows can fail at $\\eta=0.3$.","fun_headline_variants_meta":{"raw":{"variants":["Sketched GMRES cuts tensor cost with randomized rounding","Faster tensor GMRES via short recurrences and sketching","Two randomized Tucker GMRES variants beat full orthogonalization","Tensor linear solves sped up by low-rank sketching","Randomized Tucker GMRES: cheaper, streaming, self-regularizing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000646,"raw_usage":{"total_tokens":3013,"prompt_tokens":1036,"completion_tokens":1977,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":652,"completion_tokens_details":{"reasoning_tokens":1892}},"tokens_in":652,"tokens_out":1977,"duration_ms":16812,"temperature":1.0,"reasoning_tokens":1892,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T10:27:55.707727+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run either proposed solver on the paper's unpreconditioned 3D convection-diffusion benchmark with the orthogonalization window $k_{\\mathrm{trunc}}=1$ and the sketch-correction factor $\\eta=0.3$; the reported figure already shows true-residual stagnation for RHOSVD-Tucker sGMRES. If increasing the sketch size $s$ while keeping the window fixed still cannot drive the true residual below the prescribed tolerance, then the structured sketch is not a reliable stand-in for the true residual in exactly the nonsymmetric regime the method targets.","supporting_citations":[{"cited_title":"Efficient sketching-based summa- tion of Tucker tensors, 2026.arXiv:2603.13532","cited_arxiv_id":null,"evidence_quote":"It supplies the Khatri-Rao randomized tensor summation that RHOSVD-Tucker sGMRES inherits and whose baseline RHOSVD-Tucker GMRES is the main comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the multilinear Nyström approximation that defines MLN-Tucker sGMRES and its fixed-rank streaming compression."},{"cited_title":"Subspace gradient de- scent method for linear tensor equations, 2026.arXiv:2602.21974","cited_arxiv_id":null,"evidence_quote":"It provides the symmetric Tucker subspace-gradient baseline that the Poisson benchmark compares against."},{"cited_title":"On Krylov projection methods and Tikhonov regularization.Electronic Transactions on Numerical Analy- sis, 44:83–123, 2015","cited_arxiv_id":null,"evidence_quote":"It provides the hybrid projection and Tikhonov regularization framework used inside the projected least-squares problem."}],"review_version":1}