{"id":"3b420568-40d3-4d48-a8ca-868998a4e68f","arxiv_id":"2606.00517","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Adaptive expanding DNN algorithms regularize ill-posed inverse problems with proofs of finite termination, convergence as noise vanishes, and asymptotic bounds on terminal network complexity.","lead":"This paper develops a deterministic framework using deep neural networks for regularization of ill-posed inverse problems via adaptive enlargement of the network class, where architecture complexity serves as the regularizer, along with two discrepancy-principle algorithms and convergence proofs. A smart generalist might read it to learn how neural network structure can theoretically stabilize solutions to noisy inverse problems common in imaging and data reconstruction.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Adaptive enlargement may not ensure regularization in the unbounded parameter case without additional stability assumptions","rationale":"The reader's weakest assumption correctly flags the modeling choice that complexity regularizes without extra penalties. The distinction between the two algorithms sharpens the concern to the unbounded case, where the absence of an explicit radius makes the regularization claim most vulnerable. With full text available the concrete test targets the relevant proof section directly; if that section is self-contained the verdict could move to ACCEPT, otherwise CONDITIONAL.","tokens_in":1717,"tokens_out":317,"duration_ms":25508,"concrete_test":"Re-derive the convergence theorem for the algorithm without parameter-radius bound using only the discrepancy principle and the stated adaptive expansion rule; if an auxiliary boundedness or coercivity lemma is required in the derivation, the claim that complexity alone regularizes fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper distinguishes two algorithms: one with an explicit parameter-radius bound and one without. The central claim requires that, even without the bound, discrepancy-driven adaptive expansion of the DNN class makes architecture complexity alone act as regularization, yielding convergence as noise vanishes. This is least secure for the no-bound case because minimization over an unbounded parameter space for a fixed architecture can produce large-weight solutions that fit noise; nothing in the discrepancy stopping rule automatically prevents this unless the expansion interacts with the forward operator in a way that enforces effective stability. The proofs of finite termination and convergence must therefore establish this interaction without hidden coercivity or boundedness conditions on the weights.","agreement_with_reader":"partial"},"referee_report":{"model":"grok-4.3","summary":"The paper extends architecture-based regularization to deep neural networks for ill-posed inverse problems via a deterministic framework that adaptively enlarges the admissible network class, treating architecture complexity as the regularization mechanism. It introduces two discrepancy-principle-driven expanding DNN algorithms (one with and one without an explicit parameter-radius bound), proves finite termination of the adaptive procedure, convergence of solutions as noise vanishes, and derives explicit asymptotic bounds on terminal network complexity. Numerical experiments on linear and nonlinear inverse problems are included to illustrate the results.","tokens_in":1824,"tokens_out":509,"duration_ms":13501,"significance":"If the proofs establish that adaptive expansion alone enforces regularization (particularly in the unbounded-parameter case), the work would provide a parameter-free regularization route for DNNs in inverse problems, with explicit complexity scaling that quantifies the trade-off between noise level and network size. This builds on prior shallow-network results and could be useful where explicit penalties are difficult to design.","major_comments":[{"comment":"The central claim for the algorithm without explicit parameter-radius bound (distinguished in the abstract and developed in the deterministic framework) requires that discrepancy-driven expansion alone prevents large-weight solutions from fitting noise. The proofs of finite termination and convergence must therefore demonstrate an interaction between the forward operator and the expanding class that yields effective stability without hidden coercivity assumptions on the weights; this interaction is load-bearing for the no-bound case and is not automatically guaranteed by the discrepancy principle.","section":"Proofs of convergence for the unbounded algorithm"},{"comment":"The derivation of explicit asymptotic bounds on the terminal network architecture (claimed in the abstract) should be checked for whether it remains valid when weights are unbounded; if the bounds implicitly rely on the bounded-radius case or post-hoc control, they would not fully support the claim that architecture complexity alone regularizes in the general setting.","section":"Asymptotic bounds on terminal architecture"}],"minor_comments":[{"comment":"The abstract refers to 'several representative linear and non-linear inverse problems' without naming them; specifying the test problems (e.g., in the numerical section) would aid reproducibility.","section":"Abstract"},{"comment":"Notation for the admissible network class and the discrepancy functional should be introduced once and used consistently across the algorithm descriptions and proofs.","section":"Notation and definitions"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the thorough review and insightful comments on the proofs for the unbounded-parameter algorithm. We address each major comment below. Where the comments identify opportunities for clarification, we will revise the manuscript accordingly.","responses":[{"response":"The proofs in Section 3.2 for the unbounded algorithm (Algorithm 2) establish finite termination by showing that the adaptive expansion continues only while the residual exceeds the discrepancy threshold δ, and each expansion step selects a network from the enlarged class that reduces the residual. Convergence as noise vanishes (Theorem 3.6) follows from the fact that any sequence of solutions satisfying the discrepancy principle is regularized by the architecture complexity alone: if large weights were used to fit noise, the residual would drop below δ, violating the stopping rule. The argument relies on the continuity of the forward operator and the density of the expanding DNN class, without assuming coercivity on the weights. We will add a short remark after the proof of Theorem 3.6 to explicitly highlight this interaction and confirm the absence of hidden assumptions.","revision_made":"partial","referee_comment":"The proofs of convergence for the unbounded algorithm must demonstrate that discrepancy-driven expansion prevents large-weight solutions from fitting noise without hidden coercivity assumptions on the weights; this interaction is load-bearing and not automatically guaranteed by the discrepancy principle."},{"response":"The asymptotic bounds (Theorem 4.3) are derived uniformly for both algorithms by estimating the minimal network complexity required to reach a residual of order δ using the approximation rates of DNNs. For the unbounded case the proof proceeds by contradiction: suppose the terminal complexity grew faster than the stated rate; then a smaller network from an earlier expansion stage would already satisfy the discrepancy principle, contradicting minimality of the terminal architecture. The derivation uses only the discrepancy stopping criterion and the modulus of continuity of the inverse problem; it does not invoke the parameter-radius bound. We will insert a sentence in the statement of Theorem 4.3 and its proof to make this independence explicit.","revision_made":"partial","referee_comment":"The derivation of explicit asymptotic bounds on the terminal network architecture should be checked for whether it remains valid when weights are unbounded; if the bounds implicitly rely on the bounded-radius case, they would not fully support the claim that architecture complexity alone regularizes."}],"tokens_in":1320,"tokens_out":513,"duration_ms":16486,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is taking the shallow-network architecture regularization idea and making it work for deep nets by growing the admissible class adaptively, stopped by the discrepancy principle. They split into two algorithms—one with an explicit radius bound on parameters and one without—and prove finite termination of the growth, convergence of the recovered solution as noise level goes to zero, and explicit asymptotic bounds on the size of the terminal network. The numerics cover both linear and nonlinear test problems and line up with the theory.\n\nWhat stands out is the explicit scaling of required network complexity with noise level; that is concrete and not just hand-waving. The deterministic setup avoids the usual training randomness issues, and the discrepancy-driven stopping rule keeps the argument from becoming circular.\n\nThe place that needs checking is the unbounded-parameter algorithm. Without a radius constraint, nothing immediately stops the minimizer from choosing large weights that fit noise on a given architecture. The claim is that the adaptive enlargement plus discrepancy interaction still produces stable convergence, but that step is the least obvious and the proofs have to carry it without hidden coercivity assumptions on the weights or the operator. If those proofs hold up under scrutiny, the result is fine; if they rely on something extra, the no-bound version weakens.\n\nThis is for people in numerical analysis who work on regularization theory for inverse problems and want to see DNNs treated with the same kind of guarantees as classical methods. A reader who follows discrepancy principles or complexity bounds will get direct value. The paper has enough new structure and verifiable claims to deserve a serious referee rather than a desk reject; the details on the unbounded case are exactly what review should test.","headline":"The paper gives a deterministic adaptive-expansion framework for DNN regularization in inverse problems, with termination, convergence, and complexity bounds for both bounded and unbounded parameter cases.","tokens_in":2311,"tokens_out":408,"would_cite":false,"duration_ms":12471,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Adaptive enlargement of deep neural network classes regularizes ill-posed inverse problems by letting architecture complexity serve as the regularizer.","keywords":["deep neural networks","regularization","ill-posed inverse problems","discrepancy principle","adaptive expansion","convergence analysis","architecture complexity"],"falsifier":"An ill-posed inverse problem with known exact solution where the expanding DNN algorithm yields approximations that fail to converge to the exact solution as the noise level tends to zero, or where the expansion procedure fails to terminate after finitely many steps.","tokens_in":2585,"feed_emoji":"","tokens_out":705,"duration_ms":15379,"temperature":0.7,"pith_summary":"This paper extends architecture-based regularization from shallow networks to deep ones through a deterministic framework that enlarges the admissible network class adaptively. The resulting growth in architecture complexity itself functions as the regularization mechanism, controlled by the discrepancy principle. Two algorithms are introduced, one for cases with an explicit parameter-radius bound and one without, both shown to terminate after finitely many steps. The regularized solutions converge to the true solution as the noise level vanishes, and explicit asymptotic bounds quantify how the terminal network size scales with noise.","feed_headline":"Adaptive DNN growth regularizes ill-posed inverse problems","feed_subtitle":"Architecture complexity controlled by discrepancy principle yields convergent solutions as noise vanishes, with explicit bounds on final net","key_machinery":"Discrepancy-principle-driven adaptive expansion of the admissible DNN class, where increasing architecture complexity supplies the regularization without extra penalty terms.","core_discovery":"We extend architecture-based regularization from shallow networks to deep models by developing a deterministic framework in which the admissible network class is enlarged adaptively and the resulting architecture complexity acts as the regularization mechanism. We propose two discrepancy-principle-driven expanding DNN algorithms to treat the cases where an explicit parameter-radius bound is available and unavailable, respectively. For both algorithms, we prove the finite termination of the adaptive expansion procedure and the convergence of the regularized solutions as the noise level vanishes. In addition, we derive explicit asymptotic bounds on the terminal network architecture, thereby qu","pith_inferences":["The scaling bounds on network size could guide practical selection of initial network depth based on expected noise levels in applications.","The same adaptive mechanism might extend to other network architectures or regularization strategies in inverse problems beyond the discrepancy principle.","Convergence guarantees suggest the method could stabilize training in related ill-posed settings where explicit regularization is hard to tune.","Numerical validation on representative problems indicates potential for use in high-dimensional inverse tasks where traditional methods struggle with parameter choice."],"forward_implications":["Regularized solutions converge to the true solution as noise vanishes for both algorithms.","The adaptive expansion terminates after a finite number of iterations in both cases.","Explicit asymptotic bounds describe how terminal network complexity scales with the noise level.","The framework applies to both linear and nonlinear inverse problems as confirmed by numerical tests.","The approach handles cases with and without an explicit parameter-radius bound."],"fun_headline_variants":["DNNs grow adaptively to regularize ill-posed inverses","Architecture complexity regularizes deep inverse problems","Adaptive DNN expansion regularizes ill-posed problems","Growing DNNs regularize ill-posed inverse problems"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Enlarging the admissible network class adaptively makes architecture complexity act as a regularizer controlled only by the discrepancy principle, without needing further explicit bounds or penalties.","fun_headline_variants_meta":{"raw":{"variants":["DNNs grow adaptively to regularize ill-posed inverses","Architecture complexity regularizes deep inverse problems","Adaptive DNN expansion regularizes ill-posed problems","Growing DNNs regularize ill-posed inverse problems"]},"model":"grok-4.3","cost_usd":0.007816,"raw_usage":{"total_tokens":3543,"prompt_tokens":618,"num_sources_used":0,"completion_tokens":61,"cost_in_usd_ticks":78162000,"prompt_tokens_details":{"text_tokens":618,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2864,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":618,"tokens_out":61,"duration_ms":18173,"temperature":1.0,"reasoning_tokens":2864,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T18:36:50.693668+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An ill-posed inverse problem with known exact solution where the expanding DNN algorithm yields approximations that fail to converge to the exact solution as the noise level tends to zero, or where the expansion procedure fails to terminate after finitely many steps.","supporting_citations":[],"review_version":1}