{"id":"6e501ab4-d23d-4c45-a69f-906838d09d7d","arxiv_id":"2504.18830","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A dictionary of known and some newly computed closed-form kernel mean embeddings, with derivation tools and a tested Python library.","lead":"This paper collects closed-form formulas for kernel mean embeddings, the integrals of kernels against probability distributions, and packages them in a Python library. It is a reference for practitioners in Bayesian quadrature and kernel-based statistics who need these integrals without rederiving them.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Wendland/uniform K_PP in Eq. (34) is not scale invariant as printed, and its branch conditions disagree with integration of Eq. (33).","rationale":"The central claim is that the paper is a reliable dictionary of closed-form embeddings, with a companion library that implements them. Most entries are standard and cited, so the overall collection is useful. However, the Wendland section is explicitly a new contribution, and Eq. (34) as printed fails elementary scale-invariance and branch-logic checks. This is exactly the kind of entry a user would copy into a Bayesian quadrature or MMD implementation, so a demonstrable error attacks the central claim. The reader's identified weakest assumption, correctness of dictionary entries and reliance on numerical tests, is the same broad concern, but the strongest concrete instance is not the missing subscript in Eq. (12); it is the internally inconsistent Wendland branch in Eq. (34). One scale-invariance test on two parameter settings would settle whether this is a genuine formula error or a typesetting artifact. I therefore keep the conditional verdict: the paper remains a useful reference, but the Wendland entry must be corrected and the test suite should be augmented with dimensional and branch-regime checks.","tokens_in":17103,"tokens_out":16944,"duration_ms":160896,"concrete_test":"Recompute K_PP for the Wendland-0 kernel with uniform P on [0, r] by numerically integrating Eq. (33) for (r, ell) = (3, 1) and (6, 2). Scale invariance requires identical values, and the correctly derived branch gives 8/27 in both cases. As printed, Eq. (34) gives 8/27 for (3, 1) but 2/27 for (6, 2). If the numerical integration matches 8/27 in both instances, the typeset formula is corrupted and must be corrected to ell(3r - ell)/(3 r^2).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.3's Wendland/uniform entry, one of the paper's newly computed contributions, is internally inconsistent as typeset. Eq. (34) gives the second branch as (3r - ell)/(3 r^2 ell) for ell < r, r != 2ell. This expression is not invariant under simultaneous scaling of r and ell by a common factor, although K_PP for this kernel and the uniform distribution depends only on r/ell. Direct integration of Eq. (33) in the ell < r regime yields ell(3r - ell)/(3 r^2), not the printed formula. The third branch condition is also wrong: 1 - r/(3ell) is the value for r < ell, not for r > ell, so the displayed case logic overlaps the second branch and omits the actual small-support regime. Because the paper explicitly positions itself as a dictionary from which formulas are meant to be copied, a wrong branch in a new entry is a load-bearing correctness failure rather than a cosmetic typo. The 'numerical proofs' in the library do not protect the paper unless the unit tests exercise scale invariance and every branch regime, which the printed formula suggests they did not.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper collects closed-form expressions for kernel mean embeddings K_P and their integrals K_PP for a range of kernel/distribution pairs (Gaussian, Matérn, Wendland, fractional Brownian motion, power-series, and sphere kernels), reviews generic techniques for constructing new embeddings (product and mixture rules, change of measure/variable, matrix-valued kernels, Stein kernels), and releases an MIT-licensed Python library implementing the formulas. The central claim is that the typeset dictionary and the accompanying code are a reliable, comprehensive reference that users can copy from directly.","tokens_in":17292,"tokens_out":9095,"duration_ms":82846,"significance":"If the dictionary is correct, it fills a practical gap: kernel mean embeddings are needed in Bayesian quadrature, kernel quadrature, MMD-based inference, and related areas, and the formulas are currently scattered across several literatures. The paper's strengths are its scope, the clear transformation rules in Section 4, and the companion open-source library with unit tests. The library and tests are reproducible artifacts and are a genuine asset. However, the value of the paper depends entirely on the accuracy of formulas that are explicitly intended to be copied, and the newly computed Wendland entry contains a wrong formula and wrong branch conditions. For this reason the current version cannot be accepted as a reliable dictionary.","major_comments":[{"comment":"The Wendland/uniform embedding K_PP is incorrect as printed. For K0(x,y)=(1-|x-y|/ell)_+ on [a,b] with r=b-a, the integral depends only on r/ell. The second branch, printed as (3r-ell)/(3 r^2 ell), is not scale invariant and has the wrong dimension; direct integration of Eq. (33) in the ell<r regime gives ell(3r-ell)/(3r^2). The third branch prints the value 1 - r/(3ell) under the condition r>ell, but that value is the correct result for r<ell, so the case logic sends users to the wrong value in the large-support regime. Because this is one of the newly computed entries in a paper whose stated purpose is to provide formulas to be copied, this is a load-bearing correctness failure rather than a cosmetic typo and must be corrected.","section":"3.3, Eq. (34)"},{"comment":"The statement that the library and its tests 'can be thought of as numerical proofs of the identities here' is not supported. Unit tests against numerical integration are useful verification, but they are not proofs and cannot validate the formulas as typeset unless the tests are shown to cover every branch and every scaling regime. The error in Eq. (34) is concrete evidence that the current test coverage is insufficient. The authors should replace this language with a precise account of what is tested, mark which entries are newly computed, and state the verification method used for each such entry.","section":"3, paragraph after Table 1"}],"minor_comments":[{"comment":"In Eq. (12) the argument of the remaining error function, erf(r_i/(ell sqrt(2))), uses a bare ell where ell_i is meant; this should be fixed for consistency with the product notation.","section":"3.1, Eq. (12)"},{"comment":"In Eq. (31), the prefactor of the Gaussian term in the second square bracket appears to have unbalanced parentheses; please check and re-set the formula.","section":"3.2, Eq. (31)"},{"comment":"The table uses '?' for several K_PP entries; a short note explaining that these are not currently known (or not included) would prevent readers from interpreting them as open computational challenges.","section":"Table 1"},{"comment":"The separate case r=2ell in Eq. (34) is unnecessary once the correct scale-invariant formula is used, since the value 5/12 is obtained by continuity; simplifying the branch structure would make the formula easier to verify.","section":"3.3, Eq. (34)"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things worth knowing. This is a reference contribution, not a research result: it collects closed-form kernel mean embeddings for Gaussian, Matérn, Wendland, fractional Brownian motion, sphere, power-series and Stein kernels, with a small Python library, and most entries are openly acknowledged as collation of known formulas (Briol et al. 2019b, Table 1). The genuinely new content is the Wendland and fBm entries plus the derivation-trick section. The second thing is the problem: the new Wendland/uniform K_PP, Eq. (34), is wrong as typeset. The second branch prints (3r−ℓ)/(3r²ℓ) for ℓ < r, which is not scale invariant and has units 1/ℓ²; integrating Eq. (33) directly gives ℓ(3r−ℓ)/(3r²). The third branch assigns the value 1−r/(3ℓ) to r > ℓ, but that is the r < ℓ regime, and its condition r > ℓ overlaps the second branch. The piecewise logic is internally contradictory; I checked by direct integration and the problem is real. A wrong formula in a dictionary is load-bearing, not cosmetic, because readers copy from it.\n\nCredit where due. The Matérn section is careful, with the Gaussian Matérn K_PP gaps honestly marked '?' in Table 1 rather than guessed at. The fBm formulas check out. Section 4 (product and mixture decompositions, change of measure and variable, matrix-valued kernels, Stein kernels) is well organized and practically useful. The library ships unit tests, which is real evidence and a good practice, and the citation pattern is clean, with formulas traced to primary sources.\n\nSoft spots, in proportion. The 'numerical proofs' claim in Section 3 is an overclaim: unit tests against quadrature are sanity checks, not proofs, and the Eq. (34) error is exactly the kind of thing that phrasing hides. Eq. (12) has a minor ℓ vs ℓ_i typo in the last erf argument — vexing in a dictionary but small. The new Wendland entries were 'computed with Mathematica' with no derivation sketch, and one is wrong; that combination needs fixing, not hedging.\n\nThis paper is for practitioners of Bayesian quadrature, kernel quadrature and MMD inference who want a formula without rederiving it. It deserves a serious referee; a useful reference from serious authors with shipped code should not be desk-rejected. But it needs major revision: correct Eq. (34), fix the branch logic, soften the numerical-proofs language, and double-check the other new entries. I would not accept it as-is.","headline":"Useful reference dictionary of known kernel mean embeddings, but the new Wendland/uniform entry is wrong as typeset, a real problem for a paper whose purpose is to be copied from.","tokens_in":17833,"tokens_out":18719,"would_cite":true,"duration_ms":145326,"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 compiles known closed-form kernel mean embeddings for common kernel–distribution pairs, adds new entries for Wendland and fractional Brownian motion kernels, and supplies derivation rules and a unit-tested Python library.","keywords":["kernel mean embedding","Bayesian quadrature","maximum mean discrepancy","Matérn kernel","Wendland kernel","fractional Brownian motion","Stein reproducing kernel","closed-form integration"],"falsifier":"Take Equation (35), the Wendland order-zero embedding for a centered Gaussian measure, evaluate it at a few points such as $x=0$ with $\\ell=\\sigma=1$, and compare against direct high-precision numerical integration of $\\int (1-|x-y|/\\ell)_+\\,dP(y)$; any mismatch beyond integration tolerance would refute the claim that the dictionary entry is correct.","tokens_in":16895,"feed_emoji":"🧮","tokens_out":7431,"duration_ms":70665,"temperature":0.7,"pith_summary":"The paper is trying to establish that the practical bottleneck in kernel-based integration and two-sample inference—the need for closed-form kernel mean embeddings—can be removed for a wide set of standard kernel and distribution pairs, and that further embeddings can be built from these by composition rules. It collects previously scattered formulas into one dictionary, computes new entries for Wendland and fractional Brownian motion kernels, and packages the results in a Python library with unit tests. A sympathetic reader would care because Bayesian quadrature, worst-case error bounds, and maximum mean discrepancy inference become directly implementable for the covered pairs without rederiving integrals or resorting to sampling approximations.","feed_headline":"One dictionary covers closed-form kernel embeddings","feed_subtitle":"Bayesian quadrature and MMD tests can now pull exact embeddings from one unit-tested library.","key_machinery":"The central object is the kernel mean embedding $K_P(x)=\\int_\\Omega K(x,y)\\,dP(y)$ together with its double integral $K_{PP}$; these are the two quantities that Bayesian quadrature, kernel quadrature, and MMD-based tests need in closed form. The paper's machinery is a table of explicit formulas for these objects, supplemented by four composition rules: product kernels with product distributions multiply, sum kernels with mixture distributions add, a change of measure converts an intractable embedding into a known one with weights $p/q$, and a change of variable $K^\\phi(x,y)=K(\\phi(x),\\phi(y))$ moves embeddings across pushforward distributions. Stein kernels work in the opposite direction: instead of deriving an embedding for a given kernel, they define a kernel whose embedding is identically zero, giving closed forms for unnormalised densities through automatic differentiation of the score function.","core_discovery":"On the paper's own terms, the discovery is that the pair of objects $K_P(x)=\\int_\\Omega K(x,y)\\,dP(y)$ and $K_{PP}=\\int_\\Omega\\int_\\Omega K(x,y)\\,dP(x)\\,dP(y)$ have tractable closed forms for a substantial collection of kernel and distribution pairs, and that these forms can be organized by kernel family. The dictionary covers Gaussian, Matérn, Wendland, fractional Brownian motion, power-series, sphere, and periodic Sobolev kernels against uniform, Gaussian, and spherical measures, with explicit formulas for both the embedding and its double integral where known. The paper also shows that kernels can be designed so that the embedding becomes trivial: Langevin Stein reproducing kernels satisfy $\\widetilde{K}_P(x)=\\widetilde{K}_{PP}=0$ for any sufficiently regular distribution with an available score function. The paper's claim is that the listed formulas are correct and that the accompanying implementation mirrors them, with the library's unit tests serving as numerical checks of the identities.","pith_inferences":["Beyond the paper, the change-of-variable rule points to a cheap way to obtain embeddings for distributions defined only by samplers or generative models: pair a known uniform or Gaussian embedding with the inverse cumulative distribution function or a learned bijection, then check the result against independent quadrature.","Beyond the paper, the missing $K_{PP}$ values for Matérn–Gaussian and Wendland–Gaussian pairs are a natural completion target, since the listed $K_P$ formulas appear integrable in terms of error functions and elementary functions; adding them would complete the variance term needed for Bayesian quadrature.","Beyond the paper, the constant-embedding phenomenon for periodic and sphere kernels suggests a general criterion: any stationary kernel whose covariance function integrates to zero over the sphere will have constant $K_P$ and $K_{PP}$, which would let researchers generate new dictionary entries without symbolic integration.","Beyond the paper, the dictionary's practical impact depends on whether practitioners treat it as a living collection; a community-contributed extension covering conditional distributions and kernel products would directly serve the Bayesian quadrature variants mentioned in the conclusion."],"forward_implications":["Bayesian quadrature and kernel quadrature can be applied directly to uniform and Gaussian targets from the listed formulas, without Monte Carlo approximation of the embedding or the variance term.","MMD-based two-sample and goodness-of-fit tests gain exact computable values for the covered kernel–distribution pairs, removing the sampling noise that enters when embeddings are estimated.","The product, mixture, change-of-measure, and change-of-variable rules let users assemble embeddings for distributions not explicitly listed, so the dictionary extends beyond its table of entries.","Stein reproducing kernels provide closed-form embeddings for any distribution with an available score function, including Bayesian posteriors known only up to a normalising constant, at the cost of using a kernel tailored to the target distribution.","The Python library's unit tests double as numerical checks of every formula, letting practitioners move from identity to implementation with less risk of transcription error."],"supporting_citations":[{"why":"Defines the reproducing kernel Hilbert space setting in which $K_P$ and $K_{PP}$ are well-defined.","marker":"Berlinet and Thomas-Agnan (2004)"},{"why":"Supplies many of the previously known embeddings that the dictionary collects and cross-references.","marker":"Briol et al. (2019b, Table 1)"},{"why":"Provides the Matérn $n+1/2$ uniform-distribution formula that Section 3.2 generalises with a change of variables.","marker":"Ginsbourger et al. (2016)"},{"why":"Derived the Matérn $1/2$, $3/2$, and $5/2$ embeddings against a Gaussian measure reported in equations (29)–(31).","marker":"Ming and Guillas (2021)"},{"why":"Introduced the compactly supported Wendland kernel family for which the paper computes new uniform- and Gaussian-measure embeddings.","marker":"Wendland (1995)"},{"why":"Supplies the fractional Brownian motion and periodic Sobolev kernel background, including the zero-boundary variant used in quasi-Monte Carlo.","marker":"Dick and Pillichshammer (2010)"},{"why":"Introduced Stein reproducing kernels whose embeddings vanish identically, the construction behind Section 4.2.","marker":"Oates et al. (2017)"},{"why":"Shows how a log-transformation yields embeddings for log-Gaussian distributions, cited as an extension of the Gaussian-kernel entries.","marker":"Chen et al. (2024b)"}],"fun_headline_variants":["Dictionary of closed-form kernel mean embeddings","Closed-form kernel embeddings, all in one dictionary","The kernel embedding dictionary you've been missing","Kernel mean embeddings: a dictionary of exact forms","Exact kernel embeddings: now with a full dictionary"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The dictionary's correctness stands on the exactness of every listed formula; for the newly added Wendland and fractional Brownian motion entries, the paper's own support is numerical agreement with integration routines rather than a written proof, so a single transcription error in a formula would falsify that entry.","fun_headline_variants_meta":{"raw":{"variants":["Dictionary of closed-form kernel mean embeddings","Closed-form kernel embeddings, all in one dictionary","The kernel embedding dictionary you've been missing","Kernel mean embeddings: a dictionary of exact forms","Exact kernel embeddings: now with a full dictionary"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000281,"raw_usage":{"total_tokens":1616,"prompt_tokens":851,"completion_tokens":765,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":467,"completion_tokens_details":{"reasoning_tokens":695}},"tokens_in":467,"tokens_out":765,"duration_ms":7839,"temperature":1.0,"reasoning_tokens":695,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:07:32.592302+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take Equation (35), the Wendland order-zero embedding for a centered Gaussian measure, evaluate it at a few points such as $x=0$ with $\\ell=\\sigma=1$, and compare against direct high-precision numerical integration of $\\int (1-|x-y|/\\ell)_+\\,dP(y)$; any mismatch beyond integration tolerance would refute the claim that the dictionary entry is correct.","supporting_citations":[],"review_version":1}