{"id":"a796f874-f992-4af9-97c2-698a8fcc45b3","arxiv_id":"2508.02111","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper introduces a well-posed invertible 1x1 convolution that eliminates random sampling in reversible image conversion, achieving state-of-the-art results on multiple tasks.","lead":"A new type of invertible network removes the random variables that made reversible image conversion ill-posed, replacing them with a well-posed overdetermined system. If it works, it could become a standard tool for image hiding, rescaling, and decolorization.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Gram determinant non-zero is asserted, but a well-posed inverse requires a uniform lower bound on singular values; without it near-singular inputs break stability.","rationale":"The reader's weakest assumption is exactly the non-zero Gram determinant condition. My stress-test refines it: even if non-zero, a near-zero determinant leads to ill-conditioning, so a simple non-zero determinant is insufficient for well-posedness. Since the abstract provides no lower bound or proof, the central well-posedness claim is not established from the available text. This is a concern about unverified support rather than a demonstrated contradiction, so the verdict remains UNVERDICTED. The proposed concrete test would resolve the concern by inspecting the actual singular value spectrum of the trained model.","tokens_in":708,"tokens_out":3777,"duration_ms":45446,"concrete_test":"Train WIC on the image rescaling task and compute the minimum singular value of the per-pixel Gram matrix (or the full Jacobian) across the held-out test set. If any input yields a condition number above a threshold (e.g., 10^6) or a Gram determinant within floating-point epsilon of zero, the well-posedness guarantee fails. Alternatively, analytically derive a lower bound on the smallest eigenvalue of the Gram matrix for the WIC architecture; if no such bound exists, the claim must be weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim depends on the assertion that constructing an overdetermined system with a non-zero Gram determinant guarantees a well-posed approximate left inverse. This is not sufficient. For a linear system Ax=b with A∈R^{m×n}, m>n, non-zero det(A^T A) ensures a unique least-squares solution, but Hadamard well-posedness also requires stability, i.e., a bounded condition number. If det(A^T A) can be arbitrarily close to zero, the left inverse (A^T A)^{-1}A^T amplifies noise without bound. The abstract provides no proof or architectural mechanism ensuring a uniform lower bound on the smallest singular value of the forward operator across the data distribution. Since WIC is a 1×1 convolution embedded in a nonlinear network, the Gram determinant depends on input activations; even if it is non-zero on every training example, it may be near-zero for out-of-distribution or adversarial inputs. Thus the well-posedness claim is under-supported, and the paper's central contribution rests on an unverified numerical condition.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper addresses ill-posedness in reversible image conversion (RIC). It argues that existing invertible-neural-network methods are intrinsically ill-posed because they introduce uncertainty through random variable sampling, and it proposes instead to construct an overdetermined system whose Gram determinant is non-zero, which the authors claim ensures a well-posed approximate left inverse. Based on this principle, the paper introduces a well-posed invertible 1x1 convolution (WIC) and two network architectures, WIN-Naive and WIN, and reports state-of-the-art results on reversible image hiding, image rescaling, and image decolorization. The abstract promises code availability. The full text was not provided in the review packet, so this report is necessarily based on the abstract alone.","tokens_in":916,"tokens_out":2699,"duration_ms":33336,"significance":"If the well-posedness guarantee can be established with a quantitative stability bound, the contribution is significant: it would remove the random variable sampling that complicates existing RIC methods, provide a deterministic invertible architecture, and potentially set a new benchmark across several image tasks. The promise of released code is a strength that will facilitate reproducibility and direct comparison. However, the core mathematical claim in the abstract is under-specified: a non-zero Gram determinant by itself does not ensure Hadamard well-posedness without a uniform lower bound on the smallest singular value. The significance of the work therefore hinges on whether the full manuscript provides such a bound or an architectural mechanism that enforces it.","major_comments":[{"comment":"The abstract states that constructing an overdetermined system with a non-zero Gram determinant 'ensures a well-posed solution.' For a linear least-squares problem Ax=b with A in R^{m x n}, m>n, non-zero det(A^T A) guarantees a unique solution, but Hadamard well-posedness also requires stability: if the smallest singular value of A is arbitrarily close to zero, then the pseudo-inverse (A^T A)^{-1}A^T amplifies noise without bound. The abstract provides no proof or architectural mechanism ensuring a uniform lower bound on the singular values (or on the Gram determinant) across the data distribution. Since WIC is a 1x1 convolution applied inside a nonlinear network, the resulting Gram determinant depends on input activations; non-zero on training examples does not preclude near-singular behavior on test or adversarial inputs. This is a load-bearing gap in the central claim and must be addressed, e.g., by proving a spectral bound or by explicit regularization that keeps the condition number bounded.","section":"Abstract (Gram determinant claim)"},{"comment":"The paper says it builds an overdetermined system from an underdetermined forward conversion, but the abstract does not specify how the forward operator is augmented or how WIC enforces the non-zero Gram determinant in practice. Without this description, the reader cannot judge whether the construction is general or task-specific, nor whether the determinant condition is checked or guaranteed at inference time. The full paper must provide the concrete form of the overdetermined system and a verification that the Gram determinant is non-zero and bounded away from zero for all inputs used in the reported experiments.","section":"Abstract (overdetermined construction)"},{"comment":"The abstract reports state-of-the-art performance on three RIC tasks but provides no numerical results, baselines, or experimental details. While abstracts routinely omit such details, the combination of an untested theoretical condition and unverifiable SOTA claims makes it impossible to assess the central contribution from the abstract alone. The full paper needs to report quantitative comparisons with existing RIC methods and, importantly, a direct measurement of the conditioning of the Gram matrix (e.g., condition numbers or singular values) on test data to support the well-posedness claim empirically.","section":"Abstract (unverifiable experimental claims)"}],"minor_comments":[{"comment":"The term 'well-posed invertible 1x1 convolution' is used without defining the mathematical operation; a brief description of how invertibility and well-posedness are encoded in the convolution weights would help readers understand the architecture.","section":"Abstract"},{"comment":"The phrase 'well-posed invertible networks' could be misleading: if the network is guaranteed invertible for all inputs, the term 'well-posed' adds little; if invertibility is only approximate or conditional, that distinction should be stated explicitly.","section":"Abstract"},{"comment":"The abbreviation 'RIC' is defined, but the tasks 'reversible image hiding, image rescaling, and image decolorization' would benefit from a one-sentence description in the introduction of the full paper to set the context for readers outside the immediate area.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The full text was not provided in the review packet, so the assessment is based only on the abstract. The main concern is the gap between 'non-zero Gram determinant' and 'well-posedness': the authors must supply a quantitative stability argument or evidence that the condition number is controlled. If the full paper already contains such an analysis, this would likely be a minor revision; otherwise, the central theoretical claim needs substantial rework. I recommend that the editor ensure the full paper is subsequently reviewed by someone with expertise in inverse problems and spectral analysis of neural network operators."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea is a genuine attempt to fix a real ill-posedness problem in reversible image conversion: replace random sampling in invertible nets with an overdetermined linear system whose Gram determinant is nonzero. That is a concrete, sensible direction, and the WIC layer, the two network variants, and the task coverage (hiding, rescaling, decolorization) give the paper broad reach. Code release is a plus.\n\nThe soft spot is the one the stress-test note flags, and it is load-bearing: nonzero det(AᵀA) gives uniqueness but not stability. Hadamard well-posedness needs a uniform lower bound on the smallest singular value, or a bounded condition number, across the input distribution. A 1x1 conv inside a nonlinear network produces Gram determinants that can be near zero off-distribution, and then the approximate left inverse amplifies noise. The abstract gives no proof of such a bound and no experiments probing conditioning. That might exist in the full paper—the architecture could enforce orthogonality, for instance—but I cannot verify it from the abstract. So the claim \"ensures a well-posed solution\" is under-supported as stated.\n\nThe bigger worry: even a bound on training inputs may not survive out-of-distribution examples, yet the paper's language is absolute. This needs either a real theorem with a uniform constant or a softened claim plus empirical conditioning checks across tasks. I am not calling this a fatal flaw because I only have the abstract, but it is the exact point a reviewer must probe.\n\nFor the intended audience—people working on invertible networks, image hiding, rescaling, or decolorization—the paper is worth reading, but only if the full derivation holds up. My recommendation: send it to peer review. The idea is concrete, the coverage is strong, and the stability question is precisely what a referee should test. I would also bring it to a reading group if anyone in the group cares about invertible architectures, but not for a general audience.","headline":"Clever fix for ill-posedness in invertible nets, but the abstract skips the stability proof; the Gram determinant alone doesn't make an inverse well-posed.","tokens_in":1345,"tokens_out":2346,"would_cite":false,"duration_ms":28987,"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":"A well-posed invertible convolution removes the random-sampling crutch from reversible image conversion, yielding stable left inverses for image hiding, rescaling, and decolorization.","keywords":["reversible image conversion","ill-posedness","invertible neural networks","well-posedness","Gram determinant","1x1 convolution","image hiding","image rescaling"],"falsifier":"One concrete observation: feed a set of natural images through a trained WIN model and compute the Gram determinant of the associated forward operators; if any input yields a zero or near-zero determinant, the well-posedness claim fails. A simpler operational check is to measure the condition number of the Gram matrix across a standard test set; divergence on some images would refute the stability guarantee.","tokens_in":558,"feed_emoji":"🖼️","tokens_out":3948,"duration_ms":42252,"temperature":0.7,"pith_summary":"The paper takes on a persistent problem in reversible image conversion: the forward step maps a high-dimensional image to a lower-dimensional output, so reversing it is an underdetermined inverse problem. Existing invertible-network solutions inject randomly sampled latent variables to fill the missing degrees of freedom, which the authors argue leaves the result intrinsically uncertain and ill-posed. The paper's central proposal is to construct an overdetermined system whose Gram determinant is non-zero, yielding a well-posed approximate left inverse, and to instantiate this idea as a well-posed invertible $1\\times1$ convolution (WIC) that needs no random sampling. On reversible image hiding, image rescaling, and image decolorization, the resulting networks report state-of-the-art results. The reason to care is that the approach removes the random-variable crutch and offers a concrete criterion for what makes the inversion well-posed.","feed_headline":"Well-posed invertible network beats random-variable methods","feed_subtitle":"Non-zero Gram determinant yields a stable left inverse for image hiding, rescaling, and decolorization.","key_machinery":"The load-bearing object is the overdetermined system with a non-zero Gram determinant: instead of solving the underdetermined forward equation directly, the method composes forward operators so that the number of constraints exceeds the number of unknowns, and requires the Gram matrix of those operators to have full rank. The second piece is the well-posed invertible $1\\times1$ convolution (WIC), a convolution layer whose weight matrix is constructed to be invertible without any stochastic auxiliary input; using WIC as the invertible building block turns the network into a deterministic approximation of the left inverse. The WIN architecture additionally adds skip connections that carry long-range information across layers, which the paper says improves memory during the conversion.","core_discovery":"The paper claims that ill-posedness in reversible image conversion is caused by treating the forward process as an underdetermined system, and that the standard cure of attaching randomly sampled variables inside an invertible network merely masks the problem by adding irreducible uncertainty. The authors establish that a well-posed solution can be obtained by constructing an overdetermined system with a non-zero Gram determinant, which ensures the approximate left inverse is unique and stable. They implement this principle in a well-posed invertible $1\\times1$ convolution (WIC), a layer that is invertible by construction and requires no random variables, and they build two networks, WIN-Naïve and WIN, on top of it. The reported experiments on reversible image hiding, image rescaling, and image decolorization show consistent state-of-the-art accuracy, which the authors take as evidence that the well-posedness principle resolves the bottleneck of existing reversible conversion methods.","pith_inferences":["The Gram-determinant criterion is likely to transfer to other underdetermined inverse problems in imaging, such as super-resolution or lensless reconstruction, where a deterministic well-posed left inverse could replace stochastic sampling.","A natural testable extension is to monitor the smallest eigenvalue of the Gram matrix during training; a stability guarantee would require it to stay bounded away from zero, and this quantity could be reported as a diagnostic on benchmark datasets.","A useful next step would be to compare against classical regularization approaches that also aim for well-posedness without random variables; measuring that gap would clarify whether the advantage comes from the well-posedness principle or from learned network capacity."],"forward_implications":["Reversible image conversion methods no longer need to sample random variables inside the invertible network, which removes the sampling-induced uncertainty the paper identifies as the source of ill-posedness.","The non-zero Gram determinant condition gives a concrete, checkable criterion for when a reversible conversion is well-posed, rather than relying on network capacity alone.","The WIC layer can serve as a drop-in invertible building block for other tasks that currently use random-variable-based invertible networks.","Reported results on image hiding, rescaling, and decolorization suggest the method sets a new performance baseline on all three tasks."],"supporting_citations":[],"fun_headline_variants":["No random variables: well-posed invertible image conversion","Well-posed invertible network kills random-variable guesswork","Invertible net eliminates ill-posedness with Gram determinant","Well-posed invertible convolution fixes reversible image tasks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the assumption that the overdetermined system actually has a non-zero Gram determinant for the forward operators and the image data encountered in practice, and the paper does not supply a proof or a data-dependent condition that guarantees it.","fun_headline_variants_meta":{"raw":{"variants":["No random variables: well-posed invertible image conversion","Well-posed invertible network kills random-variable guesswork","Invertible net eliminates ill-posedness with Gram determinant","Well-posed invertible convolution fixes reversible image tasks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1402,"prompt_tokens":961,"completion_tokens":441,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":373}},"tokens_in":577,"tokens_out":441,"duration_ms":5217,"temperature":1.0,"reasoning_tokens":373,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T05:07:49.703976+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete observation: feed a set of natural images through a trained WIN model and compute the Gram determinant of the associated forward operators; if any input yields a zero or near-zero determinant, the well-posedness claim fails. A simpler operational check is to measure the condition number of the Gram matrix across a standard test set; divergence on some images would refute the stability guarantee.","supporting_citations":[],"review_version":1}