{"id":"cb4a94e8-162b-4f38-8de6-ae60485ae105","arxiv_id":"2510.25354","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For random geometric hypergraphs, the standard variational semi-supervised learning problem converges in the large-data limit to a density-weighted p-Laplacian problem, making it a first-order graph method; the proposed HOHL scheme converges to a higher-order Sobolev-type seminorm.","lead":"Hypergraph-based learning connects groups of points at once, and this paper shows that in the large-data limit the standard formulation behaves exactly like a first-order graph method with reweighted connections. The authors also propose a higher-order scheme, HOHL, prove that its multiscale surrogate converges to a higher-order Sobolev-type seminorm, and report benchmark experiments.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.4 is proved for the multiscale surrogate (6), not for the HOHL energy (5); the paper's own Section 3.3 concedes the two limiting Laplacians may differ in density scaling.","rationale":"I read the classical hypergraph analysis as careful and largely convincing: the pointwise operator convergence in Theorem 3.2 is supported by a detailed Taylor expansion and two clean constant identities, and the Gamma-convergence route through Propositions 4.13-4.20 follows the established framework of [39,84]. The weak spot is exactly the one the reader identified. Theorem 3.4 analyzes the multiscale Laplace surrogate, while the proposed HOHL energy built from skeleton Laplacians is not analyzed, and Section 3.3 openly states that the surrogate does not exactly approximate (5) because the limiting Laplacians may differ in density scaling. This is not an internal inconsistency in the proof of Theorem 3.4, but it is a mismatch between the theorem and the paper's attribution of that theorem to HOHL. The gap is fixable by relabeling Theorem 3.4 as a consistency result for the surrogate, by softening the HOHL claims, and by adding experiments or an analytical limit computation for (5). The verdict should remain conditional: the first-order reduction result for classical hypergraph learning stands, while the higher-order HOHL claim requires the missing link to be supplied.","tokens_in":70364,"tokens_out":7773,"duration_ms":71600,"concrete_test":"Re-derive the continuum limit of the HOHL energy (5) in the simplest nontrivial case, q=2 with p=(1,2), on random geometric hypergraphs with weights (4). Use the expansion machinery of Section 4.2 on v^T [ lambda_1 L^{(1)} + lambda_2 (L^{(2)})^2 ] v, computing the leading-order terms in n and eps for smooth test functions v, and compare with the Gamma-limit of the surrogate (6) at scales eps(1) and eps(2). If the two limits agree up to constants, the surrogate bridge is vindicated; if the density weights or operator orders differ, Theorem 3.4 does not support the headline HOHL claim.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The load-bearing gap is the identification between the proposed HOHL energy (5) and the surrogate analyzed in Theorem 3.4. The theorem is stated for (SJ)^{(q,P)}_{n,E}, whose terms are <v, Delta_{n,eps(k)}^{p_k} v>, i.e. the multiscale Laplacian surrogate (6). It is not proved for v^T [ sum_{k=1}^q lambda_k (L^{(k)})^{p_k} ] v in (5), where L^{(k)} are the skeleton Laplacians of the random geometric hypergraph with weights (4). The only bridge offered in Section 3.3 is the sentence: \"(6) does not exactly approximate (5) with weights from (4)--since the corresponding limiting Laplacians may differ in density scaling.\" That admission sits exactly where the central higher-order claim lives. If the density scaling differs, the continuum limit of (5) may carry a different density power or a different operator structure from the limit of (6), so attributing the (6)-limit to HOHL is unsupported. Table 1 and the contribution list nonetheless attribute Theorem 3.4 to HOHL (5). The numerical section also evaluates only the surrogate (6) and never runs (5), so no empirical evidence repairs the gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops discrete-to-continuum asymptotics for variational semi-supervised learning on random geometric hypergraphs with product weights (4). For classical hypergraph learning (2), Theorem 3.2 gives a pointwise convergence of the discrete Euler-Lagrange operator to a weighted p-Laplacian, and Theorem 3.3 gives Gamma-convergence of the semi-supervised objectives in TL^p, with well-posedness when n epsilon_n^p tends to 0 and ill-posedness when n epsilon_n^p tends to infinity; the limiting energy is a density-weighted W^{1,p} seminorm. The paper then proposes Higher-Order Hypergraph Learning (HOHL), based on powers of skeleton Laplacians (5), introduces the multiscale Laplace surrogate (6), and claims in Theorem 3.4 that this surrogate converges to a W^{p_q,2}-type energy. Numerical experiments on iris, digits, Salinas A, and MNIST compare the surrogate against several graph-based SSL baselines over 100 trials each.","tokens_in":70690,"tokens_out":8692,"duration_ms":77864,"significance":"If the results hold, the first half of the paper is a substantive contribution: it gives a clean asymptotic classification of the hypergraph learning model of Zhou-Huang-Scholkopf under a natural product-weight construction, with explicit thresholds n epsilon^p -> 0 versus infinity, and it shows that the classical model is asymptotically first-order up to density re-weighting. The proofs of Theorems 3.2 and 3.3 are detailed: the concentration estimates, the sigma-identities of Lemmas 4.6-4.8, and the reduction to the p-Laplacian in Corollary 4.9 are spelled out and checkable, and the Gamma-convergence scaffolding follows the established TL^p framework of Garcia Trillos-Slepcev and Slepcev-Thorpe. The empirical study is honestly reported with means and standard deviations over 100 trials. The higher-order part, however, is conditional: Theorem 3.4 is proved only for the surrogate (6), the manuscript itself concedes in Section 3.3 that the surrogate does not exactly approximate the HOHL energy (5), and Theorem 3.4's proof is imported from [94] with the threshold conditions asserted rather than derived.","major_comments":[{"comment":"Theorem 3.4 is proved for the multiscale Laplacian surrogate (SJ)^{(q,P)}_{n,E_n}, that is, for energies built from <v, Delta_{n,epsilon^{(k)}}^{p_k} v> as in (6), and not for the HOHL energy (5) built from the skeleton Laplacians L^{(k)} of the random geometric hypergraph with weights (4). The only bridge offered in Section 3.3 is the statement that '(6) does not exactly approximate (5) with weights from (4)--since the corresponding limiting Laplacians may differ in density scaling'; this admission sits exactly at the point where the higher-order claim lives. If the density scaling differs, the continuum limit of (5) may carry a different density power or a different operator structure from the limit of (6), so attributing Theorem 3.4 to HOHL (5) in Table 1 and in contribution item 4 is unsupported as written. The authors should either prove a link between (5) and (6) under explicit conditions on the weight model (4), or restrict the theoretical claim to the surrogate (6) and re-scope the abstract, Table 1, and the conclusion accordingly.","section":"Section 3.3 and Theorem 3.4"},{"comment":"The convergence claims for the higher-order model are not proved in this manuscript: Section 4.4 states that the results are 'simple corollaries from the results in [94]', and Theorem 3.4's well-posedness threshold, n(epsilon_n^{(q)})^{p_q/2-1/2} bounded together with p_q > (5/2)d + 4, is asserted in Lemma 4.26 and used without derivation. Because the HOHL parameterization (powers p_k, scales epsilon_n^{(k)}, and the finest scale q) interacts with the spectral-convergence machinery of [94] in a dimension-dependent way, a reader cannot verify from this manuscript that the thresholds are correctly re-scaled for the present operators. A self-contained derivation of the thresholds, or at least a precise statement of which propositions of [94] are used at each step, is needed for the central higher-order claim to be checkable.","section":"Section 4.4 and Theorem 3.4"},{"comment":"The conclusion claims 'a previously undocumented discrepancy between the operator obtained via pointwise convergence and that derived from the variational limit.' On the evidence in the paper, there is no such discrepancy: Corollary 4.9, equations (48)-(49), shows that the pointwise limit Delta_infty^{(k,p)}(u) equals (sigma_eta^{(k,p)}/(2 rho)) div(rho^{k+1} ||grad u||^{p-2} grad u), which is precisely the Euler-Lagrange operator of the variational limiting energy sigma_eta^{(k)} int rho^{k+1} ||grad u||^p in (8)-(9). The difference from p-Laplacian learning in [84] is a fixed density power, exactly as (9) states. Either the claimed discrepancy should be removed, or it should be reformulated precisely (for example, as a comparison of density powers with [84]) and located with a specific equation reference.","section":"Section 6 (Conclusion)"},{"comment":"The paragraph following equation (9) states that 'the well-posedness ... is ensured if and only if epsilon_n satisfies the lower bound L.2 and the upper bound n epsilon_n^p -> 0.' Theorem 3.3 proves sufficiency of these conditions, but necessity is not established: the ill-posed part of Theorem 3.3 assumes n epsilon_n^p -> infinity and does not rule out other failure modes, and the lower bound L.2 is a standard connectivity/stabilization condition rather than a proved necessary condition in this specific setting. The 'if and only if' wording should either be proved or softened to a sufficiency statement.","section":"Section 3.7.1, after Eq. (9)"},{"comment":"The numerical section evaluates only the multiscale surrogate (6): the q- and j-experiments take Laplacians of epsilon-graphs or kNN graphs and solve energies of the form (6) with coefficients lambda_ell and powers p_ell. The HOHL energy (5) with skeleton Laplacians L^{(k)} of the random geometric hypergraph (4) is never evaluated, and the asymptotic regimes of Theorem 3.4 (n epsilon_n^{(q)} bounded versus tending to infinity) are not probed in the experiments. Consequently, the experiments cannot repair the surrogate-fidelity gap identified in the first major comment; they support only the empirical utility of the surrogate model.","section":"Section 5 and Tables 3-8"}],"minor_comments":[{"comment":"The statement of Theorem 3.4 refers to 'minimizers of (SJ)^{(q,p)}_{n,epsilon_n}', but the objects defined in Section 3.5 are (SJ)^{(q,P)}_{n,E_n} with a power set P and a scale set E_n; the notation in the theorem and in its proof should be made consistent, and the typo 'limiting enery' in the paragraph after Theorem 3.4 should be corrected.","section":"Section 3.7.2"},{"comment":"The quantity t(k) is defined informally as 'the number of terms in the product'; since the concentration proof in Theorem 3.2 uses the exponent t(k) on ||eta||_{L^infty}, the paper should state explicitly that t(k) = k(k+1)/2. Also, Lemma 4.2's statement contains the typographical error 'S^{(n,k)}(i) =<= k n^{k-1}', which should read 'S^{(n,k)}(i) <= k n^{k-1}'.","section":"Section 3.2 and Lemma 4.2"},{"comment":"Figure 5's caption says the striped regions are conjectured results, but the body text near Theorem 3.4 does not repeat this qualification; since the figure accompanies a theorem statement, the conjectural nature of the striped thresholds should be stated in the main text as well to prevent readers from attributing them to Theorem 3.4.","section":"Figure 5 and Section 3.7.2"},{"comment":"The comparison is stated as showing that the proposed models 'mostly outperform' other graph SSL methods, but several of the differences between IP-QC and the best baseline are within one standard deviation (for example, Table 7 at labeling rate 0.02). A brief statement of which differences are statistically meaningful, or a paired-test reference, would make the empirical claim more precise.","section":"Tables 3-8"}],"recommendation":"major_revision","confidential_remarks":"The paper has two distinct halves of different strength. The classical hypergraph part (Theorems 3.2 and 3.3) is rigorous, detailed, and appears sound; the higher-order part is currently built on an identification between (5) and (6) that the authors themselves concede in Section 3.3 is not exact. Because the abstract, Table 1, and the contribution list attribute Theorem 3.4 to HOHL (5), the manuscript as submitted overstates what the theorems establish. If the authors can prove the identification under stated conditions, or alternatively re-scope all claims to the surrogate and clearly mark the HOHL-on-point-clouds statement as an argued heuristic, the paper would be publishable; in the latter case the novelty relative to [65] and [94] should be clarified. I would also encourage the authors to have Section 4.4 checked by someone with the full text of [94] at hand, since several threshold conditions are imported without derivation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my read. The headline result is the asymptotic first-order reduction for classical hypergraph learning, and it holds. Theorems 3.2 and 3.3 are the real contributions: pointwise operator convergence to a weighted p-Laplacian, Gamma-convergence to the density-weighted W^{1,p} energy, and the clean threshold n epsilon^p tending to 0 versus infinity. The density weighting rho^{k+1} is new, and the proof machinery—Taylor expansion with McDiarmid concentration, the two constant identities, and Corollary 4.9's p-Laplacian algebra—checks out. The paper also does a service by showing that, asymptotically, standard hypergraph models reduce to reweighted graph models; that is a useful classification.\n\nThe soft spots are in the HOHL half. Theorem 3.4 is proved for the multiscale surrogate (6), not for the HOHL energy (5) built from skeleton Laplacians L^(k). The paper's own Section 3.3 says (6) does not exactly approximate (5), since the limiting Laplacians may differ in density scaling. That is exactly where the higher-order claim lives. If the density scaling differs, the continuum limit of (5) could be a different energy, so attributing Theorem 3.4 to HOHL is unsupported. The table of contributions does that attribution. The numerical section also evaluates the surrogate, not the HOHL energy, and never runs classical hypergraph learning (2) as a baseline, so the experiments do not repair the gap. I also found the \"well-suited for low-label regimes\" framing too strong: on digits at 2% labels the surrogate gets about 22% accuracy versus Poisson's 79%, which does not look low-label friendly.\n\nThese issues are fixable and none of them undermines the first-order reduction. The fixes are cheap: re-scope Theorem 3.4 and the contribution statements to the multiscale surrogate, or prove the bridge from (5) to (6); add the missing baseline or temper the empirical claims; and stop describing HOHL as low-label friendly without qualification.\n\nWho is this for? People working on discrete-to-continuum limits of graph and hypergraph semi-supervised learning, and practitioners choosing between hypergraph models and multiscale Laplace models. The classical part deserves a serious referee; the HOHL half needs revision before its advertised conclusion is accepted.","headline":"Solid first-order reduction for classical hypergraph learning; the HOHL higher-order claim outruns what is actually proved.","tokens_in":71188,"tokens_out":2967,"would_cite":true,"duration_ms":29512,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49J55","49J45","62G20","65N12"],"pacs":[],"model":"deepseek-v4-flash","headline":"Classical hypergraph learning is asymptotically first-order, converging to a density-weighted p-Laplacian; a proposed higher-order variant reaches a Sobolev-type limit.","keywords":["hypergraphs","semi-supervised learning","asymptotic consistency","p-Laplacian","Gamma-convergence","multiscale regularization","random geometric hypergraphs","higher-order Sobolev seminorm"],"falsifier":"Compute the continuum limit of the HOHL energy (5) directly for the product-weight hypergraph model on a point cloud with smooth density $\\rho$, for instance by $\\Gamma$-convergence or by spectral convergence of the matrix $\\sum_{k=1}^q \\lambda_k (L^{(k)})^{p_k}$. If the resulting limiting energy is not a $W^{p_q,2}$-type seminorm, or if its density scaling differs from the surrogate's limit, then the claim that HOHL is genuinely higher-order in the continuum is not established.","tokens_in":70032,"feed_emoji":"🕸️","tokens_out":10358,"duration_ms":75079,"temperature":0.7,"pith_summary":"This paper analyzes semi-supervised learning on random geometric hypergraphs and proves what happens in the large-data limit. It shows that the classical hypergraph objective, despite encoding multiway interactions, is asymptotically first-order: its discrete operators converge to a density-weighted $p$-Laplacian and its minimizers converge to minimizers of a weighted $W^{1,p}$ energy. The transition between meaningful label propagation and collapse to a constant labeling is governed by $n\\varepsilon^p$ tending to $0$ versus $\\infty$. The paper then proposes Higher-Order Hypergraph Learning (HOHL), which penalizes powers of Laplacians on hypergraph-induced subgraphs, and proves that its multiscale Laplace surrogate converges to a higher-order Sobolev-type seminorm. The results matter because they give a concrete rule for when hypergraph label propagation will work and they ground a computationally cheaper multiscale method in continuum theory.","feed_headline":"Hypergraph learning limits to a weighted p-Laplacian","feed_subtitle":"Analysis pins down when label propagation works and adds higher-order regularization with a proven Sobolev limit.","key_machinery":"The load-bearing object is the product-weight hyperedge model (4), $w_{\\varepsilon,i_0\\cdots i_k} = \\prod_{j=1}^k \\prod_{r=0}^{j-1} \\eta(|x_{i_j}-x_{i_r}|/\\varepsilon)$, which ties every hyperedge's weight to the pairwise proximity of all its members and yields the discrete $(k,p)$-Laplacians $\\Delta^{(k,p)}_{n,\\varepsilon}$. Its continuum limit is the weighted $p$-Laplacian $\\Delta^{(k,p)}_\\infty u = \\frac{\\sigma^{(k,p)}_\\eta}{2\\rho}\\,\\mathrm{div}\\big(\\rho^{k+1}\\|\\nabla u\\|^{p-2}\\nabla u\\big)$, derived through angular integral identities that collapse $k$-dimensional kernel averages into a single constant and expose the density exponent $k+1$. The variational analysis runs in the $TL^p$ space, a product space of measures and functions with a transport-based metric, using $\\Gamma$-convergence. For HOHL, the key object is the multiscale Laplace surrogate (6), $\\sum_{k=1}^q \\lambda_k \\Delta^{p_k}_{n,\\varepsilon^{(k)}}$, which is proven to converge to the continuum energy $\\langle v, \\Delta_\\rho^{p} v\\rangle_{L^2(\\mu)}$, whose domain is the Sobolev space $W^{p,2}$; the paper treats this surrogate as the tractable stand-in for the skeleton-Laplacian HOHL energy (5).","core_discovery":"The paper's central claim is that classical variational hypergraph learning on the product-weight random geometric hypergraph model is asymptotically equivalent to first-order graph-based regularization. In the continuum, the discrete hypergraph energy $\\Gamma$-converges to $\\sigma^{(k)}_\\eta \\int_\\Omega \\|\\nabla v\\|_2^p \\rho(x)^{k+1} dx$, and the associated discrete operator converges pointwise to the weighted $p$-Laplacian $\\Delta^{(k,p)}_\\infty u = \\frac{\\sigma^{(k,p)}_\\eta}{2\\rho}\\,\\mathrm{div}\\big(\\rho^{k+1}\\|\\nabla u\\|^{p-2}\\nabla u\\big)$. Hyperedge structure therefore changes only the density weighting of the limiting energy, not the order of the regularization, and the semi-supervised problem is well-posed exactly when $n\\varepsilon^p \\to 0$ and ill-posed when $n\\varepsilon^p \\to \\infty$. For higher-order learning, the paper introduces HOHL, which regularizes via powers of skeleton graph Laplacians at multiple scales; for geometric point clouds it analyzes the multiscale Laplace surrogate $v^T \\sum_{k=1}^q \\lambda_k \\Delta^{p_k}_{n,\\varepsilon^{(k)}} v$ and proves $\\Gamma$-convergence to a $W^{p_q,2}$-type seminorm, establishing that higher-order structure yields genuinely higher-order regularization in the continuum.","pith_inferences":["Inference: The proven higher-order limit applies to the multiscale Laplace surrogate (6), not to the skeleton-Laplacian HOHL energy (5); a direct $\\Gamma$-convergence proof for (5) under the product-weight model is needed to confirm the skeleton construction itself has a $W^{p_q,2}$ limit, since the manuscript's Section 3.3 states the surrogate does not exactly match (5) in density scaling.","Inference: The density exponent $k+1$ in the classical limiting energy predicts that larger hyperedges amplify the influence of high-density regions on the learned function; a direct test would compare classification error on high- versus low-density clusters as the maximum hyperedge size increases.","Inference: The well-posedness conditions $p > d$ (classical) and $p_q > 2d$ (HOHL, via Sobolev embedding) suggest these methods need very high powers to remain well-posed in high dimension; this could be probed with synthetic data in $d \\geq 5$."],"forward_implications":["Classical hypergraph learning asymptotically reduces to reweighted graph $p$-Laplacian learning: hyperedge combinatorics affects only the density weighting and constants, not the order of regularization.","The threshold $n\\varepsilon^p \\to 0$ versus $\\infty$ gives a concrete, checkable rule: choose the interaction scale so that $n\\varepsilon^p \\to 0$ to keep the continuum problem well-posed and the minimizers smooth label interpolants.","The HOHL surrogate converges to a $W^{p_q,2}$ seminorm, so increasing the power $p_k = k$ raises the regularity order of the continuum limit, distinguishing HOHL from classical hypergraph learning.","The continuum-limit classification identifies multiscale Laplace learning as a principled proxy for HOHL on point clouds, giving theoretical footing to that method.","For HOHL, well-posedness is governed mostly by the finest scale $\\varepsilon^{(q)}$ and its power $p_q$, a simple diagnostic for when higher-order regularization will propagate labels instead of collapsing them."],"supporting_citations":[{"why":"Supplies the p-Laplacian learning framework and the well-/ill-posedness thresholds that the classical hypergraph analysis generalizes.","marker":"[84]"},{"why":"Provides the fractional Laplacian spectral convergence results and $W^{s,2}$ seminorm limits used for the HOHL and surrogate proofs.","marker":"[94]"},{"why":"Introduces the $TL^p$ space and the $\\Gamma$-convergence machinery for discrete-to-continuum variational limits on point clouds.","marker":"[39]"},{"why":"Defines the classical hypergraph learning model whose asymptotic behavior this paper analyzes.","marker":"[100]"},{"why":"Multiscale Laplace learning, the method whose energy becomes the surrogate for HOHL and whose empirical performance motivates the comparison.","marker":"[65]"},{"why":"Establishes that $v^T \\Delta^s v$ corresponds to a discrete Sobolev $W^{s,2}$ seminorm, the basis for HOHL's higher-order penalty.","marker":"[26]"}],"fun_headline_variants":["Hypergraph SSL: hyperedges change weights, not regularity","Hypergraph learning's limit: weighted p-Laplacian, not higher","Semisupervised hypergraphs: well-posedness requires nε^p → 0","Higher-order hypergraph regularization converges to Sobolev"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The higher-order convergence claim is proven for the multiscale Laplace surrogate used as a practical stand-in on point clouds, not for the HOHL energy built directly from the skeleton graph Laplacians; the paper itself notes that the surrogate does not exactly approximate that energy because the limiting Laplacians may differ in density scaling.","fun_headline_variants_meta":{"raw":{"variants":["Hypergraph SSL: hyperedges change weights, not regularity","Hypergraph learning's limit: weighted p-Laplacian, not higher","Semisupervised hypergraphs: well-posedness requires nε^p → 0","Higher-order hypergraph regularization converges to Sobolev"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000987,"raw_usage":{"total_tokens":4206,"prompt_tokens":989,"completion_tokens":3217,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":605,"completion_tokens_details":{"reasoning_tokens":3141}},"tokens_in":605,"tokens_out":3217,"duration_ms":24158,"temperature":1.0,"reasoning_tokens":3141,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:42:08.976890+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the continuum limit of the HOHL energy (5) directly for the product-weight hypergraph model on a point cloud with smooth density $\\rho$, for instance by $\\Gamma$-convergence or by spectral convergence of the matrix $\\sum_{k=1}^q \\lambda_k (L^{(k)})^{p_k}$. If the resulting limiting energy is not a $W^{p_q,2}$-type seminorm, or if its density scaling differs from the surrogate's limit, then the claim that HOHL is genuinely higher-order in the continuum is not established.","supporting_citations":[{"cited_title":"Analysis ofp-Laplacian regularization in semisupervised learning","cited_arxiv_id":null,"evidence_quote":"Supplies the p-Laplacian learning framework and the well-/ill-posedness thresholds that the classical hypergraph analysis generalizes."},{"cited_title":"Consistency of fractional graph-Laplacian regularization in semisu- pervised learning with finite labels.SIAM Journal on Mathematical Analysis, 56(4):4253–4295, 2024","cited_arxiv_id":null,"evidence_quote":"Provides the fractional Laplacian spectral convergence results and $W^{s,2}$ seminorm limits used for the HOHL and surrogate proofs."},{"cited_title":"Continuum limit of total variation on point clouds.Archive for Rational Mechanics and Analysis, 220(1):193–241, 2016","cited_arxiv_id":null,"evidence_quote":"Introduces the $TL^p$ space and the $\\Gamma$-convergence machinery for discrete-to-continuum variational limits on point clouds."},{"cited_title":"Multiscale Laplacian learning.Applied Intelligence, 53(12):15727–15746, nov 2022","cited_arxiv_id":null,"evidence_quote":"Multiscale Laplace learning, the method whose energy becomes the surrogate for HOHL and whose empirical performance motivates the comparison."}],"review_version":2}