{"id":"4ada2db2-8c9d-40f6-b9fc-8dc524c62ae9","arxiv_id":"2606.23282","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Extends approximation results for Sobolev functions from Wasserstein spaces to general metric measure spaces satisfying Hilbertianity and possessing a computable algebra of Lipschitz functions.","lead":"This paper extends a numerical approximation framework for Sobolev functions on Wasserstein spaces to a general class of metric measure spaces where Hilbertianity and a computable Lipschitz algebra hold, including weighted Riemannian manifolds and Hellinger-Kantorovich spaces. A smart generalist might read it to see how point-evaluation recovery methods could apply more broadly in infinite-dimensional settings used in optimization and geometry.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's diagnosis matches the paper's own stated contribution. With the full text available in principle, the absence of any detectable gap in the abstract-level description means the load-bearing step is the verification the authors assert they perform; no further objection is warranted on the supplied material.","tokens_in":1668,"tokens_out":247,"duration_ms":15306,"concrete_test":"Locate the sections proving Hilbertianity and the existence of a computable Lipschitz algebra for the Hellinger-Kantorovich space and for weighted Riemannian manifolds; verify that the arguments do not invoke additional unstated restrictions that would exclude the target examples.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the authors identify a general class of metric measure spaces (including the listed examples) for which the Hilbertianity and computable Lipschitz algebra hypotheses hold, thereby extending the approximation results. Because the full manuscript is stated to be available and the abstract asserts that the verification is carried out, no internal inconsistency or unsupported step can be located from the given information. The reader's weakest_assumption is precisely the point the paper claims to resolve.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript extends the numerical approximation framework for Sobolev functions on Wasserstein spaces from [FHS25] to a general class of metric measure spaces by verifying the Hilbertianity and computable Lipschitz algebra hypotheses from [FSS23]. Examples include weighted Riemannian manifolds and spaces equipped with the Hellinger-Kantorovich distance. Within this framework the core approximation results for recovering functions from random point evaluations are recovered and generalized.","tokens_in":1747,"tokens_out":329,"duration_ms":22194,"significance":"If the verifications of the two key hypotheses are rigorous and explicit, the work supplies a reusable abstract setting that enlarges the domain of applicability of the [FHS25] algorithms to several infinite-dimensional spaces currently studied in metric geometry and optimal transport.","major_comments":[],"minor_comments":[{"comment":"The abstract and introduction should state more explicitly which theorem in [FSS23] supplies each hypothesis and which new verification occupies which section of the present manuscript.","section":null},{"comment":"Notation for the computable Lipschitz algebra and the random sampling operator should be introduced once and used consistently; occasional re-use of symbols from [FHS25] without re-definition reduces readability.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript rests almost entirely on the authors' two prior works; the editor may wish to request an independent referee with expertise in the new example spaces (weighted manifolds, Hellinger-Kantorovich) to confirm that the verifications are not merely formal."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The referee's description of the manuscript's scope and contributions is accurate.","responses":[],"tokens_in":1121,"tokens_out":51,"duration_ms":8416,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper takes the approximation framework from the authors' earlier work on Wasserstein spaces and shows the same approach works on a wider set of metric measure spaces.\n\nThe authors check that weighted Riemannian manifolds and spaces equipped with the Hellinger-Kantorovich distance satisfy Hilbertianity and have a computable algebra of Lipschitz functions. With those conditions in place, the results on recovering Sobolev functions from random point evaluations carry over directly.\n\nWhat the paper does well is make the prior methodology reusable across more spaces of current interest without requiring entirely new techniques. The explicit listing of the general class and the claim that the verification is carried out gives a clear path for applying the method elsewhere.\n\nThe soft spot is the reliance on the two previous papers. The main new step is confirming the hypotheses for these additional spaces, so the value hinges on how solid those checks turn out to be. There is no shipped code or external benchmarks, which keeps the work inside the authors' own sequence of results.\n\nThis is for specialists already working in metric Sobolev spaces or optimal transport who want to see the approximation tools applied more broadly. A reader familiar with the cited papers will find the extension straightforward to follow.\n\nIt deserves a serious referee because the generalization is concrete and the spaces matter in ongoing research. The verification details are the part that needs external eyes.","headline":"This paper extends the authors' prior approximation results by verifying the needed hypotheses on weighted Riemannian manifolds and Hellinger-Kantorovich spaces.","tokens_in":2235,"tokens_out":349,"would_cite":false,"duration_ms":20477,"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":"The approximation methodology for Sobolev functions extends to a general class of metric measure spaces including weighted Riemannian manifolds.","keywords":["metric Sobolev spaces","approximation","Wasserstein spaces","Hellinger-Kantorovich distance","Riemannian manifolds","Hilbertianity","Lipschitz functions","function recovery"],"falsifier":"A calculation or example showing that function recovery from samples fails to achieve the expected accuracy in a weighted Riemannian manifold that satisfies the Hilbertianity and Lipschitz algebra conditions would falsify the extension.","tokens_in":2548,"feed_emoji":"","tokens_out":495,"duration_ms":23117,"temperature":0.7,"pith_summary":"This paper establishes that a numerical framework for approximating Sobolev functions on Wasserstein spaces from finite samples extends to other infinite-dimensional spaces. The extension works by identifying metric measure spaces that satisfy Hilbertianity and have a computable algebra of Lipschitz functions. Such spaces include weighted Riemannian manifolds and those equipped with the Hellinger-Kantorovich distance. The core approximation results for recovering functions from random point evaluations are recovered and generalized in this abstract framework. A reader would care because it unifies approaches to function approximation across spaces of interest in analysis.","feed_headline":"Sobolev approximation extends beyond Wasserstein spaces","feed_subtitle":"Hilbertian metric measure spaces with computable Lipschitz algebras support function recovery from samples.","key_machinery":"The general class of metric measure spaces that are Hilbertian and possess a computable algebra of Lipschitz functions, enabling the extension of approximation techniques.","core_discovery":"The combination of theoretical foundations and algorithmic strategies is robust enough to apply to a wide variety of infinite-dimensional spaces of current interest. We identify a general class of metric measure spaces for which the key hypotheses hold and within this framework recover and generalize the approximation results.","pith_inferences":["This suggests the method could be implemented for approximation tasks on manifolds in applications like machine learning.","Connections to other distances in optimal transport may yield further extensions.","Verification of the Lipschitz algebra computability in new spaces would expand the applicable domains."],"forward_implications":["The core approximation results apply to weighted Riemannian manifolds.","The results apply to spaces of measures with the Hellinger-Kantorovich distance.","The methodology generalizes to recover functions from random point evaluations in these spaces.","The framework is robust for various infinite-dimensional spaces."],"fun_headline_variants":["Sobolev approximation in broad Hilbertian metric spaces","General metric spaces allow Sobolev function approximation","Sobolev recovery from samples on diverse measure spaces","Extending approximation theory to abstract metric Sobolev spaces"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The key hypotheses of Hilbertianity and the existence of a computable algebra of Lipschitz functions hold for the identified class of metric measure spaces.","fun_headline_variants_meta":{"raw":{"variants":["Sobolev approximation in broad Hilbertian metric spaces","General metric spaces allow Sobolev function approximation","Sobolev recovery from samples on diverse measure spaces","Extending approximation theory to abstract metric Sobolev spaces"]},"model":"grok-4.3","cost_usd":0.005241,"raw_usage":{"total_tokens":2486,"prompt_tokens":564,"num_sources_used":0,"completion_tokens":57,"cost_in_usd_ticks":52412000,"prompt_tokens_details":{"text_tokens":564,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1865,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":564,"tokens_out":57,"duration_ms":11492,"temperature":1.0,"reasoning_tokens":1865,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T06:38:06.118482+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A calculation or example showing that function recovery from samples fails to achieve the expected accuracy in a weighted Riemannian manifold that satisfies the Hilbertianity and Lipschitz algebra conditions would falsify the extension.","supporting_citations":[],"review_version":1}