{"id":"3b161ba5-6be4-49d9-b840-68d80b99865b","arxiv_id":"1908.01755","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The Rashomon ratio, the fraction of near-optimal models in a hypothesis space, is proposed as a practical gauge for when accurate-yet-simple models exist.","lead":"This paper introduces the Rashomon ratio, a measure of how many different models fit a dataset almost equally well, and argues it predicts when simple, interpretable models can match complex ones. It also documents a characteristic Rashomon curve relating training error to this ratio across 52 UCI datasets.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Empirical identification of large Rashomon sets rests on an unvalidated surrogate; depth-7 trees measured with a fixed proposal may not reflect the union space's Rashomon set.","rationale":"The strongest claim of the paper is that a large empirical Rashomon set certifies the existence of simple accurate models (Theorem 9). The theorem itself is correct as a conditional statement, but its application to practice requires two unverified links: (i) the measured Rashomon ratio of depth-7 trees approximates the Rashomon ratio of the union space F2, and (ii) the simpler classes used in practice are δ-covers of that Rashomon set. The paper explicitly flags (i) as an empirical observation and provides no evidence that depth-7 trees reach the same training accuracy as the best of the five algorithms; if they do not, the 'Rashomon set' measured is not the set of models within θ of the union's optimum. The importance-sampling weights are a constant 2^{-128}, making the reported percentages a rescaled acceptance rate; the paper concedes alternative proposals change the scale. This is not a fatal flaw for the theory, but it means the empirical universality of the Rashomon curve and the reliability of the elbow selection rule rest on a measurement procedure whose validity and precision are not demonstrated. A targeted validation—comparing depth-7 tree training risk to the union's best and re-estimating ratios under an alternative proposal—would settle whether the concern lands. Pending such validation, CONDITIONAL remains the appropriate verdict: the conceptual framework is valuable, but the headline empirical claims are conditional on assumptions that can and should be tested.","tokens_in":55606,"tokens_out":8330,"duration_ms":82968,"concrete_test":"For a subset of at least 10 datasets spanning the reported large-to-small ratio range: (1) train depth-7 trees by greedy or exhaustive search and record the minimum training risk; compare it to the minimum training risk among the five algorithms (LR, CART, RF, GBT, SVM). If for any dataset the depth-7 minimum exceeds the union minimum by more than θ=0.05, the surrogate Rashomon set is empty relative to the union baseline and the measured ratio does not estimate the intended quantity. (2) Recompute the depth-7 Rashomon ratio using an alternative proposal that randomizes leaf labels (e.g., flip each leaf label with probability 1/2) and report 95% confidence intervals from the importance weights. If the large-vs-small classification or the ordering of datasets changes materially, the empirical basis of the Rashomon curve and elbow is not robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central empirical claim—that large Rashomon sets occur in practice and that similar performance across algorithms signals their presence—depends on treating depth-seven decision trees as a surrogate for F2, the union of the hypothesis spaces of logistic regression, CART, random forests, gradient boosted trees, and SVMs (Section 6). The paper acknowledges this is an empirical observation over 38 tabular datasets but does not validate that depth-7 trees achieve training risk within θ=0.05 of the union's best, nor that simpler classes (LR, CART) δ-cover the Rashomon set as required by Theorem 9. The importance-sampling estimator compounds this: every sampled tree carries the same weight 2^{-128}≈3e-39 (Supplementary Information H), so reported ratios are rescaled acceptance rates; the authors admit a different proposal would change estimated sizes. Without confidence intervals or effective sample sizes, the distinction between 'large' (≈10^-37%) and 'small' (<10^-40%) ratios—and hence the Rashomon curve and elbow—is not shown to be robust. Because Theorem 9's existence guarantee for simple models is conditional on a δ-cover condition that is never checked for the actual algorithm classes, the paper's headline practical conclusion is supported only under unverified assumptions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the Rashomon set and Rashomon ratio as data-dependent measures of model multiplicity, and argues that a large Rashomon set for a complex hypothesis space can certify the existence of simpler, accurate models that generalize. The theoretical part defines the empirical and true Rashomon sets, proves generalization and existence results for simpler classes (Theorems 3, 4, 6, 9, 10), gives a closed-form Rashomon volume for ridge regression (Theorem 11), and introduces the pattern Rashomon ratio (Definition 12, Proposition 13). The experimental part estimates Rashomon ratios for depth-seven decision trees on 38 UCI classification datasets, reports that large estimated ratios correlate with similar performance across five algorithms, observes a universal Gamma-shaped Rashomon curve over tree-depth hierarchies, and proposes the Rashomon elbow as a model-selection criterion.","tokens_in":55803,"tokens_out":8510,"duration_ms":93354,"significance":"If the empirical claims hold, the paper provides a genuinely new, data-dependent perspective on model simplicity and multiplicity, distinct from VC dimension, stability, margins, and Rademacher complexity. The closed-form ridge-regression volume and the approximating-set generalization bounds are coherent and useful contributions, and the Rashomon curve is a falsifiable empirical prediction. The significance is currently conditional, however, because the headline empirical conclusions rest on an unvalidated surrogate hypothesis space and on an importance-sampling estimator whose absolute scale and large/small dichotomy are not shown to be robust. The paper is transparent about several of these limitations, which is a strength, but the experiments as presented do not yet support the strength of the practical conclusions drawn in Sections 6, 8, and 9.","major_comments":[{"comment":"The importance-sampling estimator is the sole basis for the Rashomon-ratio values used in Section 6 and Figure 5, yet the paper reports no confidence intervals, effective sample sizes, or proposal-sensitivity results. As the authors state in Supplementary H, every sampled depth-seven tree carries the same importance weight 2^{-128} ≈ 3×10^{-39}, so the reported ratios are rescaled acceptance rates; a different proposal distribution would change the estimated size. The paper therefore does not establish that the dichotomy between 'large' ratios (around 10^{-37}%) and 'small' ratios (below 10^{-40}%) is robust, and this dichotomy drives the central empirical correlation claim. Please report effective sample sizes and confidence intervals, and verify that the large/small ordering and the Section 6 correlations are stable under alternative proposal distributions.","section":"Section 5.3, Supplementary Information H"},{"comment":"The surrogate choice of depth-seven decision trees for F2, described as the union of the hypothesis spaces of logistic regression, CART, random forests, gradient boosted trees, and SVMs, is not validated. The theorems used to interpret the experiments, particularly Theorems 9 and 10, require that a simpler class F1 delta-covers the empirical Rashomon set of F2; nothing in Section 6 checks this cover condition or checks that depth-seven trees attain training risk within theta=0.05 of the union space's best model. The paper acknowledges that the surrogate is an empirical observation about 38 tabular datasets, but the headline conclusion—that similar performance across algorithms signals a large Rashomon set—concerns the union space, not depth-seven trees. Please provide at least a partial validation of the surrogate on several datasets, for example by comparing the training risk of depth-seven trees with the best algorithm in F2 and by checking whether simpler classes satisfy the delta-cover condition on the measured empirical Rashomon set.","section":"Section 6"},{"comment":"The theoretical results that motivate the experiments, Theorems 3, 4, and 6, concern the true population Rashomon set, whereas the experiments estimate an empirical Rashomon ratio for a surrogate class. Propositions 7 and 8 only connect true and empirical anchored Rashomon sets for a fixed model, not for the full Rashomon set or for the ratios plotted in Figure 5. Thus the inference from 'the estimated empirical Rashomon ratio is large' to 'the conditions of Theorem 3 hold' is not justified by the stated results. The paper says it is using the empirical ratio as a surrogate for the true ratio, but no high-probability bound relating the two quantities for the function classes used is provided. Please either derive such a bound, or explicitly reframe the experimental conclusions as statements about the empirical Rashomon set of the surrogate class rather than about the true Rashomon set of F2.","section":"Sections 4.1 and 6"},{"comment":"The paper states that the Rashomon elbow 'either helps, or has no effect, but never achieves worse performance than other possible choices,' but the supporting evidence is visual and based on selected examples. No numerical comparison is reported across the 38 datasets, and the elbow is not compared with standard model-selection baselines such as validation-error minimization. Because the abstract and Section 9 present the elbow as a reliable model-selection criterion, please provide quantitative results, such as mean and median test error, win/tie/loss counts against alternatives, and an explicit statement of how many datasets fall into each of the three categories in Figure 9.","section":"Section 8.2, Definition 16, Figure 9"}],"minor_comments":[{"comment":"The expression for p uses binomial coefficients with (1 - Rratio(F2, gamma))|F2|, which is not an integer in general; please clarify that the formula should be read with rounding or state that Rratio is assumed to make the relevant quantities integers.","section":"Section 4.1.1, Theorem 4"},{"comment":"The proof refers to f*_1 as an optimal function in F1, but this symbol is not defined in the theorem statement; please define it explicitly.","section":"Section 4.1, Theorem 3 proof"},{"comment":"The theorem statement says the constructed functions lie in \\hat Rset(F, \\theta) without specifying F; from the context this should be \\hat Rset(F2, \\theta), and item 1 should clarify that they are in F1. Please fix the notation.","section":"Section 4.2, Theorem 9"},{"comment":"The text says all Rashomon curves for all 38 datasets follow the same trend, but then notes that some curves exhibit only the vertical or only the horizontal part; please consistently state that the observed curves are full or partial Gamma-shaped trends.","section":"Section 8.1"},{"comment":"The text refers to 'the pattern Rashomon ratio defined in (12)', but Definition 12 is not numbered as an equation; please refer to Definition 12 instead.","section":"Section 5.2"},{"comment":"There is a typo, 'Occham’s razor', which should be 'Occam’s razor'.","section":"Section 4.1"}],"recommendation":"major_revision","confidential_remarks":"The theoretical core is coherent and the topic is well suited to the journal's scope. The main risk is that the empirical section overclaims relative to what the estimator and surrogate validation actually support. I recommend requiring the proposal-sensitivity analysis, surrogate validation, and quantitative elbow evaluation before acceptance. I saw no evidence of citation or novelty-disclosure problems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThis paper is worth reading if you care about interpretable ML and model multiplicity. What is actually new: the Rashomon ratio as a volume measure over the set of near-optimal models, a pattern-based variant that avoids some parameterization issues, a closed-form Rashomon volume for ridge regression (Theorem 11, a real derivation), and the empirical Rashomon curve with its elbow as a model-selection heuristic. The theory in Section 4 is honestly conditional: existence of simple models in the Rashomon set follows from smoothness plus a delta-cover condition, and the paper says plainly that these conditions are not checked in practice. That is a weakness, but an acknowledged one; the theorems themselves are coherent.\n\nThe soft spot is the empirical core. The Rashomon ratio for depth-7 decision trees is estimated by importance sampling where every sampled tree carries the same weight, about 3e-39 (Supplement H). So the reported ratios are essentially acceptance counts rescaled by a constant, and the distinction between \"large\" (10^-37%) and \"small\" (<10^-40%) is a distinction between sampling a handful of hits versus one hit or none, without confidence intervals. The paper says a different proposal would change the estimates. That is true, and it means the absolute scale of the ratio is not meaningful; what survives is an ordinal comparison under one fixed proposal. The use of depth-7 CART as a surrogate for the union of logistic regression, SVM, forests, and boosting is a practical choice, but it is never validated against any richer space. Both are real limitations, and the reader's stress test is fair on them.\n\nThat said, the paper's central qualitative claim—that when many algorithms perform similarly, simple accurate models often exist—is plausible and consistent with a lot of practical experience. The correlation the authors report is evidence, not proof. The \"Rashomon elbow never worse than other choices\" phrasing is too strong: the figures show it often is among the better model classes, but there is no systematic paired comparison or uncertainty estimate.\n\nWho this is for: a researcher building a model-selection workflow for tabular data, or a theorist interested in why simple models exist, will find the framework useful. The estimator needs recalibration before the quantitative ratio is used as a gauge; the ordinal \"similar performance\" heuristic is fine as a rule of thumb.\n\nRecommendation: send it to a serious referee. The theory is sound, the question is important, and the empirical flaws are fixable in revision. I would not cite the absolute ratio values, but I would cite the framing and the ridge volume.","headline":"New framing for why simple models exist, with honest conditional theory; the empirical estimator is too fragile to support the headline quantitative claims, but the paper deserves a serious referee.","tokens_in":56386,"tokens_out":2518,"would_cite":true,"duration_ms":27499,"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":"This paper argues that simple, accurate models are likely to exist whenever the Rashomon set—the set of almost-equally-accurate models—is large, and introduces the Rashomon ratio and curve to detect this before searching.","keywords":["Rashomon set","Rashomon ratio","model multiplicity","model simplicity","generalization bounds","interpretable machine learning","model selection","decision trees"],"falsifier":"Recompute the Rashomon ratios for the same 38 data sets using a different proposal distribution (for example, random leaf labels as well as random splits, or trees of depth 9 instead of 7) and check whether the ordering of data sets by ratio—and especially the large-versus-small boundary near $10^{-37}\\%$—survives; if the ordering flips for even a handful of data sets, the empirical link between large ratios and similar cross-algorithm performance is an artifact of the estimator rather than a property of the data.","tokens_in":55329,"feed_emoji":"🧩","tokens_out":8249,"duration_ms":71071,"temperature":0.7,"pith_summary":"The paper sets out to show that simple, accurate models are not a lucky accident: they are to be expected whenever the Rashomon set—the set of models whose training loss is within a small tolerance $\\theta$ of the best model in a class—is large. As its central tool it defines the Rashomon ratio, the volume fraction of the hypothesis space occupied by the Rashomon set, and argues that a large ratio acts as a certificate that simpler models with comparable accuracy and good generalization exist before anyone searches for them. The main theorems show this under two conditions: the loss is $K$-Lipschitz and the simpler class $\\mathcal{F}_1$ forms a $\\delta$-cover of the complex class $\\mathcal{F}_2$'s Rashomon set; then many $\\mathcal{F}_1$ models lie inside the set and all of them inherit a generalization bound that uses only $\\mathcal{F}_1$'s complexity. Empirically, over 38 classification and 14 regression data sets, plotting empirical risk against the log Rashomon ratio across a hierarchy of decision-tree depths gives a recurring $\\Gamma$-shaped Rashomon curve whose elbow is a reliable model-selection point. If the paper is right, a practitioner can tell cheaply—by seeing whether several standard algorithms perform similarly—that a search for interpretable models is likely to pay off.","feed_headline":"When many models fit equally well, a simple one is likely nearby","feed_subtitle":"A new 'Rashomon ratio' measures the volume of near-optimal models and predicts when interpretable models can match black boxes.","key_machinery":"The Rashomon set and its normalized version, the Rashomon ratio, are the load-bearing objects. The Rashomon set is $\\hat{R}_{\\mathrm{set}}(\\mathcal{F},\\theta)=\\{f\\in\\mathcal{F}: \\hat{L}(f)\\le \\hat{L}(\\hat{f})+\\theta\\}$—the models within $\\theta$ of the best training loss—and the ratio is $V(\\hat{R}_{\\mathrm{set}}(\\mathcal{F},\\theta))/V(\\mathcal{F})$, the fraction of the hypothesis space that is almost-optimal. The argument runs on two levers: approximation and smoothness. If the simple class $\\mathcal{F}_1$ $\\delta$-covers the Rashomon set of $\\mathcal{F}_2$ and the loss is $K$-Lipschitz, then each packing ball in the Rashomon set contains an $\\mathcal{F}_1$ model, which is what yields the existence and generalization statements of Theorems 9 and 10. To make the ratio measurable, the paper supplies a closed-form ellipsoid volume for ridge regression, a pattern-based ratio that groups models by prediction vectors and converges to a binomial-entropy limit as the model class grows, and importance-sampling estimators for tree classes; the empirical Rashomon curve then connects empirical risk to log Rashomon ratio along a hierarchy and locates the Rashomon elbow.","core_discovery":"The paper's central claim is that the size of the Rashomon set is a data-dependent gauge of how easy it is to find simple, accurate models. Formally, for finite hypothesis spaces $\\mathcal{F}_1\\subset\\mathcal{F}_2$, if the true Rashomon set of $\\mathcal{F}_2$ at tolerance $\\gamma$ contains even one model from $\\mathcal{F}_1$, then the best empirical model in $\\mathcal{F}_1$ has training risk within $\\gamma$ plus a small sample term of the best true risk in $\\mathcal{F}_2$ (Theorem 3); if $\\mathcal{F}_1$ is sampled randomly from $\\mathcal{F}_2$, a sufficiently large true Rashomon ratio makes this event likely (Theorems 4 and 6). For the empirical Rashomon set, Theorem 9 proves that when the loss is $K$-Lipschitz and every model in $\\hat{R}_{\\mathrm{set}}(\\mathcal{F}_2,\\theta)$ is within $\\delta$ of some $\\mathcal{F}_1$ model, then at least $\\mathcal{B}(\\hat{R}_{\\mathrm{set}}(\\mathcal{F}_2,\\theta),2\\delta)$ models from $\\mathcal{F}_1$ lie in the Rashomon set and each has $|L(\\bar{f}_1)-\\hat{L}(\\bar{f}_1)|\\le 2K R_n(\\mathcal{F}_1)+ b\\sqrt{\\log(2/\\varepsilon)/(2n)}$. Theorem 10 extends the same reasoning to bound generalization for every model in $\\mathcal{F}_2$'s Rashomon set using only the complexity of $\\mathcal{F}_1$, meaning a large Rashomon set with a good simple approximating class makes the whole problem simpler than a naive analysis suggests. The paper also derives a closed-form Rashomon volume for ridge regression, an entropy-bounded pattern Rashomon ratio for classification, and the empirical Rashomon curve.","pith_inferences":["A direct test of the paper's practical thesis would be to run a large library of data sets through two cheap probes—cross-algorithm accuracy agreement and average pairwise prediction disagreement—and compare those probes with a directly estimated Rashomon ratio; if the correlation is strong, the ratio itself becomes unnecessary for routine model selection.","Because the absolute magnitudes of the reported Rashomon ratios are set by the importance-sampling proposal (per-tree weights around $3\\times10^{-39}$), the robust content is the ordering of data sets and the within-hierarchy shape of the Rashomon curve; cross-paper comparisons of raw ratio values will require a normalized estimator.","The $\\Gamma$-shape suggests a decision rule for benchmark progress: problems for which all modern algorithms saturate in accuracy sit on the vertical part of the curve, where effort should go into finding simpler models inside the Rashomon set, while problems where accuracy still climbs with model complexity sit on the horizontal part, where more complex classes still have room."],"forward_implications":["A practitioner who sees several standard algorithms agree in training and test accuracy can treat that agreement as an indirect signal of a large Rashomon set, making it worth searching for interpretable or sparse models before assuming they do not exist.","Under $K$-Lipschitz loss and a $\\delta$-covering simple class, a large empirical Rashomon set guarantees not just one but many simple-yet-accurate models, each with a generalization bound that uses only the simple class's Rademacher complexity.","The Rashomon elbow, chosen from training data alone, balances empirical risk against Rashomon ratio and, in the 52 data sets studied, generalized as well as or better than the other hypothesis spaces in the hierarchy.","Because the Rashomon ratio depends on both the data set and the hypothesis space and tracks multiplicity of models, it captures aspects of simplicity that VC dimension, stability, margins, and Rademacher complexity leave out.","In ridge regression, the Rashomon volume can be written in closed form as an ellipsoid whose size depends on the feature matrix's singular values but not on the target values, so the simplicity gauge can be computed exactly for this class."],"supporting_citations":[{"why":"Supplies the term and the definitional lineage of the Rashomon effect as the existence of many different almost-equally-accurate models.","marker":"Breiman et al., 2001"},{"why":"Provides the Rademacher-complexity generalization bound used in Theorems 9 and 10 to bound the difference between true and empirical loss for models in the simpler class.","marker":"Bartlett and Mendelson, 2002"},{"why":"Supplies the 52 UCI data sets on which the Rashomon ratio measurements, the Rashomon curve, and the elbow model-selection experiments are run.","marker":"Dua and Graff, 2019"},{"why":"Contributes the related Rashomon-set framework and the lemma used in Proposition 8 that links membership in empirical and true anchored Rashomon sets.","marker":"Fisher et al., 2019"},{"why":"Provides the loss-restricted close-to-optimal model class that the Rashomon set generalizes, and the local Rademacher complexity perspective that the paper contrasts with its ratio.","marker":"Srebro et al., 2010"},{"why":"Supplies a concrete example of a simpler class—supersparse linear integer scoring systems—that plays the role of $\\mathcal{F}_1$ when discussing interpretable models inside the Rashomon set.","marker":"Ustun and Rudin, 2016"},{"why":"Provides the approximation-theory results used in Table 4 to show that piecewise-constant and polynomial classes can serve as $\\delta$-covering approximating sets for larger function classes.","marker":"DeVore, 1998"}],"fun_headline_variants":["Rashomon ratio predicts simple model existence","Many equal models mean a simple one is likely","New gauge finds when simple models are possible","Large Rashomon sets hint at simple models","Simplicity emerges from many near-equal models"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The empirical claim that large Rashomon sets are common rests on treating depth-seven decision trees as a stand-in for the true union of the five algorithms' hypothesis spaces and on an importance-sampling estimate whose proposal weights set the scale for what counts as large; if either is unrepresentative, the observed ordering of large versus small ratios could change.","fun_headline_variants_meta":{"raw":{"variants":["Rashomon ratio predicts simple model existence","Many equal models mean a simple one is likely","New gauge finds when simple models are possible","Large Rashomon sets hint at simple models","Simplicity emerges from many near-equal models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000289,"raw_usage":{"total_tokens":1864,"prompt_tokens":1290,"completion_tokens":574,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":906,"completion_tokens_details":{"reasoning_tokens":506}},"tokens_in":906,"tokens_out":574,"duration_ms":5974,"temperature":1.0,"reasoning_tokens":506,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:03:50.310873+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the Rashomon ratios for the same 38 data sets using a different proposal distribution (for example, random leaf labels as well as random splits, or trees of depth 9 instead of 7) and check whether the ordering of data sets by ratio—and especially the large-versus-small boundary near $10^{-37}\\%$—survives; if the ordering flips for even a handful of data sets, the empirical link between large ratios and similar cross-algorithm performance is an artifact of the estimator rather than a property of the data.","supporting_citations":[{"cited_title":"Statistical modeling: The two cultures (with comments and a rejoinder by the author)","cited_arxiv_id":null,"evidence_quote":"Supplies the term and the definitional lineage of the Rashomon effect as the existence of many different almost-equally-accurate models."},{"cited_title":"Rademacher and G aussian complexities: Risk bounds and structural results","cited_arxiv_id":null,"evidence_quote":"Provides the Rademacher-complexity generalization bound used in Theorems 9 and 10 to bound the difference between true and empirical loss for models in the simpler class."},{"cited_title":"All models are wrong, but many are useful: Learning a variable's importance by studying an entire class of prediction models simultaneously","cited_arxiv_id":null,"evidence_quote":"Contributes the related Rashomon-set framework and the lemma used in Proposition 8 that links membership in empirical and true anchored Rashomon sets."},{"cited_title":"Smoothness, low noise and fast rates","cited_arxiv_id":null,"evidence_quote":"Provides the loss-restricted close-to-optimal model class that the Rashomon set generalizes, and the local Rademacher complexity perspective that the paper contrasts with its ratio."},{"cited_title":"Supersparse Linear Integer Models for Optimized Medical Scoring Systems","cited_arxiv_id":null,"evidence_quote":"Supplies a concrete example of a simpler class—supersparse linear integer scoring systems—that plays the role of $\\mathcal{F}_1$ when discussing interpretable models inside the Rashomon set."},{"cited_title":"Nonlinear approximation","cited_arxiv_id":null,"evidence_quote":"Provides the approximation-theory results used in Table 4 to show that piecewise-constant and polynomial classes can serve as $\\delta$-covering approximating sets for larger function classes."}],"review_version":1}