{"id":"49da6abc-1dfc-4d8e-aa9b-91aa1fcaf4aa","arxiv_id":"2607.28981","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Random objects in arbitrary metric spaces are embedded as distance-profile functions, and HSIC/KCI-type tests on the embedded space yield independence and conditional-independence tests with analytic null asymptotics, including object-valued conditioning variables.","lead":"This paper introduces the Distance Profile Embedding (DPE), which converts random objects living in arbitrary metric spaces into functions in a Hilbert space, and builds independence and conditional-independence tests on top of that embedding. The tests promise analytic p-values, work without geometric restrictions such as isometric embedding, and allow the conditioning variable itself to be a random object.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Recommended empirical reference measure violates Assumption 3's full-support condition; validity of the implemented default DPE test is unproven.","rationale":"The reader's weakest_assumption identifies the full-support requirement as the central gap, and I agree. The DPE theorems under a fixed full-support reference measure are internally coherent: continuity plus injectivity (Theorem 1) yields a Borel isomorphism onto the image, so Corollaries 2–3 are plausible. But the paper explicitly recommends the empirical reference measure as the default implementation, and this choice violates Assumption 3. Since all asymptotic results—Theorem 7's weighted chi-square null, Theorem 13's conditional null—are proven for a fixed λ, the data-dependent empirical λ leaves the validity of the recommended test unproven. This is the most load-bearing concern because it directly affects the paper's central claim of providing valid analytic tests for general metric spaces, and it is not merely a gap in edge cases; it is the exact configuration used in the microbiome application. Other potential concerns (Gaussian kernel characteristic on Hilbert spaces, Assumptions 5–8 plausibility, missing supplement) are either supported by citations or standard KCI-type conditions, and the novelty claim can be softened without undermining correctness. Therefore the verdict remains CONDITIONAL (no change), and the concrete test proposed would empirically probe whether this theoretical gap translates into size distortions.","tokens_in":25398,"tokens_out":11146,"duration_ms":453254,"concrete_test":"Conduct a Monte Carlo null simulation on S^2 with geodesic distance: generate n independent pairs (X_i,Y_i) for n=50,100,200,400, with X⊥Y, over at least 10,000 replications. Implement the DPE test exactly as in §7.2 and §8.1 using the empirical reference measure λ_X=n^{-1}Σδ_{X_i} and λ_Y=n^{-1}Σδ_{Y_i}, the Gaussian kernel, and the Theorem 7 surrogate null approximation, and record the empirical type I error at α=0.05. Then repeat with a fixed full-support reference measure (e.g., normalized surface measure). If the empirical-reference version shows size substantially different from 0.05, the missing asymptotic bridge has practical consequences; if size is nominal, the gap is theoretical but the central claim as stated remains unproven.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central equivalence X⊥Y iff Φ(X)⊥Φ(Y) (Corollary 2) and the conditional version (Corollary 3) rest on Theorem 1's injectivity, which requires supp(λ)=Ω (Assumption 3). Section 6 recommends the empirical reference measure λ_X = n^{-1}Σδ_{X_i} as 'always available' and §8.1 uses it for the microbiome application. This λ_X has finite support, so Assumption 3 is violated and Theorem 1(i) does not apply: the map is non-injective on the full metric space, as points not in the observed sample that share identical distances to all X_i are identified. Consequently, the stated equivalence between original and embedded independence is not established for the empirical-reference DPE. Moreover, Theorems 7 and 13 give asymptotic null distributions for a fixed λ; with λ_n random and data-dependent, T_n and S_n are not the same statistics and no asymptotic bridging (e.g., a uniform convergence of the empirical embedding to a population embedding) is provided. The claim in §6 that the empirical measure 'preserves the characteristic properties' is informal and unsupported. This is the weakest load-bearing link between the theory and the recommended practical implementation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the Distance Profile Embedding (DPE), which maps a random object in a general metric space to a distance function in L2 of a reference measure. Under Assumptions 2–3 the map is claimed to be injective and measurable, so that independence of X and Y is equivalent to independence of their DPEs (Corollary 2), and similarly for conditional independence (Corollary 3). The paper then builds Hilbert–Schmidt cross-covariance operators on these embedded objects and, using characteristic Gaussian kernels, claims that vanishing of the operators characterizes (conditional) independence (Theorems 4 and 10). Test statistics T_n and S_n are given explicit trace forms, and asymptotic null distributions are stated as weighted chi-square limits (Theorems 7 and 13), with fixed and local alternative results in Theorems 8–9 and 14–15. Numerical experiments on spheres, SPD matrices, and Wasserstein Gaussian distributions, plus microbiome and mortality applications, are reported. All proofs are deferred to a supplement that is not included in the submitted manuscript.","tokens_in":25608,"tokens_out":2633,"duration_ms":24995,"significance":"If the central theoretical claims are correct, DPE would be a substantial contribution: it would provide independence and conditional-independence tests for random objects in general metric spaces without negative-type, isometric-embeddability, or one-to-one-correspondence assumptions, and it would be the first framework to allow an object-valued conditioning variable. The analytic asymptotic null distributions are a practical improvement over permutation-based procedures. The paper also gives explicit trace-based statistics that are easy to implement. However, the validity of the recommended default implementation is currently not covered by the stated theorems, and the missing supplement prevents verification of the proof of every central result. The theoretical framework is plausible and follows the standard kernel-operator template, but the gap between the full-support Assumption 3 and the recommended empirical reference measure is load-bearing.","major_comments":[{"comment":"The recommended default reference measure λ_X = n^{-1} Σ δ_{X_i} has finite support, so Assumption 3 (supp λ_X = Ω_X) is violated. Consequently Theorem 1(i) does not apply, and the injectivity-based equivalences Corollaries 2–3, Theorem 4, and Theorem 10 are not established for the implemented statistic. The claim that the empirical measure 'preserves the characteristic properties' is informal, and no asymptotic argument (e.g., uniform convergence of the empirical embedding to a population embedding, or a separate proof of injectivity up to P_X-null sets with data-dependent λ_n) is supplied. Since §8.1 uses exactly this empirical reference measure, the validity guarantee of the default implementation is unproven. This is the weakest load-bearing link between theory and practice and must be addressed—either by restricting the implementation to fixed full-support reference measures, or by","section":"§6, 'The Empirical Reference Measure'; Assumption 3; Theorem 1(i); Corollary 2"},{"comment":"Theorem 12 requires ϵ_n ≍ n^{-η(β∧1)/(2η(β∧1)+η+1)} for its CLT, but the simulations fix ϵ_n = 0.005 independent of n. The asymptotic null distribution of S_n in Theorem 13 is therefore not directly applicable to the simulated procedure. A fixed ϵ_n may be viewed as a finite-sample approximation, but the paper does not provide any result showing that the test with fixed ϵ_n is asymptotically valid (e.g., that the effect of regularization vanishes or that the distribution of S_n with fixed ϵ_n is stochastically bounded by the theoretical null). This gap should be closed or explicitly discussed.","section":"§5.3, Theorem 12; §7.3, 'The tuning parameter ϵ_n … is fixed at 0.005'"},{"comment":"Every central theorem—Theorem 1 (injectivity/measurability), Theorems 6–7 (CLT and null distribution), Theorems 10 and 12–13 (conditional independence characterization and CLT)—is deferred to a supplement that is not included in this submission. As a referee, I cannot verify the correctness of the proofs, and several claims (e.g., the Gaussian-product-kernel condition in Theorem 10, the spectral decomposition arguments in Theorem 9(ii), and the matrix trace representation for S_n) are nontrivial. The authors must provide the supplement as part of the submission for review. This is a blocking issue for acceptance, though it is fixable by supplying the missing supplement.","section":"General; 'All technical proofs are presented in the supplement'"}],"minor_comments":[{"comment":"The notation in (13) is slightly ambiguous: the integral over Ω_X with respect to λ_X is written as ∫_{u∈Ω_X} ... dλ_X(u); if λ_X is a probability measure, this is fine, but for general finite measures the normalization should be explicit (e.g., λ_X(Ω_X) factor).","section":"§6, Eq. (13)"},{"comment":"Reference 'Dubey, P., , Y. & Müller' has a typo; the author list appears corrupted. Also, 'Hoffmann-Jorgensen' is spelled in multiple ways; use the standard 'Hoffmann-Jørgensen'.","section":"References"},{"comment":"The figures are referenced but not included in the text; the captions are informative but actual plots are needed to assess the empirical claims.","section":"§7, Figures"},{"comment":"The identification of T_n as hSIC is useful; however, the authors should note that the Gaussian kernel on the DPE space uses the L2(λ) distance, so the Gram matrix entries are exp(-γ times squared DPE distances), which should be stated explicitly to avoid confusion with a kernel on the original space.","section":"§4.4, Eq. (6)"},{"comment":"The p-values for 'Ball' and 'dCov' are reported to four decimals, which is fine, but the number of Monte Carlo or permutation replications used for those methods is not stated; please give the computational details for reproducibility.","section":"§8.1, Table 3"}],"recommendation":"major_revision","confidential_remarks":"The paper has a plausible and potentially important theoretical core, but the discrepancy between the full-support Assumption 3 and the recommended empirical reference measure is severe: it affects the validity of the default implementation and the microbiome application. The missing supplement makes it impossible to verify the proofs of all central results. I would need to see the supplement and either a new asymptotic treatment of the empirical-reference DPE or a change in the recommended implementation to fixed full-support reference measures before I could support acceptance. There is also a concern that the paper makes very strong 'first to ...' claims without a thorough comparison of the exact assumptions needed for existing methods (e.g., PA) versus DPE; that is secondary, though."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real news here is the conditional independence test with an object-valued conditioning variable. That is genuinely new in the metric-space literature: everyone else either needs Euclidean Z or a Hilbert structure. The DPE construction is simple — embed each object as its distance function in L2(Ω, λ) — and the paper is honest that the marginal test reduces to HSIC on those features. What is added is a careful justification that the embedding preserves independence and conditional independence without negative-type or one-to-one-correspondence assumptions, plus operator-level CLTs and local-power analysis. The simulations on the sphere, SPD matrices, and Wasserstein space are well designed and the real-data applications are appropriate.\n\nThe main soft spot is the one the stress-test flags: the paper recommends the empirical reference measure λ_X = n^{-1} Σ δ_{X_i} as 'always available' and uses it in the microbiome application, but that measure has finite support and violates Assumption 3, which requires full support for the injectivity that drives the whole equivalence. The claim in §6 that the empirical measure 'preserves the characteristic properties' is informal, and no asymptotic bridging from the empirical embedding to the population DPE is supplied. This leaves the default implementation without a proven validity guarantee. That is a load-bearing gap, though not necessarily fatal: it could be patched with a separate asymptotic argument or by restricting the theoretical claims to the fully supported reference measures used in the simulations.\n\nTwo smaller things. First, all proofs are deferred to a supplement that is not in this submission, so the rate conditions in Theorem 12 and the null-distribution derivations cannot be checked. Second, the heavy Assumptions 6–8 for the conditional test are standard for KCI-type work but are not verified in the applications. The paper also overstates novelty slightly: since the marginal test is HSIC and the conditional test is KCI with Tikhonov regularization, the 'first' claim should be framed as first for object-valued conditioning, not first in general.\n\nWho should read it: anyone working on non-Euclidean data and kernel testing, especially for causal discovery or object-valued conditioning. It deserves a serious referee, but the authors need to address the empirical-reference gap and provide the supplement. I would not desk-reject this.","headline":"A serious, mostly sound kernel-based framework for independence and conditional independence on metric spaces, but the recommended empirical reference measure falls outside the theory, so the default implementation lacks a proven validity guarantee.","tokens_in":669,"tokens_out":735,"would_cite":true,"duration_ms":29031,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H15","62H20","62G10","62G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Distance-profile embedding yields analytic independence tests on any metric space","keywords":["distance profile embedding","independence test","conditional independence test","random objects","metric spaces","reproducing kernel Hilbert space","characteristic kernel","object-valued conditioning"],"falsifier":"Construct a metric space and two distinct distributions that agree on distance profiles to all sample points but differ elsewhere; with the empirical reference measure, DPE would fail to distinguish dependence. Concretely, simulate X and Y as independent while Y depends on a rare-but-influential region absent from the sample, then check whether the analytic test keeps its nominal size.","tokens_in":25196,"feed_emoji":"📊","tokens_out":3083,"duration_ms":27364,"temperature":0.7,"pith_summary":"The paper introduces the Distance Profile Embedding (DPE), which maps every point of a metric space to the function of its distances to all other points, viewed as an element of a Hilbert space. The central claim is that this map is injective and measurable under mild conditions, so that independence and conditional independence of random objects—such as microbiome compositions, brain networks, or probability distributions—are exactly equivalent to independence and conditional independence of the embedded Hilbert elements. Because the embedded space is Hilbertian, the paper applies reproducing-kernel machinery to build tests whose statistics have explicit weighted-chi-square limits, giving analytic p-values without permutation. If correct, this provides the first conditional independence test in which the conditioning variable itself may be an object in a general metric space, with no need for negative-type metrics, isometric embeddings, or one-to-one correspondence assumptions.","feed_headline":"One map puts independence tests within reach on any metric space","feed_subtitle":"Distance profiles become Hilbert elements, yielding closed-form p-values and object-valued conditioning without permutations.","key_machinery":"The distance profile embedding Φ(x) = dX(·, x), landing in the Hilbert space L2(ΩX, λX). Its injectivity and measurability let the authors transfer independence and conditional independence questions from any Polish metric space to a Hilbert space, where RKHS cross-covariance operators and their Moore–Penrose inverses provide exact characterizations and tractable null distributions.","core_discovery":"DPE maps each random object X to the distance function u ↦ d(u, X) in L2(ΩX, λX). Under a reference measure with full support, the map is injective (Theorem 1), so X⊥Y iff Φ(X)⊥Φ(Y) (Corollary 2) and X⊥Y|Z iff Φ(X)⊥Φ(Y)|Φ(Z) (Corollary 3). With characteristic Gaussian kernels on the embedded space, the cross-covariance operator vanishes exactly under the null (Theorems 4 and 10), and the test statistics Tₙ and Sₙ have explicit weighted-chi-square null limits (Theorems 7 and 13).","pith_inferences":["The recommended empirical reference measure (λ = average of Dirac masses at the observed sample) has finite support and does not satisfy the full-support assumption behind the injectivity theorems; the paper's validity guarantees may not cover the default implementation, and no asymptotic argument connects the empirical DPE to the population DPE.","The equivalence results suggest a general recipe: any injective, measurable distance-based representation of metric-space objects could inherit RKHS testing machinery, making DPE one instance of a broader 'reference-measure embedding' class.","The conditional test's regularity assumptions (range inclusions, kernel eigenvalue decay, representability of conditional expectations) are substantial; applying the method to non-smooth distributions on metric spaces will require checking these conditions case by case.","One could test the empirical-reference shortcut directly: simulate independent X and Y but with a dependence driven by a region not covered by the reference sample; if the analytic null distribution fails to control size, the shortcut is invalid."],"forward_implications":["Independence and conditional independence tests now apply to metric spaces that admit no isometric Hilbert embedding, including spheres with geodesic distance, SPD matrices with affine-invariant Riemannian metric, and Wasserstein spaces of distributions.","Conditioning variables can be object-valued; for example, testing whether female mortality is conditionally independent of fertility given the male mortality distribution.","Analytic p-values replace permutation tests, enabling fast inference for large samples.","The distance profile representation itself becomes a bridge between metric-space statistics and Hilbert-space operator theory, independent of the testing application."],"fun_headline_variants":["Distance profiles turn any metric space into a testable Hilbert space","Closed-form p-values for independence testing of random objects","Object-valued conditioning now possible in independence tests","Inject distance profiles into Hilbert space for exact independence tests"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The reference measure used to build the distance embedding must put mass on every region of the metric space; the paper's recommended default, an empirical measure on the observed sample, does not, so the central equivalence is unproven for the default implementation.","fun_headline_variants_meta":{"raw":{"variants":["Distance profiles turn any metric space into a testable Hilbert space","Closed-form p-values for independence testing of random objects","Object-valued conditioning now possible in independence tests","Inject distance profiles into Hilbert space for exact independence tests"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000248,"raw_usage":{"total_tokens":1369,"prompt_tokens":717,"completion_tokens":652,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":461,"completion_tokens_details":{"reasoning_tokens":589}},"tokens_in":461,"tokens_out":652,"duration_ms":6072,"temperature":1.0,"reasoning_tokens":589,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T16:01:22.629724+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a metric space and two distinct distributions that agree on distance profiles to all sample points but differ elsewhere; with the empirical reference measure, DPE would fail to distinguish dependence. Concretely, simulate X and Y as independent while Y depends on a rare-but-influential region absent from the sample, then check whether the analytic test keeps its nominal size.","supporting_citations":[],"review_version":1}