{"id":"64e46e61-754a-43bb-ab3d-acefde0569cb","arxiv_id":"2504.13390","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Filtered least squares and ADMM both accelerate INR-based sparse-view CT reconstruction, with ADMM giving the lowest final error on a simulated breast phantom.","lead":"This paper tests two ways to speed up training of implicit neural representations that reconstruct CT images from sparse X-ray measurements: a filtered least-squares loss and an ADMM splitting algorithm. Both reduce image error much faster than standard least-squares training on a simulated breast CT phantom.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The ADMM speed-up claim rests on a projection-operator iteration count that omits the 50 INR Adam steps per outer iteration; wall-clock may not support the reported acceleration.","rationale":"The reader's weakest assumption—that ADMM's speed advantage may shrink under wall-clock accounting—is the right one to stress. The reported iteration count equates one Adam step for LS/FLS with one CGLS step for ADMM, and the 50 Adam INR updates per ADMM outer iteration are not reflected in the same way. Even if the total number of Adam updates and P/P^T products per 1000 counted iterations is similar, the plotted MSE curves for ADMM may not represent the convergence of the INR output q, and no wall-clock data are given. This directly undermines the paper's headline quantitative claim. I also note the FLS loss's DC nullspace as a secondary correctness concern, but the paper's FLS results are still plausible and the nullspace may be benign for this phantom. Because the issue is a missing comparison rather than a demonstrated failure, a CONDITIONAL verdict is appropriate. The authors can settle it with wall-clock and per-outer-iteration reporting. No change to the reader's verdict is recommended.","tokens_in":6439,"tokens_out":9292,"duration_ms":90370,"concrete_test":"Run all three methods for the three architectures on the same CPU/GPU and record wall-clock time (and total FLOPs if feasible) to reach target image MSEs, e.g., 1e-4 and 1e-5; additionally plot ADMM's q=E(f_theta) MSE at every outer iteration (20 points) instead of per CGLS step. If ADMM does not reach the target MSE faster than LS/FLS in wall-clock, or if its per-outer-iteration q MSE is substantially worse than the x-MSE plotted in Fig. 2, the acceleration claim for ADMM should be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that FLS and ADMM 'significantly accelerate' INR-based sparse-view CT. For ADMM, the comparison in Fig. 2 uses the footnote definition of one iteration as one P and P^T multiplication, i.e., one CGLS step. The algorithm in Sections 3.2 and 4, however, performs 50 CGLS steps for the x-update and 50 Adam steps on the INR for the theta-update per outer iteration. In 1000 counted 'iterations' the INR receives 1000 Adam updates whose cost (rasterization and backpropagation through a 512x512 network, e.g., the ReLU architecture with hundreds of Fourier features) is excluded from the count. The plotted MSE curves can therefore show progress of the x-subproblem (or an average that hides the fact that q=E(f_theta) changes only every 50 CGLS steps) rather than the actual INR reconstruction error. If the two approaches are compared at equal wall-clock time or equal total FLOPs, it is unclear whether ADMM retains its apparent MSE advantage. Since acceleration is the paper's main quantitative claim, the missing wall-clock/total-cost comparison is load-bearing. A secondary issue is that the FLS loss uses a ramp filter F with a zero at DC (Section 3.1), so a constant sinogram offset lies in the nullspace of the loss; this is not discussed and could bias reconstructions in settings where the DC component matters.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two modifications to accelerate the optimization of implicit neural representations (INRs) for sparse-view CT reconstruction: (1) a filtered least squares (FLS) loss that applies a ramp-filter preconditioning to the data fidelity term, and (2) an ADMM algorithm that splits the problem into a linear projection-domain subproblem and a pixel-space INR fitting subproblem. The authors analyze the conditioning of the linearized least-squares problem with a fixed INR feature matrix, reporting that FLS reduces the Gram matrix condition number by 1-3 orders of magnitude at initialization for ReLU, SIREN, and hash-encoding architectures. In experiments on one simulated sparse-view breast CT phantom with 128 views and Poisson noise, they show that FLS and ADMM substantially reduce image MSE over 1000 iterations compared with standard least-squares training, achieving reconstructions visually similar to a tuned TV-regularized baseline.","tokens_in":6719,"tokens_out":9466,"duration_ms":98755,"significance":"If the reported acceleration holds under a fair cost comparison, the paper makes a useful contribution: FLS is a simple, architecture-agnostic preconditioning idea that can be plugged into any INR-based CT reconstruction, and the ADMM formulation decouples the expensive projection operator from the INR training, which could enable reuse of fast INR fitting machinery. The linearized conditioning analysis is the strongest element, being theoretically motivated and empirically verified over 100 random initializations. The experimental protocol is also careful in avoiding the inverse crime by simulating projections on a 2048x2048 grid while reconstructing on 512x512, and the comparison against a tuned TV baseline provides a meaningful reference. The paper is clearly written and the three INR architectures give some breadth to the empirical claims.","major_comments":[{"comment":"The computational cost comparison is not balanced for ADMM. For LS/FLS, one counted iteration equals one full gradient step, i.e., one application of P and P^T plus one Adam update. For ADMM, one counted iteration is a single CGLS step of the x-subproblem; each outer ADMM iteration additionally performs 50 Adam updates of the INR, whose forward/backward cost through the rasterized network is not represented on the x-axis. Moreover, the figure caption does not state whether the plotted MSE is the current x_k from the x-update or the INR output q_k = E{f_{theta_k}}, which matters because theta is updated only after each block of 50 CGLS steps. The claim that ADMM gives the lowest final MSE after 1000 iterations should be re-evaluated at equal wall-clock time or equal total FLOPs; please provide a timing comparison and clarify exactly which quantity is plotted.","section":"Section 4, Fig. 2 and footnote"},{"comment":"All reconstruction MSE curves are obtained on a single synthetic breast CT phantom and a single noise realization, with no error bars or multiple seeds reported. The conditioning ratios in Table 1 are averaged over 100 initializations, but the central acceleration claim rests on the MSE curves, which are not. To support the conclusion that both approaches 'significantly accelerate' INR-based CT reconstruction, please report results over several noise realizations or phantoms, with statistical summaries (e.g., mean and standard deviation of final MSE or of MSE vs iteration).","section":"Section 4, Fig. 2"},{"comment":"The ramp filter F used in the FLS loss has a zero at DC, so the quadratic form in Eq. (4) is a seminorm: constant offsets in z-y are not penalized. The manuscript does not discuss this nullspace or its effect on the linearized condition number analysis in Section 3.1.1 and Table 1. Please analyze this issue explicitly; either modify F to include a small DC penalty, or provide a numerical demonstration that the nullspace does not bias reconstructions in the reported setting.","section":"Section 3.1, Eq. (4)"},{"comment":"The ADMM updates for the nonconvex constraint x = E{f_theta} are not guaranteed to converge; the paper cites residual monitoring from [11] but reports no primal/dual residuals or stopping criterion. Since the experiments fix 20 outer iterations and 50 inner Adam/CGLS steps, it is unclear whether the 1000-iteration results correspond to convergence of the algorithm or to an arbitrary cutoff. Please report convergence diagnostics or otherwise justify the stopping rule.","section":"Section 3.2"}],"minor_comments":[{"comment":"The title contains unwanted spacing ('ACCELERA TED OPTIMIZA TION', 'REPRESENTA TIONS'); please fix the typesetting.","section":"Title"},{"comment":"The expression f_theta(r) = PW i=1 ... appears to have a rendering artifact (the summation sign is missing); please ensure the equation is typeset correctly.","section":"Section 3.1.1, Eq. (6)"},{"comment":"The learning rates and ADMM penalty parameters are tuned separately per architecture, and the results may be sensitive to these choices; please add a brief sensitivity discussion or ablation.","section":"Section 4, hyperparameters"},{"comment":"Since the y-axis is log-scale, it would be helpful to mark the outer ADMM iterations or to use distinct symbols for ADMM so that the reader can see where the INR subproblem is updated.","section":"Figure 2"}],"recommendation":"major_revision","confidential_remarks":"This is a concise empirical study that is within the scope of the journal, but the fairness of the ADMM iteration count is essential to the central acceleration claim and must be addressed with a wall-clock or total-FLOP comparison. A small multi-seed or multi-phantom experiment would also materially strengthen the paper. I would not require new theory, but the current evidence is not sufficient to support the strongest claims as written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a useful, incremental study on making INR-based CT reconstruction faster. The new combination is real: putting an FBP-type preconditioner directly into the loss (FLS) and using ADMM to separate the projection subproblem from the INR fitting. Neither idea is brand new, but the pairing with three INR architectures is a sensible empirical contribution. The linearized conditioning analysis is the strongest part—clean, reproducible in spirit, and the condition-number ratios (1–3 orders of magnitude improvement at initialization) genuinely explain why FLS helps. That part deserves credit.\n\nThe ADMM side is where I have a real concern. The paper defines one iteration as one P/Pᵀ multiplication. That is fine for LS and FLS, where each gradient step costs exactly that. For ADMM, though, each outer iteration also runs 50 Adam steps on the INR in pixel space. Those steps are not free: they involve rasterizing the network and backpropagating through it. The plotted MSE curves therefore partly reflect progress in the x-subproblem, not the actual reconstruction cost. The stress-test note is right that this could overstate ADMM's advantage. That said, the Adam steps in ADMM do not require forward/backward projections, so each is cheaper than a full LS gradient step; the question is by how much. Without wall-clock times or total-FLOP comparisons, the acceleration claim is not fully supported. This is load-bearing for the paper's main quantitative message.\n\nOther soft spots are minor by comparison. The main MSE comparison is on one phantom, one noise realization, no error bars. That limits certainty but does not invalidate the trend. The FLS ramp filter has a DC nullspace; a constant sinogram offset is invisible to that loss. The paper does not discuss this, and while it may not matter much for the fan-beam geometry used here, it is worth acknowledging.\n\nThe baselines are fair: TV-regularized reconstruction with tuned regularization strength, FBP, and the inverse-crime avoidance is a good touch. The methods are described clearly enough to reproduce, though no code is released. The nonconvex ADMM convergence caveat is stated honestly.\n\nWho is this for? People working on INR-based inverse problems in CT or similar tomographic settings. It is not a breakthrough, but it is a practical recipe that likely works. I would send it to peer review, with the request that the authors report wall-clock time (or total FLOPs), repeat over multiple noise realizations, and address the FLS nullspace. Even with those fixes, the contribution remains modest—but it is an honest, useful one.","headline":"A solid, incremental optimization recipe for INR-based CT: the conditioning analysis is clean, but the ADMM speed-up claim hinges on an iteration count that omits the INR Adam steps, so wall-clock comparison is needed before believing the headline acceleration.","tokens_in":7236,"tokens_out":1494,"would_cite":true,"duration_ms":16995,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Two optimizer changes—a filtered least-squares loss and ADMM—speed INR-based CT reconstruction to TV-level quality.","keywords":["Implicit Neural Representations","Coordinate Based Neural Networks","CT Reconstruction","Model-based Iterative Reconstruction","filtered least squares","ADMM","sparse-view CT","preconditioning"],"falsifier":"Run the same sparse-view breast CT reconstructions to a fixed target image MSE on identical hardware while measuring wall-clock time and counting every Adam and CGLS inner iteration; if ADMM is not faster to target MSE than the filtered least-squares loss, the ADMM acceleration claim fails.","tokens_in":6228,"feed_emoji":"🩻","tokens_out":8612,"duration_ms":72952,"temperature":0.7,"pith_summary":"This paper tries to make implicit neural representation (INR) reconstruction for sparse-view CT practical by attacking the slow optimization. It proposes two fixes: a filtered least-squares loss that preconditions the ill-conditioned projection operator, and an ADMM algorithm that splits the problem into a linear least-squares step and an INR training step in pixel space. On a simulated sparse-view breast CT phantom, both fixes converge in hundreds of counted iterations rather than thousands, with image MSE comparable to a total-variation-regularized baseline. The claim matters because INR-based CT is unsupervised and does not need ground-truth images, but its training cost has so far been a barrier.","feed_headline":"Two optimizer changes speed neural CT reconstruction","feed_subtitle":"Filtered least-squares loss and ADMM match TV-level quality on sparse-view breast CT in far fewer iterations.","key_machinery":"The load-bearing objects are the filtered least-squares (FLS) loss and the nonlinear constrained ADMM split. FLS is the quadratic loss $L_{\\mathrm{FLS}}(z,y)=(z-y)^\\top F(z-y)$ with $F$ a positive-semidefinite ramp-filter convolution, equal to $\\|F^{1/2}(z-y)\\|^2$; it acts as a preconditioner on the inner matrix $P^\\top F P$ and improves the conditioning of $Q^\\top P^\\top F P Q$ relative to $Q^\\top P^\\top P Q$. The ADMM machinery rewrites the fitting problem as $\\min \\tfrac{1}{2}\\|P x-y\\|^2$ subject to $x=\\mathcal{E}\\{f_\\theta\\}$, then alternates a CGLS $x$-update, an INR $\\theta$-update trained in pixel space without forward/back projections, and a multiplier update. Inexact solves with 50 CGLS and 50 Adam steps keep each outer iteration cheap. A linearized analysis treats the INR feature matrix $Q$ as fixed and reduces the convergence-rate question to Gram matrix condition numbers, which is what the empirical condition-number ratios in Table 1 measure.","core_discovery":"The central claim is that the bottleneck in INR-based CT reconstruction is the ill-conditioning of the projection matrix, not the INR architecture. Even when the INR feature matrix $Q$ is well-conditioned, the Gram matrix $Q^\\top P^\\top P Q$ inherits ill-conditioning from $P^\\top P$. The paper proposes two cures: replacing the least-squares loss with filtered least squares, $L_{\\mathrm{FLS}}(z,y) = (z-y)^\\top F (z-y)$, where $F$ is a positive-semidefinite ramp-filter matrix from filtered back-projection, and solving the constrained problem $\\min_\\theta \\tfrac{1}{2}\\|P x-y\\|^2$ subject to $x = \\mathcal{E}\\{f_\\theta\\}$ by ADMM. At initialization the filtered loss reduces the Gram condition number by one to three orders of magnitude across ReLU, SIREN, and hash-encoding architectures. On a simulated 128-view sparse-view breast CT phantom, both FLS and ADMM produce reconstructions visually similar to a total-variation-regularized baseline with smaller or comparable MSE within 1000 counted iterations, whereas plain least-squares training is much slower and worse; ADMM achieves the lowest final MSE in all three architectures.","pith_inferences":["A natural next step is to benchmark wall-clock time with all inner ADMM Adam updates counted; the paper's counted-iteration convention may make the ADMM gain look larger than a fair timing would.","Combining FLS with ADMM—for example, using a filtered loss for the pixel-space INR subproblem—is a direct test that could compound the two accelerations.","The same conditioning argument should apply to other linear inverse problems where an FBP-like preconditioner exists, such as sparse-view MRI or deblurring, and the paper's approach could be ported there.","Since the constraint $x=\\mathcal{E}\\{f_\\theta\\}$ is nonlinear, ADMM convergence is not guaranteed; monitoring residuals is a stopgap, and local convergence analysis would settle when the method is safe to use."],"forward_implications":["FLS and ADMM bring INR-based sparse-view CT into the same iteration budget as classical iterative reconstruction: enough to match or beat a TV-regularized baseline in 1000 counted iterations.","The gain transfers across three structurally different INR architectures (ReLU with Fourier features, SIREN, and hash encoding), so the acceleration is not tied to one network design.","ADMM gives the lowest final image MSE in every architecture tested, while FLS gives the fastest early error reduction for two of the three.","Because the FLS improvement is a preconditioning of $P^\\top P$, it should stack with other conditioning fixes for the INR feature matrix $Q$.","The ADMM formulation moves INR training into pixel space, so each inner INR step avoids costly forward and back projections, which is the main source of its per-iteration efficiency."],"supporting_citations":[{"why":"Supplies the known result that gradient convergence for least squares depends on the Gram matrix condition number, motivating the FLS preconditioner.","marker":"[7]"},{"why":"Supplies the ADMM algorithm that the paper adapts to the constrained INR fitting problem.","marker":"[9]"},{"why":"Provides the CGLS method used to solve the ADMM x-update linear least-squares subproblem.","marker":"[10]"},{"why":"Source of the breast CT phantom that is modified for the simulated sparse-view experiments and used for ground truth.","marker":"[12]"},{"why":"Provides the primal-dual algorithm used to compute the TV-regularized baseline reconstruction.","marker":"[13]"},{"why":"Defines the Fourier feature embedding used in the ReLU INR architecture.","marker":"[14]"},{"why":"Defines the SIREN periodic-activation architecture.","marker":"[15]"},{"why":"Defines the multiresolution hash encoding architecture.","marker":"[16]"}],"fun_headline_variants":["Filtered loss and ADMM slash INR CT training iterations","Neural CT reconstruction gets faster with conditioned losses","ADMM and filtered loss accelerate INR-based CT","Conditioning fix speeds implicit neural CT reconstruction"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The speed comparison counts one iteration as a single matrix-vector product with $P$ and $P^\\top$; for LS and FLS that is one gradient step, but for ADMM it is one CGLS step, and the 50 Adam updates on the INR inside each ADMM outer iteration are not counted, so if wall-clock time is the real metric the ADMM speedup may be smaller than reported.","fun_headline_variants_meta":{"raw":{"variants":["Filtered loss and ADMM slash INR CT training iterations","Neural CT reconstruction gets faster with conditioned losses","ADMM and filtered loss accelerate INR-based CT","Conditioning fix speeds implicit neural CT reconstruction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000599,"raw_usage":{"total_tokens":2805,"prompt_tokens":955,"completion_tokens":1850,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":571,"completion_tokens_details":{"reasoning_tokens":1790}},"tokens_in":571,"tokens_out":1850,"duration_ms":13132,"temperature":1.0,"reasoning_tokens":1790,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T12:09:50.437241+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same sparse-view breast CT reconstructions to a fixed target image MSE on identical hardware while measuring wall-clock time and counting every Adam and CGLS inner iteration; if ADMM is not faster to target MSE than the filtered least-squares loss, the ADMM acceleration claim fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the known result that gradient convergence for least squares depends on the Gram matrix condition number, motivating the FLS preconditioner."},{"cited_title":"NeRP: implicit neural representation learning with prior embedding for sparsely sampled image reconstruction,","cited_arxiv_id":null,"evidence_quote":"Supplies the ADMM algorithm that the paper adapts to the constrained INR fitting problem."},{"cited_title":"PINER: Prior- informed implicit neural representation learning for test- time adaptation in sparse-view CT reconstruction,","cited_arxiv_id":null,"evidence_quote":"Provides the CGLS method used to solve the ADMM x-update linear least-squares subproblem."},{"cited_title":"An attenuation field network for dedicated cone beam breast CT with short scan and offset detector ge- ometry,","cited_arxiv_id":null,"evidence_quote":"Source of the breast CT phantom that is modified for the simulated sparse-view experiments and used for ground truth."},{"cited_title":"A fast regularized iterative algorithm for fan-beam CT reconstruction,","cited_arxiv_id":null,"evidence_quote":"Provides the primal-dual algorithm used to compute the TV-regularized baseline reconstruction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Fourier feature embedding used in the ReLU INR architecture."},{"cited_title":"Distributed optimization and statistical learning via the alternating direction method of multipliers,","cited_arxiv_id":null,"evidence_quote":"Defines the multiresolution hash encoding architecture."}],"review_version":1}