{"id":"bd9a552a-e3be-4ebf-91fc-831b872ff2e6","arxiv_id":"2505.22641","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A spectral method for CoxPH survival analysis that computes hazard scores via a Markov chain steady state, enabling deep models to train on ultra-high-dimensional data.","lead":"This paper adapts a spectral method from ranking regression to speed up Cox proportional hazard survival analysis, enabling deep models to train on high-dimensional data like full CT scans. If it works, it could make survival predictions in medicine and advertising faster and more accurate without expensive dimensionality reduction.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"App. G replaces Δ_i=1 for censored samples in the implementation, but Theorem 1 and Eq. (8) analyze the true event indicators; the experiments therefore do not validate the claim that Spectral solves the CoxPH/DeepSurv objective.","rationale":"The paper's central claim is that Spectral decouples and solves the weighted CoxPH/DeepSurv negative log-likelihood Eq. (8); Theorem 1 is the theoretical load-bearing result. The implementation section App. G explicitly changes the event indicators to 1 for all samples. Since Δ_i appears multiplicatively in the objective and in the Markov-chain transition rates that Theorem 1 derives, the executed algorithm solves a different optimization problem. The reader's weakest-assumption identifies exactly this gap, and I agree. I also note that the memory savings and scalability measurements in Tables 3/10 are genuine engineering achievements and are not called into question by this objection; the issue is with the claim that the method is a spectral solver for the Cox objective. The paper itself flags the Δ_i=1 choice as an implementation note, which is good transparency, but it does not analyze it. No ad hominem is intended; the critique is on the internal consistency between Sec. 4 and App. G. Because the central claim is unsupported by the execution, the reader's REJECT verdict remains appropriate; the method could become publishable in revised form either by proving the modified loss is an acceptable surrogate or by demonstrating that the true-Δ variant works.","tokens_in":34772,"tokens_out":8272,"duration_ms":89554,"concrete_test":"Run the released code with a single change: pass the true Δ_i (0 for censored, 1 for events) into the IterativeSpectralRanking updates in Eqs. (12)-(13) instead of the current all-ones assignment, and compare the final intrinsic scores, learned θ, and test C-index on DBCD and DLBCL against both the default all-ones run and a full-batch CoxPH/DeepSurv fit of Eq. (8). If the true-Δ variant collapses (censored scores near zero, CI drop) or otherwise differs materially, the reported results are produced by the modified loss, not the Cox partial likelihood; if it reproduces the reported performance, the all-ones assignment can be presented as an analyzed surrogate rather than a silent change.","verdict_should_be":"UNCHANGED","load_bearing_attack":"App. G states: 'we set Δ_i=1 for all samples and treat it as regularization that keeps the intrinsic scores away from 0.' In the objective Eq. (8), Δ_i is precisely the event indicator: it multiplies each Cox/DeepSurv partial-likelihood term, and Theorem 1's transition rates in Eqs. (12)-(13) inherit it through the factors Δ_ℓ in μ_ji. Setting Δ_i=1 for censored samples converts the objective from the Cox partial likelihood into a different loss in which censored observations contribute numerator log-terms as though they were events. The theoretical derivation in Sec. 4 and App. E does not model this replacement, so Algorithm 1 as implemented does not compute stationary points of Eq. (8) and the empirical results in Tables 2-7 do not support the paper's central claim. This is not a minor numerical detail: the modification is structural, and the paper itself flags it only as an implementation note in App. G. A revised version would need either to prove properties of the modified loss (e.g., that its stationary points coincide with, or bound, those of Eq. (8)) or to present experiments with the true censoring indicators.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes Spectral Survival Analysis, an ADMM-based method that adapts spectral ranking regression to the Cox proportional hazards partial likelihood and to DeepSurv. The central claim is that the intrinsic scores obtained in the π-update are the stationary distribution of a continuous-time Markov chain, so that the CoxPH/DeepSurv loss can be decoupled from network fitting and solved by spectral iterations plus standard regression. The paper derives a weighted extension, applies it to several CoxPH variants and counting processes, and reports experiments on eight datasets including the full 3D CT-scan LUNG1 dataset, where it claims superior predictive performance and memory usage relative to existing deep survival methods.","tokens_in":34988,"tokens_out":9743,"duration_ms":113363,"significance":"The connection between the Cox partial likelihood and Plackett-Luce ranking is a genuine and interesting observation, and the reported ability to train on full LUNG1 CT scans with modest memory is striking. The paper ships public code and includes a substantial experimental comparison. However, the central claim is not supported as stated, because the implementation changes the objective by setting Δ_i=1 for all samples in Appendix G, and the theoretical result is not self-contained. If the objective mismatch can be resolved, the spectral approach could be a valuable contribution to scalable survival analysis; as it stands, the empirical results do not validate the claim that the method solves the stated CoxPH/DeepSurv objective.","major_comments":[{"comment":"Appendix G states 'we set Δ_i=1 for all samples and treat it as regularization that keeps the intrinsic scores away from 0.' In the objective (8), Δ_i is the event indicator: it multiplies every numerator log-term, and Theorem 1's transition rates in Eq. (13b) inherit it through the factors Δ_ℓ in μ_ji. Setting Δ_i=1 for censored samples converts the Cox partial likelihood into a different loss in which censored observations contribute numerator log-terms as though they were events. Therefore Algorithm 1 as executed does not compute stationary points of Eq. (8), and the experimental results in Tables 2-7 do not validate the paper's central claim. A revision would need either to prove properties of the modified loss (e.g., that its stationary points coincide with, or bound, those of Eq. (8)) or to present experiments with the true censoring indicators.","section":"App. G and Eq. (8)"},{"comment":"The proof of Theorem 1 in App. E derives the balance equations only for the simplified problem in Eq. (45), and then invokes 'Theorem 4.2 from [81]' without stating that theorem or showing how it applies. Since Theorem 1 is the paper's main theoretical contribution, the proof should be self-contained or should include a full statement of the cited theorem. In addition, Eq. (13b) defines the denominator as sum_{t∈R_ℓ} W_{t,i} π_t, but the derivative computation in Eq. (42) and the rearrangement in Eqs. (46)-(48) give sum_{t∈R_ℓ} W_{t,ℓ} π_t; as written, the transition rates are incorrect.","section":"App. E and Theorem 1"},{"comment":"The complexity analysis states that Algorithm 1 has memory complexity O(n^2)+O(PB), but Table 10 reports Spectral using only 1424 MB on MovieLens with n=100,000. A dense n×n transition matrix would require tens of gigabytes. The implementation must therefore exploit a sparse or implicit representation of the transition matrix, but this is not described in Sec. 4.3 or App. G. The O(n^2) memory claim is misleading as a scalability argument for large n, and the reported memory numbers cannot be reconciled with the stated complexity.","section":"Sec. 4.3 and Table 10"}],"minor_comments":[{"comment":"There are numerous typographical errors, including 'staionary' in Sec. 1, 'Ror completeness' in Sec. 2, 'inlcuding' in Sec. 2, 'compareable' in Sec. 5.1, 'textsc-Spectral' in Sec. 5.2, and 'event' for 'even' in the Table 4 caption.","section":"Throughout"},{"comment":"Reference [62] is a self-citation with a placeholder arXiv identifier (arXiv:XXXX.XXXXX); it should be removed or replaced with a proper citation.","section":"References"},{"comment":"The notation Δ_ji(π) in Eq. (12b) is easily confused with the event indicator Δ_i; renaming the transition-rate correction term would improve readability.","section":"Eq. (12b)"},{"comment":"The statement that direct optimization of the partial likelihood forces censored samples to have intrinsic score 0 is the motivation for the Δ_i=1 deviation; this claim should be made precise and analyzed formally, since it is the basis for a structural change to the objective.","section":"App. G"},{"comment":"The caption of Fig. 3 refers to 'ADS100K' but the corresponding dataset in Table 1 and Table 2 is ADS10k; please check and make the notation consistent.","section":"Fig. 3 caption"}],"recommendation":"reject","confidential_remarks":"The paper relies heavily on the authors' own prior spectral ranking work, and the cited Theorem 4.2 is used without a statement; this is not inappropriate per se, but it makes the main proof non-self-contained. The Δ_i=1 modification is acknowledged only in an appendix, yet it changes the objective that is actually optimized; this discrepancy is serious enough that the current experimental contribution cannot be taken as evidence for the stated method. If the authors can either justify the modified loss or rerun the experiments with true censoring indicators, a resubmission may be worth considering."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has a genuinely useful idea and the best scalability results I've seen for deep Cox models on high-dimensional data, but the central claim that the spectral update solves the CoxPH/DeepSurv objective is not supported by the implementation. App. G sets Δ_i=1 for all samples, silently converting the loss into one where censored samples are treated as events. That's a load-bearing flaw, but it is fixable, and the empirical contribution is real.\n\nWhat's actually new: adapting spectral ranking to censored, weighted survival data is nontrivial—weights enter the Plackett–Luce transition rates, and censoring is handled through the risk-set structure. The ADMM decomposition into a spectral score update and a per-sample regression step is clean and, if it works, removes the siamese-network memory bottleneck of partial likelihood fitting. On LUNG1 they run a 6-layer 3D CNN on full CT scans in 15GB versus >80GB for baselines, with better concordance. That's a practical win regardless of the theoretical claim.\n\nThe soft spots: the Δ_i=1 replacement is the big one. Eq. (8), Theorem 1, and App. E all treat Δ_i as the true event indicator; the transition rates in Eq. (13) inherit it. Setting it to 1 for censored samples changes the objective, so the experiments do not validate the claim of solving the Cox partial likelihood. The authors call it regularization, but it is structural and unanalyzed. A revision could fix this by proving that the modified loss has stationary points related to the original, or by rerunning with true censoring indicators to see whether the method still works. Minor issues: Theorem 1's proof relies on prior Theorem 4.2 without a self-contained derivation, and Eq. (12a) has a notational clash with Δ_j i. On the plus side, code is public, experiments are extensive, and hyperparameter sensitivity is reported.\n\nWho this is for: researchers working on scalable survival analysis or deep Cox variants. The paper deserves a serious referee; a good reviewer could push the authors to either correct the claims or analyze the actual loss being optimized. I would not desk-reject it.","headline":"Promising spectral scaling for deep Cox models, but the App. G Δ_i=1 choice means the experiments don't validate the claim that the method optimizes the Cox partial likelihood.","tokens_in":35533,"tokens_out":3172,"would_cite":false,"duration_ms":34336,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62N01","60J27","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Spectral rank-regression methods reach stationary points of CoxPH and DeepSurv losses: a Markov-chain stationary distribution yields the per-sample scores, and the network is fit to those scores by regression.","keywords":["survival analysis","Cox proportional hazards","DeepSurv","spectral methods","rank regression","Plackett-Luce model","high-dimensional data","censoring"],"falsifier":"A deciding experiment: take a small, heavily censored dataset, run the spectral method to convergence, and evaluate the gradient of the true Cox negative log-partial-likelihood (Eq. 4, with genuine indicators $\\Delta_i$) at the returned scores and parameters; if that gradient norm is far from zero, the method is converging to a different objective than the one claimed. A sharper check compares the spectral solution with the exact minimizer of Eq. 5 on the same data to see whether any concordance gap is explained by the censoring surrogate.","tokens_in":34537,"feed_emoji":"🕒","tokens_out":18930,"duration_ms":177026,"temperature":0.7,"pith_summary":"Survival models such as CoxPH and its deep variant DeepSurv (a neural-network Cox model) are hard to scale because their partial likelihood couples every sample with every other at-risk sample, so each training step must embed whole risk sets in memory. This paper claims that the Cox partial likelihood is essentially a Plackett-Luce ranking likelihood — the standard model for learning from ranked choices — and that spectral methods developed for rank regression therefore transfer to survival analysis. The authors show that per-sample intrinsic scores can be computed as the stationary distribution of a continuous-time Markov chain, with the neural network subsequently fit to those scores by plain regression, converting likelihood optimization into a regression problem. On eight datasets, including full-resolution CT scans with roughly 17 million features per sample, the method reports better concordance than six competing survival models while using less memory, and it is the only tested method that can process the CT scans in full.","feed_headline":"A Markov-chain theorem recasts Cox survival fitting as regression","feed_subtitle":"It decouples likelihood optimization from the neural net, so deep survival models can run on full 17M-feature CT scans.","key_machinery":"The carrying mechanism is Theorem 1: the stationary point of the intrinsic-score subproblem satisfies the balance equations (flow into each state equals flow out) of a continuous-time Markov chain with transition rates $\\boldsymbol{P}_{ji}(\\boldsymbol{\\pi}) = \\mu_{ji} + \\Delta_{ji}(\\boldsymbol{\\pi})$, where $\\mu_{ji}$ sums weighted at-risk contributions over the winner/loser sets induced by the observed event order, and $\\Delta_{ji}$ adds the KL-divergence and dual corrections that keep the chain recurrent. This turns the score update into a spectral computation — the power method applied to the chain's transition matrix — and leaves the parameter update in Eq. (11b) as a max-entropy regression of the network onto the scores. Because the transition matrix depends on the scores themselves, the procedure iterates between computing the stationary distribution and updating the rates, with the weights $\\boldsymbol{W}_{ji}$ carrying the generality to CoxPH variants including DeepSurv.","core_discovery":"On the paper's own terms, the central claim is that minimizing the CoxPH and DeepSurv negative log-partial-likelihood can be decoupled from fitting the neural network. The objective is reformulated with auxiliary per-sample intrinsic scores $\\boldsymbol{\\pi}_i$ — quantities proportional to the probability that sample $i$ experiences an event before other at-risk samples — and solved by ADMM. Theorem 1 proves that the score-update subproblem has a stationary point satisfying the balance equations of a continuous-time Markov chain with transition rates $\\boldsymbol{P}_{ji}(\\boldsymbol{\\pi}) = \\mu_{ji} + \\Delta_{ji}(\\boldsymbol{\\pi})$, where $\\mu_{ji}$ collects weighted contributions from the risk sets and $\\Delta_{ji}$ is a corrective term from the KL-divergence and dual constraints. The scores can therefore be computed by power iteration on the chain, while the network update reduces to a max-entropy regression against those scores. The introduction of weights $\\boldsymbol{W}_{ji}$ is the technical departure from prior spectral ranking work, and it is what lets the same theorem cover Weighted CoxPH, heterogeneous Cox models, Deep Heterogeneous Hazard models, the Accelerated Failure Time model, and counting-process arrival rates, in addition to plain CoxPH and DeepSurv.","pith_inferences":["Extension: because the Cox partial likelihood is the Plackett-Luce ranking likelihood in survival disguise, other exchangeable ranking models (Bradley-Terry, Mallows, generalized Plackett-Luce) may admit the same Markov-chain spectral treatment for censored or partially observed orders.","Extension: the alternating scheme used for DHH and AFT — fix baseline hazards as weights, solve the weighted spectral problem, re-estimate baselines with a Breslow estimator — is a general template, so any hazard model whose likelihood becomes a weighted Cox loss once baseline rates are fixed could reuse the solver.","Extension: the intrinsic scores are per-sample quantities proportional to relative event probability, which suggests testing whether they transfer across datasets or tasks, for instance as initialization or pretraining signals when outcome data are sparse."],"forward_implications":["Deep Cox models can be trained in full batch on datasets that previously forced subsampling or patch extraction, because the memory bottleneck moves from the Siamese partial likelihood to score computation plus a standard regression fit.","The mini-batch bias in partial-likelihood training, quantified in the paper's Lemma 1 and Corollary 1, is sidestepped: the score step consumes the full data, and the fitted loss is a decomposable regression loss.","Because the formulation is weighted, the same solver covers Weighted CoxPH, heterogeneous Cox models, Deep Heterogeneous Hazard models, the Accelerated Failure Time model, and counting-process arrival-rate regression.","On the LUNG1 CT dataset, competing methods exhaust 80 GB of memory even at 20% mini-batches, whereas the spectral method fits the entire scan in 15 GB and reports higher concordance, including against legacy methods that used hand-crafted patches and radiomic features."],"supporting_citations":[{"why":"Supplies the original spectral inference scheme for Plackett-Luce models — iteratively computing the stationary distribution of a Markov chain — that the paper transplants to survival analysis.","marker":"[52]"},{"why":"Deep Spectral Ranking, the ADMM-with-KL framework whose Markov-chain stationary-point theorem (B.1) Theorem 1 extends by adding censoring and weights.","marker":"[83]"},{"why":"Provides Theorem 4.2, cited in the proof of Theorem 1 to connect the simplified score equations to the stationary point of the full augmented Lagrangian.","marker":"[81]"},{"why":"DeepSurv, the deep neural-network Cox model whose negative log partial likelihood (Eq. 7) is the objective the spectral method scales to large datasets.","marker":"[39]"},{"why":"The Cox proportional-hazards partial likelihood (Eqs. 3-4), the semi-parametric objective the paper sets out to optimize.","marker":"[57]"},{"why":"ADMM, the alternating-direction multiplier framework (Eqs. 10-11) that separates the intrinsic-score step from the network-fitting step.","marker":"[5]"}],"fun_headline_variants":["Survival analysis gets a spectral speedup","Markov-chain scores decouple Cox from networks","Spectral method scales Cox to deep models","Cox fitting simplified via Markov chain trick","Speeding up survival models with spectral ranking"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the implementation's treatment of censored samples as events — setting $\\Delta_i = 1$ for all samples to keep censored intrinsic scores away from zero, as described in App. G — is a faithful stand-in for the true Cox partial likelihood; the theorem covers the unmodified objective, so if this surrogate shifts the optimum, the spectral method is not solving the loss the paper claims to solve.","fun_headline_variants_meta":{"raw":{"variants":["Survival analysis gets a spectral speedup","Markov-chain scores decouple Cox from networks","Spectral method scales Cox to deep models","Cox fitting simplified via Markov chain trick","Speeding up survival models with spectral ranking"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000148,"raw_usage":{"total_tokens":1180,"prompt_tokens":927,"completion_tokens":253,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":186}},"tokens_in":543,"tokens_out":253,"duration_ms":3846,"temperature":1.0,"reasoning_tokens":186,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:02:53.967416+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A deciding experiment: take a small, heavily censored dataset, run the spectral method to convergence, and evaluate the gradient of the true Cox negative log-partial-likelihood (Eq. 4, with genuine indicators $\\Delta_i$) at the returned scores and parameters; if that gradient norm is far from zero, the method is converging to a different objective than the one claimed. A sharper check compares the spectral solution with the exact minimizer of Eq. 5 on the same data to see whether any concordance gap is explained by the censoring surrogate.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the original spectral inference scheme for Plackett-Luce models — iteratively computing the stationary distribution of a Markov chain — that the paper transplants to survival analysis."},{"cited_title":"Peter Campbell, Michael F","cited_arxiv_id":null,"evidence_quote":"Deep Spectral Ranking, the ADMM-with-KL framework whose Markov-chain stationary-point theorem (B.1) Theorem 1 extends by adding censoring and weights."},{"cited_title":"Peter Campbell, Michael F","cited_arxiv_id":null,"evidence_quote":"Provides Theorem 4.2, cited in the proof of Theorem 1 to connect the simplified score equations to the stationary point of the full augmented Lagrangian."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"DeepSurv, the deep neural-network Cox model whose negative log partial likelihood (Eq. 7) is the objective the spectral method scales to large datasets."},{"cited_title":"Prentice","cited_arxiv_id":null,"evidence_quote":"The Cox proportional-hazards partial likelihood (Eqs. 3-4), the semi-parametric objective the paper sets out to optimize."}],"review_version":1}