{"id":"3d75af2e-f22f-4ebf-bc78-c1305188b141","arxiv_id":"2412.19212","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"DSSW applies data-adaptive weights to spherical projection directions and outperforms unweighted spherical sliced-Wasserstein baselines on several learning tasks.","lead":"The paper introduces a weighted version of the spherical sliced-Wasserstein distance, giving more importance to projection directions where two distributions on a sphere differ most. It reports better results than prior sliced-Wasserstein methods on gradient flows, earth density estimation, and self-supervised learning.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The implemented f in Definition 2 is normalized over the sampled directions, so the Monte Carlo estimator is not an estimator of the Definition 1 integral; the proofs of Prop. 3 and Thm. 1 rely on boundedness alone and do not close this gap.","rationale":"The reader's weakest assumption identifies exactly the definitional mismatch in Definition 1 vs Definition 2 and the invalid universal-constant step in Prop. 3 and Thm. 1. This is the most load-bearing concern because the paper's stated contribution includes 'complete theoretical guarantees' for the DSSW distance; if the implemented quantity is not the integral being analyzed, the guarantees do not cover the method that is actually evaluated. The empirical results may still indicate a useful heuristic, but the central theoretical claim is unsupported as written. My read does not change the reader's conditional verdict: the paper needs a reconciled definition (e.g., a continuum normalization f(W)=g(W)/∫g(W)dσ, with a separate analysis of the sampled estimator) and a repaired proof of the boundedness-to-Lipschitz step before the guarantees can be accepted.","tokens_in":30839,"tokens_out":4910,"duration_ms":53365,"concrete_test":"Take L=2 and g=exp. For any fixed pair of directions with projected distances W_1 and W_2, W_1≠W_2, the Definition 2 contribution is f_1 W_1 + f_2 W_2 with f_1=e^{W_1}/(e^{W_1}+e^{W_2}) and f_2=1−f_1. Recompute the expectation of \\widehat{DSSW}_{p,2} over the random direction pair and compare it with Eq. (7). Show that no choice of fixed f:(0,1) reproduces this expectation for all μ,ν; equivalently, the estimator is biased for the Definition 1 integral. This single computation settles whether Theorem 1's Monte Carlo claim can hold as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theoretical object is Eq. (7), an integral over V_{d,2} with a fixed weight function f(P^U_# μ, P^U_# ν). Definition 2 defines f for the ℓ-th sampled direction by normalizing over the L directions actually drawn, so f depends on the entire random set {U_k}_{k=1}^L and on all projected Wasserstein distances. The estimator \\widehat{DSSW}_{p,L} in Theorem 1 is therefore not an average of independent copies of a fixed integrand: its expectation is not the Eq. (7) integral, and the claimed sample-complexity and Monte-Carlo guarantees (Prop. 3 and Thm. 1) do not apply to the implemented quantity. The proofs of Prop. 3 and Thm. 1 use only 0<f<1 to replace fW by CW (Eq. (A.19) and Eq. (A.23)); boundedness alone does not yield |f_n W_n − f W| ≤ C|W_n − W| without a Lipschitz control on f, and f_n here is normalized over a changing sample, so even continuity fails in general. This is a definitional mismatch between the continuum theory and the sampled algorithm, not a mere notational issue. A secondary symptom is that Algorithm 2 minimizes L(ψ), the same weighted average, which for flexible h_ψ will concentrate weight on small projected distances, the opposite of the claimed discriminative emphasis.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a data-adaptive Discriminative Spherical Sliced-Wasserstein (DSSW) distance that weights projection directions in the spherical sliced-Wasserstein integral. Two weight mechanisms are introduced: a non-parametric projected energy function (exponential, identity, or polynomial of the projected Wasserstein distance) and a parametric neural-network induced weight (linear, nonlinear, or attention). The authors claim topological and statistical guarantees, including positivity, symmetry, asymptotic continuity, dimension-free sample complexity, and Monte-Carlo projection complexity, and they report experiments on gradient flows, spherical density estimation, sliced-Wasserstein autoencoders, and self-supervised learning where DSSW variants outperform SSW, S3W, and other baselines.","tokens_in":31151,"tokens_out":3341,"duration_ms":34054,"significance":"If the theoretical guarantees were valid, the paper would offer a simple and useful extension of spherical sliced-Wasserstein distances, with a cheap non-parametric variant and a more flexible learned variant. The experimental study is broad, includes several standard benchmarks, and reports consistent improvements with t-test significance markers; the non-parametric variants also add negligible runtime overhead. The main weakness is that the stated theoretical results do not match the quantity actually implemented: the normalized weight in Definition 2 depends on the full sample of L projection directions, while the integral in Definition 1 uses a fixed function of the two projected measures. This mismatch invalidates the sample-complexity and Monte-Carlo guarantees as written, and it also affects the interpretation of the parametric training objective. The empirical contribution is likely salvageable, but the theoretical claims need to be reworked substantially.","major_comments":[{"comment":"The f in Eq. (7) is a fixed function of the two projected measures, but the f in Eq. (8) is normalized over the L sampled directions, so it depends on the entire random set {U_k}_{k=1}^L and on all L projected Wasserstein distances. Therefore the Monte Carlo estimator in Theorem 1 is not an average of independent copies of the integrand in Eq. (7), and its expectation is not DSSW_p^p(μ,ν;f). This is a definitional mismatch between the continuum object and the sampled algorithm, not a notational issue.","section":"Definition 1 (Eq. 7) and Definition 2 (Eq. 8)"},{"comment":"The proofs of Proposition 6 and Theorem 1 replace |f W - f W| with C|W - W| using only 0<f<1. Boundedness alone does not imply Lipschitz continuity of the product f·W. With the sample-dependent normalization of Eq. (8), f changes with the empirical measures and with the set of sampled directions, so the inequality in Eq. (A.19) and its Monte-Carlo counterpart in Eq. (A.23) do not follow. The sample-complexity and projection-complexity guarantees therefore lack support.","section":"Appendix A, Eq. (A.19) and Eq. (A.23)"},{"comment":"The training objective L(ψ) for the parametric weight network is exactly the same weighted average that defines the DSSW distance. Since the softmax weights can concentrate mass on the smallest projected Wasserstein distances, minimizing L(ψ) over ψ for a fixed pair (μ,ν) drives the discrepancy toward small values, which is the opposite of the claimed 'discriminative' emphasis. The paper should specify what property of hψ is being optimized and explain why this does not trivialize the discrepancy in downstream minimization tasks.","section":"Algorithm 2, line 10"}],"minor_comments":[{"comment":"The proposition and theorem numbers in the main text (Proposition 1, 2, 3; Theorem 1) do not match the numbers in the appendix (Proposition 4, 5, 6; Theorem 2); this makes cross-referencing the proofs unnecessarily confusing.","section":"Naming and numbering"},{"comment":"In the proof, the argument of f is written as f(P^U_#μ_k, P^U_#ν), but the statement concerns DSSW_p^p(μ_k, μ; f); the second argument should be P^U_#μ. This appears to be a typo, but it should be corrected.","section":"Proof of Proposition 2 (Appendix A)"},{"comment":"The paper uses 'S9', 'S2', and 'Sd−1' inconsistently; for instance, Table 4 writes 'S9' where S^{9} is meant, and the text writes 'S2' where S^{2} is meant. Please use consistent superscript notation for spheres.","section":"Notation for spheres"},{"comment":"There are several typographical errors, including 'Molleweide' for Mollweide, 'preform' for perform, 'followd' for followed, 'propsoed' for proposed, 'the the' for the, and 'Scipy' should be 'SciPy'. These do not affect the science but should be cleaned up.","section":"Typos"}],"recommendation":"major_revision","confidential_remarks":"The empirical section is substantial and likely publishable after the theory is corrected. The central issue is that the implemented f in Definition 2 is sample-dependent, while the proven results treat f as fixed. The authors should either redefine DSSW as an L-dependent quantity and prove the corresponding laws, or prove uniform Lipschitz/continuity properties of the normalized weight and then show the theorems hold. The parametric training objective also needs a principled justification. This is a load-bearing problem, but it is fixable within the scope of the paper, so major revision rather than rejection seems appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the idea is a straightforward spherical port of Energy-Based Sliced Wasserstein, the experiments are consistent and the non-parametric version costs almost nothing, but the theory as written does not prove what it claims. The gap is Definition 1 vs Definition 2, and it is not cosmetic.\n\nWhat is new: weighting the projection directions in SSW by a function of the projected Wasserstein distance (or a learned network) is new on the sphere. It is a direct adaptation of Nguyen and Ho's EBSW, but the circular projection geometry is non-trivial and the paper works through the details. The non-parametric form (exp/identity/poly) adds negligible cost, which the runtime plots support. The experiments cover the standard benchmarks for spherical sliced distances—gradient flows, earth density estimation, SWAE, SSL—and the DSSW variants are consistently better than SSW and S3W baselines. That is a solid empirical package even if the gains are incremental.\n\nThe soft spot is real and load-bearing. Definition 1 defines DSSW as an integral over V_{d,2} with a fixed weight function f(P^U_# mu, P^U_# nu). Definition 2 defines the weight for the l-th sampled direction by normalizing over the L directions actually drawn. Those are different objects. The Monte Carlo estimator in Theorem 1 is not an average of independent copies of the integrand in Eq. (7); its expectation is not that integral. The proofs of Proposition 3 and Theorem 1 replace f·W by C·W using 0<f<1, but boundedness alone does not give |f_n W_n - f W| ≤ C|W_n - W|. The normalized f depends on the entire sample of directions and distances, so without a uniform Lipschitz bound the sample-complexity and Monte Carlo bounds do not follow. This is a definitional mismatch, not a typo. The parametric training objective in Algorithm 2 is also suspect: it minimizes the same weighted sum, so a flexible h_psi can concentrate weight on small projected distances, which is the opposite of the claimed discriminative emphasis. The authors should reconcile the definitions and either prove the missing Lipschitz control or soften the guarantees.\n\nMinor issues: no code release, baseline numbers mostly cited from prior papers rather than re-run, and some proposition numbering is inconsistent between main text and appendix.\n\nBottom line: the empirical finding is plausible and useful, and the non-parametric variant is cheap enough to be adopted. But the 'complete theoretical guarantees' are not there as written. I would send it to review with a request for major revision: fix the estimator definition, fix the proofs, and clarify the parametric objective. It deserves referee time, not a desk reject.","headline":"A useful empirical variant of spherical sliced-Wasserstein with a cheap non-parametric weighting, but the theory overclaims: the estimator in Definition 2 is not an estimator of the integral in Definition 1, and the proofs do not close that gap.","tokens_in":31654,"tokens_out":1934,"would_cite":false,"duration_ms":20034,"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 claims that weighting each projection direction by a data-adaptive projected energy function yields a spherical sliced-Wasserstein distance that better discriminates distributions and outperforms equal-weight SSW in gradient…","keywords":["spherical sliced-Wasserstein distance","data-adaptive projection direction","projected energy function","optimal transport on the hypersphere","gradient flows","density estimation","self-supervised learning","Stiefel manifold"],"falsifier":"Fix two distributions on $S^2$ with one dominant projection direction, compute the DSSW Monte Carlo estimator in Definition 2 for $L = 10$, $100$, and $1000$, and compare the empirical mean squared error against the bound $C^2 \\operatorname{Var}(W_p^p)/L$ from Theorem 1; if the error fails to decay at rate $1/L$ with a constant independent of $L$, the bounded-$f$ assumption behind the proof is violated.","tokens_in":30591,"feed_emoji":"🌐","tokens_out":6648,"duration_ms":60664,"temperature":0.7,"pith_summary":"The paper proposes replacing the uniform average over projection directions in the Spherical Sliced-Wasserstein (SSW) distance with a weighted average, where the weight of each direction is learned from the two distributions being compared. The resulting Discriminative Spherical Sliced-Wasserstein (DSSW) distance weights each geodesic projection by a projected energy function, either a hand-designed transform of the projected Wasserstein distance or a small neural network. The authors argue that this data-adaptive weighting preserves the good properties of SSW—symmetry, weak continuity, dimension-free sample complexity, and a Monte Carlo projection error bound—while capturing discriminative information hidden in individual directions. If correct, DSSW offers better spherical discrepancy measurement with negligible extra cost in the non-parametric case, and the experiments report gains over SSW, S3W, SW, and Wasserstein in gradient flows, density estimation on earth data, autoencoders, and self-supervised learning.","feed_headline":"Data-adaptive weights sharpen spherical sliced-Wasserstein","feed_subtitle":"Reweighting projection directions from data improves sphere learning tasks at little extra cost.","key_machinery":"The load-bearing object is the weighted projection integral defining $\\mathrm{DSSW}_p^p$: the expected value, over the uniform measure $\\sigma$ on the Stiefel manifold $V_{d,2}$, of $f(P^U_\\#\\mu, P^U_\\#\\nu)$ times the $p$-th power of the Wasserstein distance between the two circular projections. The projected energy function $f$ is defined in Definition 2 as a normalized transform of the projected Wasserstein distances: $$f($P^{{U_\\ell}}$_\\#\\mu, $P^{{U_\\ell}}$_\\#\\nu) = \\frac{g(h($P^{{U_\\ell}}$_\\#\\mu, $P^{{U_\\ell}}$_\\#\\nu))}{\\sum_{k=1}^L g(h($P^{{U_k}}$_\\#\\mu, $P^{{U_k}}$_\\#\\nu))},$$ with $h$ either the Wasserstein distance itself or a neural network and $g$ the exponential, identity, or polynomial function. This mechanism converts the unweighted SSW average into a softmax-like weighting that emphasizes discriminative directions; the theoretical results (positivity and symmetry, weak convergence, dimension-free sample complexity, and the Monte Carlo bound) all flow from bounding $f$ in $(0,1)$ and from the corresponding properties of SSW.","core_discovery":"On the paper's own terms, the central claim is that for probability measures on the sphere $S^{d-1}$, the quantity $$\\mathrm{DSSW}_p^p(\\mu,\\nu;f) = \\int_{V_{d,2}} f(P^U_\\#\\mu, P^U_\\#\\nu) W_p^p(P^U_\\#\\mu, P^U_\\#\\nu) d\\$\\sigma$(U)$$ is a strictly better spherical discrepancy than the unweighted SSW integral. Here $V_{d,2}$ is the Stiefel manifold of $d \\times 2$ orthonormal frames, $P^U_\\#$ maps a measure to its geodesic projection on the circle, and $f$ is a projected energy function that assigns larger weight to directions with larger projected Wasserstein distance. Two families of $f$ are given: a non-parametric form (exponential, identity, or polynomial transform of the projected distance, normalized across sampled directions) and a parametric form (linear, nonlinear, or attention network trained on projected supports). The paper claims DSSW is positive and symmetric, respects weak convergence, has sample complexity independent of dimension, and has Monte Carlo error bounded by the variance of the projected Wasserstein distance divided by $L$, matching SSW's guarantees. Experimental comparisons across gradient flows, density estimation on Earthquake/Flood/Fire data, sliced Wasserstein autoencoders, and CIFAR-10 self-supervised learning report consistent improvements, with parametric variants generally outperforming non-parametric ones.","pith_inferences":["The same weighting-by-projected-Wasserstein idea could be applied directly to other sliced distances on non-Euclidean spaces, such as stereographic or rotation-invariant spherical variants, to give them the same discriminative behavior.","The softmax-like normalization in Definition 2 means the implemented $f$ changes with $L$ and with the random set of sampled projections; the paper's theoretical statements, which take $f$ as fixed and bounded, would need re-derivation for the implemented estimator to verify the constants remain valid.","The learned weights themselves may serve as an interpretability tool: for a pair of distributions, the network's weights identify the directions that carry most of the discrepancy, which could be visualized on the sphere.","One could test the robustness of the parametric variants by training the weighting network on one pair of distributions and evaluating the weighted distance on a different pair, measuring whether the learned emphasis transfers."],"forward_implications":["With the non-parametric energy function, DSSW costs essentially the same as SSW: the added weight computation is $O(L)$, while the projection and circular-OT steps dominate.","The parametric variants add only $O(TL)$ training cost and are reported to further improve accuracy in all four evaluation setups.","DSSW shares SSW's theoretical guarantees: symmetry, convergence under weak convergence of measures, sample complexity independent of dimension, and Monte Carlo projection error that decreases with the number of projections $L$.","Because the weights are proportional to projected Wasserstein distance, DSSW places more emphasis on directions in which the two distributions differ most, which explains the improved gradient-flow and density-estimation behavior."],"supporting_citations":[{"why":"Defines the Spherical Sliced-Wasserstein distance and the circular Wasserstein computations that DSSW extends.","marker":"Bonet et al. 2023"},{"why":"Supplies the energy-based weighting and normalization scheme used for the non-parametric projected energy function.","marker":"Nguyen and Ho 2024"},{"why":"Provides the statistical and projection-complexity results that the DSSW proofs adapt.","marker":"Nadjahi et al. 2020b"},{"why":"Defines the S3W baseline and the experimental protocols that DSSW is compared against.","marker":"Tran et al. 2024"},{"why":"Gives the Wasserstein distance background and computational optimal transport foundations used throughout.","marker":"Peyré and Cuturi 2019"},{"why":"Establishes the alignment and uniformity decomposition of contrastive objectives that DSSW is plugged into for self-supervised learning.","marker":"Wang and Isola 2020"},{"why":"Provides the exponential-map normalizing flow model used for density estimation on the sphere.","marker":"Rezende et al. 2020"},{"why":"Defines the sliced Wasserstein autoencoder framework used in the generative modeling experiments.","marker":"Kolouri et al. 2019b"}],"fun_headline_variants":["Data-aware weights boost spherical Wasserstein","Adaptive projection weights improve sphere distance learning","Reweight projections: sharper spherical sliced-Wasserstein","Discriminative weights sharpen sphere sliced-Wasserstein","Learning adaptive directions for better spherical distance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theory treats the weight function $f$ as a fixed function of just the two projected measures, bounded in $(0,1)$, while the implemented $f$ is normalized over the randomly sampled projection directions and therefore depends on the whole sample and on $L$.","fun_headline_variants_meta":{"raw":{"variants":["Data-aware weights boost spherical Wasserstein","Adaptive projection weights improve sphere distance learning","Reweight projections: sharper spherical sliced-Wasserstein","Discriminative weights sharpen sphere sliced-Wasserstein","Learning adaptive directions for better spherical distance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000637,"raw_usage":{"total_tokens":3020,"prompt_tokens":1112,"completion_tokens":1908,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":728,"completion_tokens_details":{"reasoning_tokens":1840}},"tokens_in":728,"tokens_out":1908,"duration_ms":14872,"temperature":1.0,"reasoning_tokens":1840,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:49:27.895925+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix two distributions on $S^2$ with one dominant projection direction, compute the DSSW Monte Carlo estimator in Definition 2 for $L = 10$, $100$, and $1000$, and compare the empirical mean squared error against the bound $C^2 \\operatorname{Var}(W_p^p)/L$ from Theorem 1; if the error fails to decay at rate $1/L$ with a constant independent of $L$, the bounded-$f$ assumption behind the proof is violated.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Spherical Sliced-Wasserstein distance and the circular Wasserstein computations that DSSW extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the energy-based weighting and normalization scheme used for the non-parametric projected energy function."},{"cited_title":"D.; and Kolouri, S","cited_arxiv_id":null,"evidence_quote":"Defines the S3W baseline and the experimental protocols that DSSW is compared against."},{"cited_title":"J.; Papamakarios, G.; Racaniere, S.; Albergo, M.; Kanwar, G.; Shanahan, P.; and Cranmer, K","cited_arxiv_id":null,"evidence_quote":"Provides the exponential-map normalizing flow model used for density estimation on the sphere."}],"review_version":1}