Efficient Sliced Wasserstein Distance Computation via Adaptive Bayesian Optimization
Authors: Manish Acharya
David Hyde
Venue: ICLR 2026 Poster (2026)
Keywords: Sliced Wasserstein Distance
Wasserstein Distance
Bayesian Optimization
Bayesian Quadrature
Quasi-Monte Carlo
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: We develop algorithms that involve using Bayesian optimization techniques for computing sliced Wasserstein distances (SW), and we show our algorithms can achieve state-of-the-art on this task.

Abstract: The sliced Wasserstein distance (SW) reduces optimal transport on $\mathbb{R}^d$ to a sum of one-dimensional projections, and thanks to this efficiency, it is widely used in geometry, generative modeling, and registration tasks.  Recent work shows that quasi-Monte Carlo constructions for computing SW (QSW) yield direction sets with excellent approximation error.  This paper presents an alternate, novel approach: learning directions with Bayesian optimization (BO), particularly in settings where SW appears inside an optimization loop (e.g., gradient flows).  We introduce a family of drop-in selectors for projection directions: **BOSW**, a one-shot BO scheme on the unit sphere; **RBOSW**, a periodic-refresh variant; **ABOSW**, an adaptive hybrid that seeds from competitive QSW sets and performs a few lightweight BO refinements; and **ARBOSW**, a restarted hybrid that periodically relearns directions during optimization.  Our BO approaches can be composed with QSW and its variants (demonstrated by ABOSW/ARBOSW) and require no changes to downstream losses or gradients.  We provide numerical experiments where our methods achieve state-of-the-art performance, and on the experimental suite of the original QSW paper, we find that ABOSW and ARBOSW can achieve convergence comparable to the best QSW variants with modest runtime overhead.  We release code with fixed seeds and configurations to support faithful replication (see supplementary material).

Published as a conference paper at ICLR 2026
EFFICIENT SLICED WASSERSTEIN DISTANCE COMPUTATION VIA ADAPTIVE BAYESIAN OPTIMIZATION
Manish Acharya and David Hyde
Department of Computer Science
Vanderbilt University Nashville, TN, USA
mmanish.acharye david. hyde. 1 (eed1devvvddbbllee rbi eeluudu
ABSTRACT
The sliced Wasserstein distance (SW) reduces optimal transport on R2 to sum of one-dimensional projections, and thanks to this efficiency, it is widely used in geometry, generative modeling, and registration tasks. Recent work shows that quasi-Monte Carlo constructions for computing SW (QSW) yield direction sets with excellent approximation error. This paper presents an alternate, novel ap- proach: learning directions with Bayesian optimization (BO), particularly in set- tings where SW appears inside an optimization loop (c.g., gradient flows). We in- troduce a family of drop-in selectors for projection directions: BOSW, one-shot BO scheme on the unit sphere; RBOSW, a periodic-refresh variant; A ABOSW, adaptive hybrid that seeds from competitive QSW sets and performs few lightweight BO refinements; and ARBOSW, restarted hybrid that periodically relearns directions during optimization. Our BO approaches can be composed with QSW and its variants (demonstrated by ABOSW/ARBOSS
and require no changes to downstream losses or gradients. We provide numerical experiments where our methods achieve state-of-the fe--hhe-n art performance, and on the experimental suite of the original QSW paper, we find that ABOSW and ARBOSW can achieve convergence comparable to the best QSW variants with modest runtime overhead. We release code with fixed seeds and configurations to support faithful replication (see supplementary material).
INTRODUCTION
Optimal transport (OT) is mathematical framework for measuring distances between probability measures (Villani et al. 2008). OT has seen increasing interest from the learning community in recent years since, for example, collections of sample data can be interpreted as empirical distribu- tions. In particular, the Wasserstein distance is popular metric for OT tasks in the learning literature (compared to, e.g., Kullback-Lieible (KL) divergence, which is not symmetric, nor does it satisfy the triangle inequality) (Solomon et al. 2014 Montavon et al.) 2016 Kolouri et al. (017))
However, evaluating the conventional Wasserstein distance (WD) is computationally prohibitive (O(n' log n) in time and O(n²) in space (Nguyen et al. 2023al), particularly for higher-dimensionaal measures, where computing Wasserstein distances can become the bottleneck of an application (Kolouri al.] 2019). The sliced Wasserstein (SW) distance (Bonneel et al.) 2015a) offers scalable alternative to the classical WD by averaging one-dimensional Wasserstein costs over projections on the unit sphere, giving Oon log n) time and (n.) memory for discrete measures via sorting, and enabling its use across geometry, imaging, and generative modeling tasks (Peyré & Cuturi 2019 Formally, the SW between μ4, v E (I (p 2 1)
Bonneel et al. 2015a
SW v) Eo [W3(03/4, 0,v)]-
Note that Bonnotte (013) showed SW equivalent to WD for compactly supported measures.

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Here, d is the ambient dimension, and p 2 1 is the order of the Wasserstein cost. Pp (R.)) denotes the set of Borel probability measures on R with finite p-th moment, i.c., J || du(x) cc. gd-1 is the unit sphere, and U((K-1) is the uniform distribution on it. For direction 0 E g the pushforward fe is the distribution of the scalar projection A XX when X ~ f (i.e., un- der x + @T 2). Wp(:: is the p- o-Wasserstein distance on I; we average its p-th power over Te (W7 (0,1,, 0yv)]. Equivalently, one may write V))
directions, i.e., VW)(
(0gH, 0.
Of course, in practice, the expectation integrates over all directions must be estimated using a finite set of directions. Given L directions ("slices") 92 = {0e)é- we estimate
θ2) 1 Σ WB((( (0e)
(2)
For fixed budget of L slices, the quality of SW estimate is therefore determined by the set O C gd-1, For notational convenience, we denote f(0; p,, v) (0,0, 0gv).
In the absence of any information, natural choice for selecting the O2 is random Monte Carlo (MC) sampling, which famously suffers from a slow OLL-1/2) error convergence rate (Nadjahi ct al. 2020). To mitigate this issue, recent work has applied quasi-Monte Carlo (QMC) sampling techniques estimate SW (Nguyen et al. 2024a). However, QMC methods still sample pscudo- randomly, merely using inte-inneeendent heuristics to generate sample sets with more uniform cov- erage of the space being sampled (i.e., lower discrepancy) (Niederreiter 1992 Dick & Pillichsham- mer 2010). Randomized counterparts (ROSW) restore unbiasedness for stochastic optimization while retaining uniformity (Nguyen al. 2024a).
However, in estimating SW, there is not an absence of information; the motivating idea of our paper is that when computing SW, every slice yields additional information that can inform the selection of further slices. For instance, if two nearby I have substantially different values of [0; μu, 1/), one may wish sample additional slices between those two slices. Put another way, given fixed budget of slices (especially for small L.). the goal of selecting slices in SW should not be uniform coverage, but rather to maximize coverage where most of the "signal" of SW is determined.
Based on this insight, the present paper develops novel family of algorithms for estimating SW, learning to select projection directions via Bayesian Optimization (BO), which proves to be espe- cially effective in settings where SW is called inside optimization loop (e.g., gradient-flow-stlle updates). BO has become popular, sample-efficient strategy for selecting informative queries un- der tight budgets across hyperparameter tuming, experimental design, and robotics (Shahriari al.) 2016 Snock et al., 2012] Garnett 2023; Vardhan et al. 2024). Unlike QSW, which provides task- agnostic uniform coverage of the sphere, BO exploit structure in the projection landscape to prioritize informative directions and adapt them as the task evolves. Our thesis is that when the objective repeatedly queries SW on evolving distributions, small set of task-adapted directions can accelerate convergence without altering downstream losses or gradients-and that BO provides a simple, black-box way to pick them.
Of course, BO is not the only possibility for adaptive refinement of SW estimates. For instance, control variates such as the up/low method of [guyenn & Ho (2024) or spherical harmonics as in Lelue et al. (024) have been shown to improve MC convergence in the context of SW. Repulsive point processes for MC have also been applied SW (Petrovic et al., 2025). showing potential advantages in higher dimensions. Random-path projecting directions (Nguyen et al.] 2024b) and Markovian SW (Nguyen et al. (2023b) both seek to select informative directions and show improved performance over baseline MC SW. Nonetheless, a recent survey on sampling for sliced OT (Sisouk ct al. 2025) confirms BO remains unexplored in this area.
Our paper ultimately introduces and evaluates four drop-in. BO-based direction selectors: BOSW:
a one-shot BO search on Sd-1 to pick L directions; RBOSW: a periodic-refresh variant that reuses BO for light retuning during optimization; ABOSW: an adaptive hybrid that seeds from strong QSW sets and applies a few lightweight BO refinements and ARBÓSW: a restarted hybrid that periodically releams directions (fresh BO) while retaining QSW seeding.
We note that our idea of leveraging BO can be complementary to QSW approaches: for instance, combining QSW and BO yields our ABOSW and ARBOSW methods, which often achieve the best
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performance in numerical results. ROSW-style randomization can still be layered when unbiased stochastic gradients are required (under the constructions proposed Nguyen et al. (2024a))
The contributions of our work include:
1. We provide family of simple, plug-and-plas selectors for SW projections driven by BO. In appropriate settings, these methods achieve state-of-the-arr performance (measured by error vs. number of iterations) for estimating SW with a finite number of slices (BO makes estimators biased and prevents convergence to the true, unbiased SW value limit).
2. We integrate these methods into the QSW/RQSW pipeline without changing downstream losses/gradients.
3. On the QSW paper's test suite Nguyen et al. 2024a our hybrid variants (ABOSW/ARBOSW) show convergence competitive with the best QSW baselines mod- est overhead; we follow their reporting protocol for one-to-one comparability.
2 BACKGROUND
Wasserstein and sliced Wasserstein. Consider cost function metric I2 x R + R++ and let Pe((d) be the set of probability measures on IR where there exists an zo such that Ja c(x, Zo (x))) < cc. Then, given two measures M.v E Pe(I). the p- o-Wasserstein distance (p 2 1) between and is defined by
/ c(x, [( v) dr(xx)))
W2,
min "Ell(W))
where [[(s4, v)) denotes couplings with marginals μa v (gguyen 2025 Peyré Cuturi 2019). The standard choice of c(z, y) is ||x- yllp- in which case the is conventionally omitted from subscripts.
Computing Wepp in high dimension can be costly (cf. Section I). The sliced Wasserstein (SW) distance mitigates this averaging one-dimensional Wasserstein costs directions the unit sphere (see Equation T.. Following the notation of Ngguyen (2025), the one-dimensional Wasserstein-p distance between μ4, V. Po(
WP ( v) dx, F Fx(x)dpp(z),
where Fμ is the cumulative distribution function (CDF) of μ. and F,1 is the generalized inverse CDF of v:
(x) inf(y R|x S F.()).
Nguyen 2025) elucidate (see their Remark 2.9) that the computational complexity of evaluating e, in the one mee-iieesional case is (n log n.) in time and O(n) in space.
As discussed in Section we use the finite-direction estimator in Equation 2 with a set of slices 0, = (0:) With i.i.d. Monte Carlo (MC) directions, the root mean square error scales as O(L-1/3)
Quasi-Monte Carlo Methods. Quasi-Monte Carlo (QC) methods replace uniform random
draws by fow-discrepancy point (Niederreiter) 1992 Dick & Pillichshammer 2010 improving integration error for sufficiently smooth integrands (e.g.. a convergence rate of O O (log L)d) vs.
(L/21 for MC (Caflisch 1998)). The idea of QMC for integrals dates back at least 50 years Zarembal 1968). In either MC QMC methods, an integral estimated as discrete average of function evaluations (cf. Equation]] 2)
I [(x)dxx f(x).
The distinction of QMC is rather than uniformly randomly sampling points from the domain of f, one uses quasi-random (i.e., cow-discrepancy sequences of points. For instance, one dimension, der Corput sequence (van der Corput (935) takes the positive natural numbers, expresses

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them in base (commonly base 2). reverses the digits, moves the digits after the radix point, and translates back into decimal (for instance, + 101₂ - 101₂ - 0.101, - 0.625). This scheme repeated using different bases in different dimensions to form Halton sequences (Halton 1964), common multi-dimensional quasi-random scheme. Sobol sequences (Sobol 1967] are related, but notably, they use only hase 2, enabling fast computational arithmetic (Pharr al. 2023).
QSW and RQSW. QSW instantiates QMC on S² via (i) equal-area transforms of Sobol points, (ii) Gaussianized Sobol points (Ökten & Göncü 2011) projected to the sphere, and (iii) structured deterministic sets such spiral sequences and energy/distance-opiiit
designs; it also studies randomized counterparts (RQSW) that either scramble the net or apply random rotations, which re- store unbiasedness while retaining uniform coverage (Nguyen et al. 2024a). Since these algorithms have cate-of-the-an performance, we evaluate our methods against the same experimental protocols (datasets, metrics, and reporting) as Nguyen et al. (2024a) for fair comparison.
Bayesian optimization (BO). BO is state-of-the-art sample-efficient strategy for optimizing expensive, black-box functions (Shahriari et al., 2016; [Snoek al.] 2012] Garnetr, 2023 Vard- han et al. 2024). BO works by first fitting a surrogate model . to set of seed observations Dn = (T), {))))>=> With M as Gaussian process (GP), for example, the posterior predictive distribution at a new point x is normally distributed:
f(z) Dn ~ N(((z).
where y1 and 2² are posterior mean and variance. 0² depends kernel (covariance) function k. For details on how the GP hyperparameters are fit to the data, we refer to the excellent tutorial and visualizations of Duque (2023).
To determine the next point to sample, BO optimizes an acquisition function a<x) and adds the resulting to D.. A popular choice of acquisition function is the upper confidence bound (UCB), which, for a GP surrogate, takes the form
aucB (z; B) ((z) + Bo(x),
where is a scalar hyperparameter that balances mean (exploitation) and variance (exploration) terms. The exploitation term favors candidate points that look promising based the GP prior, whereas the exploration term pushes BO to select points from undersampled regions where little is known (large predictive variance).
Problem view for this work. Many learning tasks repeatedly evaluate sliced Wasserstein distances between an evolving distribution and fixed target, e.g., gradient flows, registration, and neural network training. In such settings, the projection directions act purely numerical estimator and can therefore adapted to the task hand without changing the underlying optimization objective. Our BO-based methods learn such task-adapted direction sets: BOSW and ABOSW leamn fixed set once, while RBOSW and ARBOSW periodically refine the set as the distribution evolves. All variants drop into existing SW pipelines, c.g., those from Nguyen et al. (202a)). without modifying downstream losses or gradients.
3 METHODS
3.1 BAYESIAN OPTIMIZATION ON THE SPHERE
We treat f(0; (1, v) from Equation 2 as black-box function gd-1 and fit GP surrogate with covariance (kernel) :.. We use the angular RBF kernel, i.e., a Gaussian of the spherical geodesic distance:
k(0,6" 8') exp (a110000))
ds(0, 0') arccos((,
(3)
with lengthscale L (set by median heuristic pairwise geodesic distances among the currently evaluated directions). This isotropic (zonal) kernel on spheres is standard (Schoenberg)
1942 Borovitskiy al. 2020; Rasmussen & Williams, 2006).
Given a budget of L slices {0.)- the goal of Bayesian optimization in the context SW is to select the slices maximize information about the integral in Equation This is sequential
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experimental design problem selecting which projection directions E s to evaluate next, given previous observations.
At iteration t, with data D; {(0,, f(0))) (so n. (.)), the GP posterior (1e, Øz) defines an acquisition az (f); by default we use UCB with = 0.7 (Srinivas et al. 2010 Jones et al. 1998; proposed by maximizing 04 7, sampled uniformly on gd-1 (we use n. 4096 unless stated), selecting a small batch & (default b = 5). Candidates are normalized unit norm and we suppress near-duplicates by dropping any proposal whose cosine similarity to the current set exceeds 0.98. These small, parallelizable rounds keep the BO overhead modest relative to SW evaluation; per round, scoring costs O( n2) kernel evaluations and adds only b new f-evaluations.
Shahriari et al. 2016). New directions
candidate pool of size
We performed ablations choose the models and parameters listed here: see Appendix C Further- more, we discuss the theoretical guarantees on using BO for SW in Appendix D
3.2 FOUR VARIATIONS OF BAYESIAN OPTIMIZATION FOR SLICED WASSERSTEIN
BOSW: one-shot learned directions BOSW performs a single BO run to select L directions, starting from small random initialization and iteratively expanding θ2 until full. The learned set remains fixed during downstream optimization. In terms of computational cost, each BO round requires ((n,7e)) kernel evaluations, which is modest for L in the hundreds. We note that the task-adapted directions BOSW learns do not comprise unbiased quadrature rule for the spher- ical average. We include BOSW in the approximation error plot (see Section 442) only for com- pleteness and do not expect it to excel there; for unbiased estimation we rely on (R)QSW. We omit RBOSS/ABOSW/RRROO optimizationniin-tt-llooo rather than one-off integral.
from that plot since their refresh/seeding mechanisms target
periodic refresh RBOSW re-runs the BOSW selection process every R optimization steps using updated source and target distributions (i.e., the current pair of distributions whose dis- tance is being measured with SW at step t). Between refreshes, the selected directions remain fixed. This adaptation can track evolving geometry without restarting the main optimization loop. We note that "refresh" here means re-running BOSW from scratch on the current data without carrying forward the previous surrogate model.
RBOSW:
ABOSW: QSW-seeded, lightweight refinement ABOSW is variant that uses adaptive BO: standard GP-based BO (UCB) whose proposals are conditioned on the slice values already observed. ABOSW is adaptive the task initialization (QSW seed few BO refinements). but not time- adaptive; the set is refined once and then fixed.
ABOSW begins with a strong QSW set (e.g., spiral/Coulomb) and performs a few BO rounds to lightly adjust it:
1. Initialize the GP by evaluating f(f) on the QSW seed θ these evaluations form the initial dataset Do-
2. Run small number of BO rounds (in our experiments, we run 2 rounds with mini-
batch size = 5; thus at most br I 10 directions change when L 100, i.., S 10% of the set). At each round: sample candidate pool of size n. 4096 uniformly Sd-1 score by UCB (default 3 = 0.7), suppress near-duplicates (cosine similarity > 0.98), pick b proposals, and replace the b worst directions in OL.
3. Return the refined set ④ the overhead is small since only br new directions are evaluated (mere seconds of compute, in our experiments).
restarted hybrid ARBOSW periodically restarts the ABOSW process: at fixed in- tervals. it re-seeds from a QSW set and runs the short BO refinement routine on current data, The key distinction from RBOSW is that each restart begins from QSW initialization rather than the previous BO state, enabling both periodic adaptation and consistent seeding.
ARBOSW:
Compatibility with randomized estimators When unbiased stochastic gradients are needed, ran- domized QSW remains appropriate (Nguyen al. 2024a). Our BO-based selectors orthogonal
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Quadratic Landscape
0656
5068
05/9
Diot FEONDO
Figure 1: Synthetic projection-selection experiment on three fitness landscapes (higher is better). Bayesian optimization consistently and quickly finds the optimal projection directions. In the ridge example, one of the (pseudo-random) QSW methods happens identify useful slice early on; but BO clearly succeeds across all test cases.
to this choice: QSW (deterministic or randomized) provides the base set, and BO offers refinement or re-selection. We do not claim unbiasedness for COSW-basedd sets (cf. Appendix D): evaluations follow the same deterministic vs. randomized protocols as Nguyen et al. (2024a).
3.3 COMPUTATIONAL OVERHEAD
The dominant cost of sliced Wasserstein evaluation is the On log n) sorting n projected points for each of the J. slices. BO-driven selectors add additional cost of scoring ne candidate di- rections (we use 4096) and evaluating a small batch of b = 5 new slices per refinement round, giving an overhead of O (n. + log n) relative plain SW/QSW. In practice, this cost is modest compared to the core sorting operations: BOSW and ABOSW run within a few percent of QSW run- times, ARBOSW roughly 1.5-2x slower due occasional refinements, and only RBOSW incurs substantial overhead due to its repeated refreshes. This pattern is consistent across all experiments, including point-clour interpolation (Table T), image translation, and autoencoding.
4 EXPERIMENTS
Aside from Section our experiments match those run in Nguyen et al. (2024a): we use the same datasets, tasks, and reporting style, and we only "swap" the projection selector (ours vs. theirs). Across all tasks, we keep the optimizer, learning rates, and stopping criteria identical to Nguyen jet al. (2024a) to ensure one-to-one comparability. well as vanilla Monte Carlo SW (denoted MCSW in our results). We note that in all our experiments, for methods with refreshes (RBOSW and ARBOSW), we use refresh interval of R 23, which found to be balance of speed and accuracy (see Appendix
include the QSW families used the paper, as
4.1 SYNTHETIC COOUECTIOO-SELECTIIO EXPERIMENT
Settings. To verify that Bayesian optimization (BO) can adaptively identify high-quality projec- tion directions, we construct synthetic "fitness landscapes" over the sphere where certain directions yield much higher rewards. We design three landscapes: (i) Peaks, with several local maxima and one dominant peak; (ii) Ridge, where projections aligned with [1, 1, achieve high value; and (iii) Quadratic, where a single target direction maximizes fitness. Each method given small eval- uation budget L E (5, 10, 15, 20} and must return the best projection found. We compare BO against QSW baselines (EQSW, GQSW, SQSW, DQSW, CQSW) and their randomized counter- parts (RQSW, RGQSW, RSQSW, RDQSW, RCQSW) (seyen et al. (2024a) for details of each variant). Performance measured by the mean best fitness value over 5 trials.
Results. Figure summarizes the outcomes. On the Peaks landscape, BO steadily improves with budget and clearly outperforms all QSW variants L 15 and 20 (best value 1.90 versus 1.2 for the strongest QSW). On the Ridge landscape, QSW methods that happen to align with the ridge (notably GQSW) perform well, but BO rapidly adapts approach the optimum, surpassing

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randomized variants at all budgets. On the Quadratic landscape, BO dominates: it already finds near-optimal projections by -= 10 (value 2.69 vs. I 1.0 for QSW/RQSW), and reaches the global optimum ~ 2.84 by L 15, while QSW remains below 2.4. These results illustrate how BO lever- ages feedback to focus search on promising regions, whereas QSW's fixed designs (though effective for uniform coverage) cannot adapt landscape structure. This synthetic experiment confirms BO's capability as an adaptive projection selector, motivating its use in sliced Wasserstein computations.
4.2 APPROXIMATION ERROR
Setting. We select four point clouds (indices 1-4; 2048 points; 3D) from the ShapeNet Core- 55 (Chang et al. 2013) and use them as in Nguyen et al. (002a)) The population (ground truth) SW value for each pair (1-2, 1-3, 2-4, 3-4) is approximated by a high-budget MC estimate with L = 100,000. L ranges from 10 to 10, 000. We report the absolute error of: (i) MCSW, (ii) the QSW variants (GQSW, EQSW, SQSW, DQSW, CQSW), and (iii) our BOSW (one-shot BO). Stochastic methods (MCSW, BOSW) averaged over five seeds; QSW variants are deterministic.
Results. Figure 2 shows absolute error versus L. across the four pairs. Unsurprisingly. QSW vari- ants consistently yield lower errors than standard MC, with CQSW/DQSW (and SQSW/EQSW closely behind) approaching the s 10 regime once L Z 10S, In contrast, BOSW exhibits notably larger error across all L and pairs, decreasing with L but remaining above MC. This is expected: in this problem, the data are highly uniform, such that slices along each direction are roughly equally meaningful. Since BOSW learns task-adapted directions and is not designed to uniformly integrate the sphere, it does not perform competitively on approximation error for task of this nature. Conse- quently, in subsequent sections, we focus on optimizationnin-thee where learned directions via BO are shown to help.
settings (e.g., gradient flows),
4.3 POINT-CLOUD INTERPOLATION
Setting.
We evolve
(t) = - n V 2(t) [SW (S22(2(0) Pr]]
to interpolate between two point clouds X and Y. Here, Px, Py empirical distributions over X and Y, and the curve starts at 7(0)= = X (and ends at Y). We use the same Fuler scheme as Nguyen et al.] (2024a): 500 iterations with step size 0.01, and evaluate the distance between (2)(e) and Py using the 2-Wasserstein metric from POT (Flamary et al. 2021). We set 100 directions for all methods and report the mean + standard deviation over thirty seeds at steps t E {100, 200, 300, 400, 500). We compare MCSW. the QSW family, the randomized QSW (RQSW) family, and our BO-driven selectors: BOSW (one-shot), RBOSW (periodic refresh), ABOSW (hybrid seed+refine), and ARBOSW (restarted hybrid).
Results. Table reports W2 (x10²) and wall-clock time, As in Nguyen et al. 2024a), random- ized QSW variants generally yield the shortest trajectories at later steps. However, our ARBOSW
10-1
Absuitte Eror
MCSW
70-3
603W
F3SW
535W
2000
4000
6000 0020 2000
30000
40.00
65000
0020
30200
2000
4000
6000
2030
30200
2000
4202
6000
8000
10000
Figure 2: Approximation emror for SW between empirical distributions over point clouds. BO does not provide an advantage compared to QSW methods, which excel on this type of problem that relies on uniformly sampling the sphere of possible directions.
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achieves the best value step 500, with moderate overhead (6.9 vs 4.2s for QSW baselines). RBOSW is strongest early (best at steps 100 and 200) but incurs large cost due to periodic BO re- freshes. The deterministic one-shot BOSW trails (consistent with the need for fresh directions the flow evolves), while ABOSW performs similarly to the deterministic QSW group. Overall, BO hy- brids provide competitive convergence without changing the optimization loop. For completeness, we also report results at a much smaller budget (L- = 10 directions) in Appendix [A.3] The trends are consistent: ARBOSW remains among the top-performing estimators, RBOSW dominates early steps but is slower, and BOSW/ABOSW align with their deterministic counterparts.
Estimators
Step 100 (W, 4)
Step (W: ()
Step 300 (W; ())
Step 400 (W2 J)
Step S00(W; 4)
Time (s.)
MCSW
5,749 1: 0.079
0.187 0.006
0,031 1: 0.002
0.013 1: 0.002
0.006 0.002
4,06
GQSW
6.082 1: 0.0
0.251 :: 0.0
0.078 1: 0.0
0.070 1: 0.0
0.069 :: 0.0
4,23
BQSW
5,408 + 0.0
0.246 : 0.0
0.083 + 0.0
0.074 +: 0.0
0.072 = 0.0
3.85
SQSW
0,085 +
0.0
4,08
DQSW
5.771 +: 0.0
0.201 = 0.0
0.075 +: 0.0
0.065 + 0.9
0.064 = 0.0
3,96
CQSW
5.603 4: 0.0
0.183 : 0.0
0.078 4: 0.0
0.073 +: 0.0
0.071 + 0.0
3.96
RGOSW
5.713
0.001
RRGQSW
5,735 + 0.035
0.183 0.007
0,029 + 0.003
0.012 + 0.003
0.005 + 0.002
4.01
REQSW
5,705 + 0.023
0.183 0.002
0.029 +: 0.002
0.013 + 0.002
0.007 0.002
3.87
RREQSW
5.721
0.002
0.002
0.002
RSQSW
5.710 0.000
0,152 0.003
0.026 : 0.002
0.010 + 0.002
0.005 = 0.002
3.95
RDOSW
5,711 1: 0.004
0,181 0.003
0.027 1: 0.002
0.012 1: 0.002
0.005 0.001
4,00
ROQSW
5,708 + 0.005
0.181 0.002
0,027 + 0.003
0.011 + 0.003
0.002
BOSW
3,744 + 0.119
0:217 0.040
0.101 + 0.013
0.991 + 0,010
0 + 0.009
3.81
RBOSW tours)
0,083
0.004
0.005
ABOSW (ours)
6.199 1: 0.522
0.298 0.084
0.087 1: 0.012
0.077 1: 0,009
0.075 1 0.008
3.95
ARBOSW (ours)
5.717 +: 0.079
0.186 0.004
0,025 :: 0.003
0.012 + 0.001
0.003 0.001
6.91
Table 1: Summary of Wasserstein-2 distances (multiplied by 10²) from 30 different runs (L=100).
4.4 IMAGE STYLE TRANSFER
(n is the number of pixels) and are evolved along the same SW-driven curve as in Section 4.3 As in Nguyen et al. (2024a), we round RGB values to (0..... 255} at the final Euler step, use 1,000 iterations with step size 1, and set L 100 projection directions for all methods; W2, is computed with POT (Flamary) jet al., 202I). We visualize MCSW, RCQSW, and our RBOSW/RBOSS. We report RCQSW as the representative ROSW baseline because the QSW study found it consistently among the top performers. Randomized variants behave very similarly overall; see Appendix A.6 for full results.
Setting. RGB source and target image are treated as point clouds X,Y E R 3
Results. Figure 3 shows a representative transfer. Both RCQSW and our BO hybrids clearly out- perform vanilla MCSW in terms of final W2 (printed above each panel) and visual fidelity. For ease of exposition, we keep RCOSW as the representative randomized baseline in the main text, mirror- ing the recommendation of Nguyen et al. (2024a); see Appendix (x.4nddx(A.4foo for full comparisons. We note, for transparency, that other randomized variants can be marginally stronger particular, RGQSW achieves the lowest final W2 on our example at both L=10 and L-100. How- ever, the randomized family behaves very similarly overall, and RCQSW remains competitive. Our ARBOSW matches RCQSW on this example while preserving contrast and texture; RBOSW im- proves over SW but typically trails RCQSW/ARBOSW consistent with its lighter periodic refresh. Thus, while not as clear of an accuracy advantage as seen in Sections and 4.5 ARBOSW is at least competitive with the ofate-of-the-art for this example.
some instances; in
Source
MCSW M2 22
ACOSW Wy 0.72
R2OSH W2 17
ARBDOW; Mj 0G5
Taree
Figure 3: Image style transfer with L=100 and 1,000 iterations.
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Appreximation
Epoch 100
Epoch 200
Epoch 400
SWa(I)
Wa(I)
SWa(U)
Wa())
SWa(()
Wa(()
MCSW
2.25 +: 0.06
10.58
+ 0.12
2.11 +: 0.04
9.92 + 0.08
1.94 + 0.06
9.21 #: 0.06
GQSW
11.17 4:
32.58 +: 0.06
11.75 : 0.07
33.27 = 0.09
14,82 + 0.02
37.99 0.05
BOSW
2.25 0.02
10.57
+ 0.02
= 0.07
= 0.04
9.20 = 0.07
SQSW
2.25 0.01
10.57
2.08 0.01
9.90 0.04
1,90 + 0.02
9.17 0.05
DOSW
2.24:00.77
10.58
+: 0.05
2.06 0.04
9.83 + 0.01
1.86 4 0.05
9.12 = 0.07
CQSW
2
10.54 0.02
2.05
9.81 = 0.04
1.84 +.02
9.06 = 0.02
RGQSW
0.00
10.57 0.01
0.03
9,92 0.01
1.94 0.02
9.18 = 0.02
RRGQSW
2.23
10.51 1
0.05
9.44
9.16
REQSW
2.04
10,53 + 0.04
2,08 - 0.04
9.90 0.08
1.89 0.04
9.17 = 0.06
RREQSW
2.04
10.50
+: 0.04
2
9.8 0.02
188 0.05
9.15 = 0.06
RSQSW
0.05
10.53 0.01
0.06
9.2
L 0.05
9.12
RDOSW
2.21 0.03
10,50 0.02
2.03
9.82 + 0.04
1
9.12 0.02
RCQSW
022-0000
10,50 + 0.05
2.03
9.82 0.05
1.85 0.06
9.12
BOSW
2.20 0.01
10.34 + 0.02
2.02 0.04
9.78 0.03 0.03
1.00 0.01
9.01 = 0.02
RBOSW
2.28 0.03
10.54 0.04
9.84 0.05
ABOSW
2.18 = 0.01
10,27 + 0.02
2.01 + 0.03
9.76 - 0.02
1.81 + 0.02
9,01 = 0.03
ARBOSW
2.21 0.01
10.44 + 0.02
2.04 + 0.04
9.80 +: 0.03
1885 0.02
9.07 0.02
Table 2: Reconstruction losses on the autoencoder task from different approximations with 100. Losses are scaled by 102 for ease of exposition.
4.5 DEEP POINT-CLOUD AUTOENCODER
Setting.
Following Nguyen al. (2023a) and Nguyen et al. (202a)), we train deep point-cloud
autoencoders with SW on ShapeNet Core-55 (Chang et al. 2015). This amounts to optimizing an objective function:
min xx) (SW, (S Po (oo())())))]
0.3
where (xX) is the data distribution, fo is deep encoder, and Iø is a deep decoder. Both fg and Io use PointNet architecture (Qi et al., 2017). We approximate gradients using either MC, QSW, ROSW, our BO-driven selectors, and train for 400 epochs with SGD (learning rate 10-3 batch size 128, momentum 0.9, weight decay 5x 10 10-+). We set L = 100 directions for all methods. Evalu- ation on the distinct ModelNet40 dataset (Wu et al.) 2015): we report the mean reconstruction loss over three runs, measured against W2 and SW₂ (which are estimated via 10,000 MC projectionn)
Results. Table and Figure show the reconstruction losses and qualitative outputs with 100. Consistent with prior work, CQSW is strong deterministic baseline. However, BO- based methods, particularly ABOSW, achieve the lowest reconstruction losses across epochs, even surpassing CQSW in both SW2 and W2 at the final epoch. BOSW also performs competitively, while the restart variants (RBOSW, ARBOSW) lag slightly behind their non-restart counterparts. This reversal compared to gradient flows highlights the different nature of dataset-level training: here, the distribution is large and stable, so maintaining consistent, high-quality projection sets (as in BOSW/ABOSW) is more effective than frequent refreshes, which can inject unnecessary variability. We emphasize that while methods like RGQSW, RRGQSW, and REQSW were top performers for image style transfer Sectioon4.4) they are among the worst worst-performing methods in Table[[aafter 400 epochs. Overall, BO-based selectors not only remain competitive but, in this setting, outperform the previously recommended CQSW, suggesting that task adaptivityy via BO can provide tangible benefits beyond the low-discrepance sampling enabled by QSW. For clarity, include MCSW, CQSW, and our BO variants in the main text; the full visual and quantitative comparisons with all QSW/RQSW variants deferred to Appendix A.6
CONCLUDING REMARKS
We proposed BOSW, RBOSW, ABOSW, and ARBOSW: Bayesian optimization-driven selectors for projection directions in sliced Wasserstein (SW) computations that complement quasi-Monte Carlo (QSW) designs. Our hybrids combine QSW's low oow-dicrepanccy coverage with BO's task adap- tivity and drop in to existing SW-based optimization loops without changing losses or gradients. Across the QSW benchmark suite of Nguyen et al. (0024a). ABOSW and ARBOSW match the

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Published as a conference paper at ICLR 2026
best QSW baselines in convergence, RBOSW offers strong early-stage gains, and BOSW provides a simple deterministic alternative. Collectively, our methods perform competitively on the image style transfer task, and achieve state-of-the-art convergence for the point-cloud interpolation and deep point-cloud autoencoder tasks. By uniting deterministic geometrie constructions with data- driven adaptivity, work broadens the design space for SW direction sets and opens path toward principled, learned projections in large-scale optimal transport applications.
Reflecting our proposed algorithms, remark on the work of Nguyen & Ho (2023 where the authors reweight slices (directions) for SW based on the slices' ID costs, yielding a new slic- ing distribution. In the view of that work, our proposed nonparametric regression framework uses in- and out-of- out-of-sample weights-training directions and new proposed directions, respectively to go beyond simply defining set of projection directions and effectively define new slicing distri- bution (or, alternatively. data-dependent Radon transform (Bonneel et al., 2015b Kolouri et al.) 2019)). We note that while the importance sampling energy-based SW of Nguyen not an unbiased estimator of SW for finite L, it is an unbiased estimator of a new metric it induces ("importance-wiigheed SW"). Thus, as future work, we are interested in further analyzing bias of and metrics induced by BO-based methods in light of Nguyen & Ho (2023).
Ho (2023) is
common criticism of BO is its poor scaling with respect to the dimension of the data. However, recent liter- ature has suggested several variants of BO that perform well in high di- mensions (Kim al. 2021 Shen & Kingsford| [2021] Binois & Wycoff 2022 Jaquier Rozo| 2020). and in an exciting development. Hvarfner al.) (024) just found that even
NESW
CQSW
vanilla BO can perform excellently in high dimensions with a simple scal- ing of the GP lengthscale. More- over, the GP surrogate, which is pri- mary bottleneck in BO, could poten- tially be replaced with neural surro- gate for performance that scales bet- ter with dimension (Li et al.] 2023) Lim al. 2021). Thus, overall, is no longer fair to claim that BO is cursed by dimensionality. We would like to test our methods on higher- dimensional datasets in future work.
RCQSW
BOSW
ABOSW
ARBOSW
We remark that the present work is reminiscent of (adaptive) Bayesian
Figure 4: Reconstructed point clouds from the deep autoen- coder experiment, with 100.
quadrature (BQ) (Duvenaud, 2012
Kanagawa & Hennig 22099 Briol
al. Osborme et al.] 2012). However, BQ is designed to compute an integral itself, not select individual directions. In our setting, the integrand changes every step, so BQ would require refitting a GP and recomputing quadrature weights each time, for total of O runtime; see Sec- tion A.2 for a short experiment. Thus, while BQ is infeasible to directly apply, it remains closely related to the ideas in our work.
Finally, we imagine several opportunities to improve our methods. Parallelism and GPU implemen- tation, as explored by others in the field (Balandat al. 2020 Knudde al. 2017 Munawar al., 2009), could significantly accelerate the runtime of our methods. Adding constraints or information BO based knowledge of the problem, as in Eriksson & Poloczek (202T): Gelbart (2015): Vard- han et al.) (2023); Jaquier et al. (2022). would give our methods further advantage over data- blind techniques like QSW. Finally, recent variants of BO could offer higher peteerrpeffrmmacee drop-in replace- ments for the BO functions in our paper (Visser al., 2025) McLeod et al.) 2018 Kawaguchi et al, 2015). We aim to explore these extensions in future work.
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ACKNOWLEDGMENTS
This paper was inspired by Ph.D. preliminary exam of student advised by Soheil Kolouri, where the authors found that sliced Wasserstein is about approximating integrals accurately; and by the DARPA Design.R project, which introduced the authors to Bayesian optimization. D.H. acknowl- edges Peter Volgyesi for involving him in the Design.R efforts. We thank the SyBBURE Searle Un- dergraduate Research Program for providing funding and support throughout this research. This ma- terial is based upon work supported by the National Science Foundation under Grant No. 2442853.
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APPENDIX
A ADDITIONAL EXPERIMENTAL DETAILS AND RESULTS
A.1 SENSITIVITY KERNEL CHOICE AND ACQUISITION PARAMETERS
Setup. We evaluate the robustness of our BO-based direction selectors by varying the Gaussian
process kemel (RBF vs. Matémn3/22 and the UCB acquisition parameter 3 E {1, 2, 3}. We choose these kernels because they are the standard GP covariances used on the sphere (Borovitskiy 2020). Experiments follow the approximation-errv setting from Section 4.1 with L E {100, 1000J, four ShapeNet point-cloud pairs, and five random seeds. We focus on BOSW (fixed direction set learned once) and ARBOSW (periodically refined set), which represent the two BO refinement regimes used the paper. The other variants, ABOSW and RBOSW, use the same BO inner loop and exhibited identical kernel/acquisition behavior in preliminary tests, so we omit them for brevity.
al.,
Results. Across all kernel and choices, both BOSW and ARBOSW show extremely small vari- ation in approximation emor. The standard deviation remains below 4 x 10-5 for I.=100 and below x 10- for L=1000, and the relative ranking of methods unchanged. RBF and Matérn-3/2 kernels yield nearly identical results. These findings confirm that our BO-driven direction selection highly robust to kernel and acquisition hyperparameters.
Method
Kernel
B
L=100
L=1000
BOSW
RBF
1
(2.20x10-5 7.12x
BOSW
RBF
2
(9.74x10-3+4410x00
(2.20x10=$4.12x1000
BOSW
3
(9.1xx10"'4416x100))
(231I11-"11.700
BOSW
Matérn-3/2
(0,2xx0-=+4.1x100"
(2.26x10==16.11
BOSW
Matémn-3/2
2
(9.01x10~~+4111000
BOSW
Matém-332
1.62x100*+41111000
"26x110=+6.81x100""
ARBOSW
(9.74x10"344.10
2220x10"=7.12xx1000
ARBOSW
RBF
2
+7.x100-94.10x1100
1,20x10-5+..12x100
ARBOSW
RBF
(2311100-+6.77)
ARBOSW
Matémn-31
(2.2xx10$$77.2x11
ARBOSW
Matém-3/2
2
(9.74x10-3
11111000-44.01111 14.10x10-5)
(2.20x10-5 7712x
ARBOSW
Matémn-3/2
3
10-11x10"> 41.16x100
(2.31x10 $160000$$6.7.::: 6.70 10-
Table Kernel and aequisitiin-paramette sensitivity (mean + std over four ShapeNet pairs and five seeds).
A.2 FEASIBILITY OF CLASSICAL BAYESIAN QUADRATURE (BQ)
Classical Bayesian Quadrature (BQ) is conceptually related but computationally impractical for sliced- Wasserstein projection selection: each iteration requires refitting GP and recomputing quadrature weights, leading to overall (LL)) cost (see Section 5).
To illustrate this sealing, we henchmarked a standard adaptive BQ method (GP refit + UCB acquisi- tion) against SW and our ABOSW method synthetic point clouds. reports the end-to-end runtime for generating all projection directions.
Method
L=50
T=1000
T==200
I-=500
T==0000
T.=2000
I==500
MCSW
0.00
0.00
0.00
0.00
0.01
0.01
0.03
ABOSW
0.35
0.06
0.06
0.06
0.06
0.07
0.09
BQ
0.10
0.21
0.47
1.61
5.78
35.73
654.66
Table 4: Wall-clock runtime (seconds) for generating L projection directions.
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ABOSW shows slightly higher cost at L=50 because its adaptive initialization performs more refinement steps when the budget is small; for larger budgets (L 2 100), this cost is amortized and ABOSW switches quickly into its fast QSW-style fill-in stage, yielding nearly flat runtime of 0.06-0.09s across all T..
In contrast, BQ grows from 0.47s L=200 1.61s I-500, 5.78s I=1000, and over 650s I=5000, fully consistent with its O(L") scaling. These results further support our decision not to include classical BQ as a comparative baseline the main experiments.
A.3 POINT-CLOUD INTERPOLATION WITH 10 DIRECTIONS
Extimatooss
Step (W2
Step (W2 .)
Step 300 (W2 D)
Step 400 (W; J)
Step 500 (W2 L)
(8.)
MCSW
5,719 + 0.113
0.190
0.016
0.041 + 0.001
0.020 + 0.002
0.011 + 0.001
2.71
GQSW
9.207 0.000
3,602
0.000
2,461 + 0.000
2. 2.191 + 0.000
2.119 + 0.000
2.92
FOSW
4,045 0.000
0.552
0.000
0.000
0,482 0.000
SQSW
0.000
DQSW
5,897 0.000
0.856 1: 0.000
0.612 1: 0.000
0.595 1: 0.000
0.594 1: 0.000
2.79
CQSW
5,574 1 0.000
0,755 1 0.000
0.592 1 0.000
0.582 1: 0.000
0.582 1 0.000
2.87
RGOSW
6,036 1. 0.231
0,183 0.022
0.034 1 0.003
0.017 1: 0.003
0.009 0.002
2.84
RRGQSW
0.004
REQSW
1722-+-0.155
0.189 0.008
0,79 + 0.805
+: 0.004
0.809 0.002
RREQSW
5,832 0.018
0 + 0.006
0.038 + 0.002
0.022 + 0.001
0.015 + 0.000
2.83
RSQSW
5,718 0.064
0.187 0.008
1U135 + 0.004
0 = 0.062
0.007 : 0.001
2.81
RDOSW
5,694 0,044
0.182 0.020
0.033 0.002
0.007 0.002
RCOSW
0.002
BOSW (ours)
4,833 0.863
1,303 1 0.491
0.967 1. 0.375
0,912 1: 0.345
0.900 J. 0.333
2.86
RBOSW (ours)
3,457 1. 0.109
0,175 1. 0.044
0.062 1 0.009
0.059 1: 0.005
0.047 1 0.004
22.63
ABOSW (ours)
9.086 3.294
3.005 1.499
1.637 :: 0,520
197 :: 0.305
1,216 :: 0.139
3.09
ARBOSW (ours)
0.307 0.015
0.002
Table 5: Summary of W distances (multiplied by 102) from three different runs (L=10),
Table reports results for point-cloud interpolation with a smaller projection budget of L 10. As expected, variance across methods is larger under such limited directions, but the relative trends later
mirror those observed for = 100: randomized QSW variants achieve fast convergence steps, while ARBOSW remains within the top four methods overall, offering competitive final ac- curacy. RBOSW continues to excel early in the trajectory but incurs higher runtime due to periodic refreshes.
A.4 COMPLETE IMAGE STYLE TRANSFER COMPARISONS
Figures and visualize the full method grid. Across both budgets, the randomized family performs very similarly; within that family RGQSW attains the lowest final W2 on our example for both T=10 and 100 This is consistent with the QSW observation that randomized designs are closely clustered quality. Our BO hybrids remain competitive: ARBOSW is comparable to RCQSW visually and in W2 at 1-100, while RBOSW improves over SW but generally trails the strongest RQSW variants, reflecting its lighter refresh. These results support our choice to use RCQSW as the representative randomized baseline in the main figures while providing full transparency here.
A.5 COLOR-TRANSLATTON EXPERIMENTS AT 128 128
We additionally include a controlled translatonot study in which the source and target images differ only through a global additive RGB shift. This setting preserves the spatial structure of the image and isolates the effect of a distributional shift in color space. For given 128 x 128 source image X, we construct the target image X via
(t,j) clip(X(i,,) 88, 0, 0,255),
where 5 E R is fixed collortraanslatoon vector. We then the same syle-transte procedure as in Section 4.4 using both L=100 (Figure ? and L=10 (Figure & projection directions.
Across both projection budgets, methods that employ randomized QSW constructions or BO-based refinement accurately recover the color shift and yield very small final W2 distances. while deter- ministic QSW variants show larger variability. As in the main experiments, ARBOSW behaves
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Searce
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comparably to the strongest randomized baselines at L-100, and RBOSW improves over vanilla SW though it is typically outperformed by RCQSW and RGQSW. These controlled translation re- sults further highlight the stability the BO-refined direction sets and their consistency across varying projection budgets.
A.6 FULL COMPARISON FOR DEEP POINT-CLOUD AUTOENCODERS
Setting. We follow the same setup as Section 4.5 ShapeNet Core-55 training, PointNet en- coder/decoder, I, = 100 projection directions, SGD with learning rate 10-3 momentum 0.9, weight decay 10-4, batch size 128, and training for 400 epochs. Evaluation is performed on Model- Net40, reporting SW₂ and W2 reconstruction losses averaged over three runs. Here, we expand the comparison to include the entire QSW family (BQSW, GQSW, SQSW, DQSW, CQSW) and their randomized counterparts (RQSW, RRGQSW, REQSW, RREQSW, RSQSW, RDQSW, RCQSW), alongside our BO-based methods (BOSW, RBOSW, ABOSW, ARBOSW).
Results. Figure 9 shows qualitative reconstructions across all baselines and our BO variants. While randomized QSW methods produce consistent and visually plausible outputs, BO-based designs, especially ABOSW, stand out with sharper reconstructions and more stable geometry. No- tably, CQSW remains strong deterministic baseline, but ABOSW surpasses it in final reconstruc-
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ScuCc
PIW M. 037
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BNOOSA My -000
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Figure 7: Color-translation experiment at 128 x 128 with L 100.
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Figure 8: tolosttranslttion experiment at 128 128 with L 10.
tion loss while maintaining visual quality. This contrasts with gradient flow experiments, where refresh-style variants (RBOSW/ARBOSW) had stronger performance; in autoencoders, the stable large-scale dataset favors consistent shot or hybrid projection sets (BOSW/ABOSW) These ob- servations reinforce our conclusion that the optimal projection strategy task-dependent: stability- driven tasks benefit from non-refresh BO, while dynamic flows gain from refresh hybrids.
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MESW
REQSW
KREQSW
DQSW
RDOSW
CQsw
RCQSW
BOSW
ABBOSW
Figure 9: Full reconstructed point-clouds from MCSW, QSW, ROSW, and BO variants with 100.
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Convergence: Iterations 100)
Convergence: 100)
True Waczestaan Dbrance
Thue Wasterstin 200.stttn--000mmmmm Delance
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BOSW
BOSW
BOSW
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BOSW (Thompsor)
20.7-
10
10
10
Iteraion
Wal-Click (seconds)
Convergence with increasing iterations.
(b) Convergence with increasing wall-clock time.
Figure 10: Ablation study of different acquisition functions for Bayesian optimization, including upper confidence bound (UCB), expected improvement (ED, Thompson sampling, and log-EI, UCB consistently performs best in terms of number of iterations and wall-clock time required to converge.
Convergence: Distance vs. Iterations (L = 1001
Convergence: Distance vs. Time (L 100)
Thue Waserstain Detance
Thue Songeeaee Detarce
zonk.
BDSW (1-0))
1014
ROSW (1-3)
BDSW (1-25))
BoSw (1-25)
BOSW (5-0.))
BOSW (6=2.)
BOSW (6-2)
BOEW (2=2)
BOSW (3-3)
BOSW (3=3)
23 Wal-Clock Time (seconds]
(a) Convergence with increasing iterations.
(b) Convergence with increasing wall-clock time.
Figure 11: Ablation study of different values of 3 (the weight parameter for the UCB acquisition function). Although the different values of 3 yield similar results, B = 0.7 consistently demonstrates the best performance.
B COMPUTATIONAL INFRASTRUCTURE
We use dual NVIDIA RTX A6000 GPUs conduct experiments training deep point-cloud au- toencoders. Other applications are run on a MacBook Pro 14-inch (Nov 2023) equipped with an Apple M3 Max chip and 36 GB memory.
C ABLATION STUDIES
Section 3.1 describes several modeling and parameter choices made for our implementation of
Bayesian optimization. We justify these choices here via ablation studies. Each study is evalu- ated on the point-cloud interpolation (gradient flow) example of Section 4.3 For consistency, we also select = 100 for all our ablation studies unless otherwise noted.
First, we experiment with different possible acquisition functions for Bayesian optimization. We test popular functions such as upper confidence bound (UCB), expected improvement (EI), Thompson sampling [Thompson] 1933). well as the recentlyprrooosee log-EI -I (Ament et al. 2023). Figure IO shows that UCB consistently outperforms other acquisition functions. Thus, we select UCB for all of examples in the paper. Interestingly, that LCB (identical to UCB, but for minimization rather than maximization) was the most efficient acquisition function choice, in completely different application domain (design optimization).
note that recent work (Vardhan et al.) 2024 also found
Next, we consider different values of B, the parameter that tunes exploration vs. exploitation in a constant {8; adaptive
the UCB acquisition function. (Note that in experiments,
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Convergence: Iterations
Convergence: Time 100)
Tue Waseerstnn Disance
mar Cereaae 1 2 Disannce
aomt..
BCSW Icou=0.7)
PCSW Ico
RCON (ros-0.8)
ROOW tras 0.40
BCSW 1cod 3.39
BC5W 1co2-0.3) 0.39
BCSW (cas-0.95) .5]
BC5W lcns
aunk-
BORW (coso0
20
RORW trosen
10"2
10"2
Peration
Woll-Clock Time (scconds)
Convergence with increasing iterations.
(b) Convergence with increasing wall-clock time.
Figure 12: Ablation study of different values of the cosine-similarity cutoff threshold used in our method. The plots motivate our usage of 0.98 for all our experiments.
Corvergence
AJOOA
ARROSS
12.05 2000 X392 12:14:
Figure 13: Effect of varying ne on the convergence of our BO-based methods. For most meth- ods, n. 4096 is optimal, except for BOSW, which attains minimal error after 500 steps using n. = 1024. Note that the vertical axis is scaled logarithmically to make the data points easier to distinguish.
schedules are certainly possible and may benefit bias (see Appendix D As demonstrated in Figure 11 the constant value 3 = 0.7 that we select for all our experiments outperforms all other choices considered (both larger and smaller).
We also explore different thresholds for the cosine-similarity cutoff (used to reject the selection of directions that are too close to each other). We remark that this value should typically be chosen closer to as L increases, to facilitate large numbers of slices becoming dense on the unit hyper- sphere. For L = 100. though (as generally used in our experiments), Figure [2]shows that our choice of 0.98 is near-optimal; while sometimes a slightly lower value (0.9 or 0.95) helps achieve a slightly lower error at particular time, a value of 0.98 results in the lowest error after sufficient iterations or time. An annealing scheme for a dynamic threshold value may unlock marginally better perfor- mance (particularly for large L), but feel this would be a very minor optimization, especially since L is usually kept relatively small to maintain computational efficiency of SW.
Next, we consider different values of Te, the number of candidate directions considered on any iteration of our core BO routine. We chose n. = 4096 to copy the number of candidate directions used in the QSW methods of Nguyen et al. (2024)). Interestingly, as Figure I3 shows. this value turns out to be optimal for all methods except BOSW (where 1024 is optimal). Nonetheless, we found that values from 512 to 16384 did not significantly change the error in our methods after 500 steps, despite the linear-time computational complexity associated with n.- Thus, in practice, it may be appropriate to use smaller values of ne, accepting a modest change in accuracy for a potentially substantial change in performance. We note that, expected, for sufficiently small values of ne, method converged well (neither QSW methods nor our BO-based methods).
Finally, consider different values of R for the variants of our method that periodic BO refreshes. We evaluate E {5, 10, 25, 50) for both RBOSW and ARBOSW. Smaller R values adapt more aggressively but incur substantial GP and BO overhead, while larger values refresh less
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Efficiency Frontier Refrech Rates
Efficiency Frontier Refresh Rates
Tue Wasersten Dbsance
10
Thus Wesserston Dissance
10
RROOW (R-1)
ARTOSN (A-10
aunt.
R805W (R=10)
ARBOSW (A-10)
RS2S5N [I0=25]
ARBISSIN (0-22]
ARROSN (R-50)
10-2
10
Wall Clock (secondss
Clock Time (seconds)
(a) Efficiency frontier for RBOSW
(b) Efficiency frontier for ARBOSW
Figure 14: Ablation study on refresh frequency R for BO-refined SW estimators. Across both RBOSW and ARBOSW, refresh rates in the range R {5, 10, 25, 50) provide the best accu- racy-efficiency balance, with R = 25 achieving the strongest performance overall.
often and risk using stale direction sets. As illustrated in 14 R 25 consistently offers the best accuracy -fffrccceffcc trade-off: it converges quickly while maintaining low overhead. A slightly less firqueett refresh, = 50, performs comparably and is a reasonable secondary choice when computation time is constrained. In contrast, the more frequent refresh schedules (Rt 5 and R = 10) incur significant overhead and have the poorest wall-clock efficiency. Motivated by these results, adopt R 25 for all of our numerical experiments.
D THEORETICAL PERFORMANCE OF BAYESIAN OPTIMIZATION FOR SLICED WASSERSTEIN
Although our paper primarily intended to provide numerical evidence of the efficacy of using Bayesian optimization for sliced Wasserstein distance computations, we briefly remark the pos- sibility of theoretical performance bounds.
It is known that under i.idd. MC sampling and sufficient regularity assumptions, SW converges at rate of O(L-1/2). By the triangle inequality, we can relate the error of our BO methods to the error of MCSW:
(Wr( - SW w - (WE(Or) - (WC) + [WT((Mc) - (SP
BO-MC Error
MC-SW Error
The latter term is bounded above by O while bounding the former term remains an open question. The primary reason bounding that term is problematic is that BOSW may not perform as well as MC SW the limit, because BOSW is not generally an unbiased estimator. For instance, with our UCB acquisition function as used in our paper.
(0) = (0) + 3
we take a constant 3 0.7. Thus, ag will always be encouraged to select yielding high f(0; p. v). In the limit, this suggests that
L S(.))
IO0)) sup f(0) > Eg [[()]]
bega
This is strict inequality assuming f is non-constant on 5d-1; and when sampling & uniformly (MC), as mentioned above, Eg [f()] = O. This rough sketch demonstrates that BOSW, at least with UCB with a constant 8, will not converge in the limit-despite its attractive numerical perfor- mance (demonstrated in our experiments in the main body) during early iterations. Bounding the performance of BOSW for finite L is an interesting open question. It may be possible to approach this question using the concepts of information gain and regret minimization; see, e.g., Vakili et al. 2027). This would be impacted by our choice of using angular RBF kernel.
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Despite gaps in the current theory, note that GP-UCB sampler towards eventual uniform coverage of the sphere. In spirit, this follows the idea of using BOSW for early iterations, then switching to MC other variants for later iterations idea that could harmonize the benefits of fast early convergence with BO-based methods with the benefits of uniform coverage of MC samplers. For instance, if we define
annealing schedule for 3 to drive
Er (0) - (,0 + Ex) |
we can choose = = (0 < 1). Then as - speaking the second Porel-Cantelli lemma would suggest that every measurable region of the hypersphere is sampled infinitely often with probability 1. It follows that the bias will match that of uniform MC sampling, which is O in the limit; and under this scheme, the convergence rate would reach (LL-1/2) asymptotically (whether it is faster or slower during initial iterations).
- O, but E oo, so informally
E
USE OF LARGE LANGUAGE MODELS
Large language models (LLMs) were used in the preparation of this paper. Specifically, Perplexity was prompted to act peer reviewer for our draft. In this context, the model identified several recent references for us to discuss in our related work that we previously had not included; and the model reminded to perform a few of the ablation studies seen Appendix C
"The lemma assumes each "event" (direction selection) independent, which not the case, by design. Nonetheless, this assumption would hold in the limit of e, + 0. Generalizations of the Borel-Cante Cantelli lemma that do not require strict independence have been studied (Chandral 2008, Biró & Curbelo 2021). and it is possible that those may apply here. For the present work, though, only characterize our arguments as informal.
23


Reviews:
[{"review_id": "LcKMXJj5tO", "reviewer": "Reviewer_EFeU", "summary": "This work introduces a new family of simple, plug-and-play methods for selecting projections in the Sliced-Wasserstein (SW) distance. The key contributions are seamless Integration and novel selectors.", "questions": "1) This is the critical trade-off. How much more computational time does their BO-driven selection add compared to simpler random or linear selection methods? The definition of \"modest\" can be subjective, and the overhead might be acceptable for some applications but not others.\n\n2) Could you provide expeirmtns with translation with resolution 128x128?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1) The fact that your method can be integrated into existing QSW/RQSW pipelines without changing downstream losses or gradients is a huge advantage. It means adoption is easy. \n\n2) Using Bayesian Optimization  to drive projection selection is a very logical and promising direction. I would like to point out this solution \n\n3) High-dimensional experiments", "weaknesses": "1) Comparison with other image-to-image models like Cycle GAN, DSBM or Optimal transport solvers", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7u5fPyezir", "reviewer": "Reviewer_EUMw", "summary": "The authors tackle the problem of choosing the right directions to use in the sliced Wasserstein (SW) distance when it is used inside an optimization loop. They propose using Bayesian optimization (BO) to select these directions, with the goal of adapting them to the evolving probability distributions. To this end, they propose four algorithms: BOSW, RBOSW, ABOSW, and ARBOSW. The first two algorithms learn a Bayesian process to compute $L$ SW directions, which are then kept fixed during the subsequent learning task. For the small set of initial points used to initialize the Bayesian process, BOSW uses random points sampled uniformly on the sphere, while ABOSW uses points sampled via a quasi–Monte Carlo method. The RBOSW and ARBOSW algorithms periodically refit the Bayesian process during the learning task, using BOSW and ABOSW, respectively, to adapt to the evolving probability distributions throughout training. The authors tested the four algorithms on different tasks (including learning with SW loss) and compare them to quasi–Monte Carlo approaches.", "questions": "- Can you provide references for the problems you mention in the paragraph “Problem view for this work” that use SW in an “outer optimization” and keep the same set of $\\Theta$ directions during the learning task? Can you also explain what you mean by “outer optimization”?\n\n- Can you compare your approach to the natural baseline that would uniformly sample new directions on the sphere at each time step for Experiments 4.3, 4.4, and 4.5? Is there a reason why you did not include this baseline?\n\n- For your algorithms RBOSW and ARBOSW, you periodically rerun BOSW and ABOSW. It would be helpful to derive a criterion for when the Gaussian process should be updated (and new directions generated). How do you decide when to perform this update? Did you try to measure the impact of the timing of this update on the performance you obtained?\n\n- Do you know why your methods (except ARBOSW) perform poorly in Experiments 4.3 and 4.4 but achieve very good performance in Experiment 4.5?\n\n- Ideally (although infeasible because it is too costly), we would use BO to choose each new $\\Theta$ direction and resample new directions at each time step. Since this is too computationally expensive, it seems necessary to design a method that achieves a trade-off between sampling random directions, keeping some directions fixed, and selecting new directions with BO. Do you think that, instead of keeping the directions fixed between BO updates, uniformly sampling over the sphere could improve performance?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "- The idea of carefully selecting directions for SW during the learning task, when probability distributions evolve, is a difficult but very interesting problem, as it would permit the use of SW as a surrogate for the Wasserstein distance (in a learning context), which is expensive to compute.\n\n- The experiments conducted by the authors permit a comparison between QSW methods and BO methods on various tasks to gain an intuition about how the BOSW algorithm behaves.", "weaknesses": "- The paper lacks clarity on the type of learning problems that would be solved using BO inside SW.\n\nIn Section 2, paragraph “Problem view for this work” (l.168), the authors define the learning tasks targeted by their method as:\n\n“When SW is used inside an outer optimization (e.g., gradient flows or registration), the same direction set \n$\\Theta$ is queried repeatedly on evolving point clouds.”\n\nFrom this sentence, it seems that the problems targeted are certain bilevel optimization problems, although in their applications (see Sections 4.3, 4.4, 4.5), the problems solved involve a direct optimization of the sliced Wasserstein distance between an optimized distribution and a target distribution.\n\nMoreover, this statement also suggests that, for the problems considered, the learned directions are fixed during optimization, although the authors propose two algorithms, RBOSW and ARBOSW, that update the set of directions during the learning task.\n\n- For problems that aim to optimize a parameterized distribution by minimizing the SW distance between this distribution and a target distribution, the authors should compare their approach to the most natural use of SW, which consists of randomly sampling new directions on the sphere at each time step. I do not see why keeping the same set of directions $\\Theta$ would be a good idea, as the model risks optimizing only along these fixed directions. \n\n- The paper is not easy to read. I suggest that the authors add more background on BO and quasi–Monte Carlo methods. While there is a long recap of the background on Wasserstein and sliced Wasserstein distances, the section on BO is very brief.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4YgeiZTcGA", "reviewer": "Reviewer_eyCr", "summary": "This paper proposes several Bayesian optimization (BO)–driven methods to improve the efficiency and adaptivity of sliced Wasserstein (SW) distance computations. The key idea is to learn projection directions adaptively—rather than sampling them quasi-uniformly as in Quasi-Monte Carlo SW (QSW)—by using Gaussian-process-based BO on the sphere. Four variants are introduced: (i) BOSW – one-shot BO; (ii) RBOSW – periodically refreshed BO; (iii) ABOSW – QSW-seeded lightweight refinement; (iv) ARBOSW – restarted hybrid (periodic QSW seeding + BO refinement). Experiments on synthetic benchmarks, point-cloud interpolation, image style transfer, and deep autoencoders show that the adaptive BO variants (especially ABOSW and ARBOSW) achieve competitive or superior convergence to QSW and randomized QSW (RQSW) baselines with modest overhead.", "questions": "1. Could the GP surrogate be replaced by a neural surrogate to improve scaling?\n2. How sensitive is performance to kernel choice or acquisition parameters?\n3. Is there a way to enforce unbiasedness while retaining adaptivity?\n4. How does performance evolve with dimensionality (e.g., $d>10$)?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The idea of combining Bayesian optimization with sliced Wasserstein direction selection is novel and natural. The approach bridges deterministic geometric constructions (QSW) with data-driven adaptivity, potentially broadening the design space of SW estimators; \n\n2. The authors reproduce the entire QSW benchmark suite (Nguyen et al., 2024a) for fair comparison. Tasks span synthetic projection selection, approximation accuracy, dynamic interpolation, style transfer, and autoencoding—demonstrating both theoretical and practical relevance; \n\n3. The methods require no change to downstream objectives or gradients, making them easy to plug into existing OT pipelines. \n\n4. RBOSW provides strong early-stage performance. ARBOSW matches or slightly exceeds state-of-the-art RQSW on long-run optimization tasks. ABOSW improves deterministic settings such as autoencoder training; \n\n5. The paper provides clear mathematical exposition, fixed random seeds, and references to public code.", "weaknesses": "1. Although an appendix sketches guarantees, there is no formal convergence proof or error bound for the BO-selected projections compared to the true spherical integral; \n\n2. The method relies on GP-based BO, whose cubic cost in the number of evaluations becomes prohibitive as $L$ or dimensionality $d$ increases. This is acknowledged but not resolved beyond citing recent BO-in-high-dimension literature; \n\n3. In static estimation tasks, BOSW performs worse than QSW since it is not designed for uniform coverage. The benefits only appear when SW is nested inside an optimization loop; \n\n4. While conceptually appealing, the improvements over strong QSW baselines are modest in some tasks (e.g., style transfer) and might be sensitive to hyperparameters; \n\n5. The reported improvements lack variance/error-bar plots or significance testing.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jA3HcA5Chj", "reviewer": "Reviewer_AKwQ", "summary": "Many modern machine learning and data analysis tasks rely on computing the sliced Wasserstein (SW) distance to compare probability distributions. By treating the SW distance as a function of the projection direction, this paper introduces a method that uses Bayesian Optimization (BO) to automatically find the most informative projection directions for these tasks. Compared with traditional methods such as quasi-Monte Carlo (QSW) and randomized QSW (RQSW), which sample directions more evenly but do not adapt to the specific structure of the task, the BO approach identifies the most effective directions during the optimization process, allowing it to choose directions that provide the most useful information.", "questions": "1, Could the authors quantify the estimation bias, either through empirical experiments or theoretical analysis?\n\n2, If not, I would suggest the authors to revise the claim \"these methods achieve state-of-the-art efficiency for estimating SW\", as the proposed BO-based estimators are data-dependent and therefore biased, not designed for unbiased SW estimation.\n\n3, Could the authors repeat the experiments more times to minimize random fluctuations or numerical instabilities?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "While Bayesian Optimization (BO) has been widely used for computing expectations, applying it to guide projection direction selection in sliced Wasserstein (SW) distance estimation for machine learning tasks is a new idea.", "weaknesses": "1, BOSW (and even its hybrid variants) may yield biased SW estimates since the BO-selected directions depend on the specific data. As a result, it may converge to a different value from the true SW. This issue is also noted in the paper (see line 299). However, a contradictory statement appears on line 109, where the authors state that these methods achieve state-of-the-art efficiency for estimating SW.\n\n2, For simulation in subsection 4.2, the evaluation metric (maximum fitness value) favors BO by design, since BO is built to maximize scalar objectives, not to approximate integrals, making the comparison against QSW conceptually unfair. In addition, BOSW performs worse than both Monte Carlo and all QSW variants across all budgets L, confirming that BO-selected directions hurt uniform integration accuracy. \n\n3, In subsection 4.3, the improvements are small and sometimes disappear in later steps. BOSW shows high variance and weak convergence. RBOSW and ARBOSW perform better early on but take much longer to run, taking ten times the runtime of other methods.\n\n4, Most results are averaged over only three to five runs.\n\n5, The experiments omit comparisons against other adaptive sampling or Bayesian-quadrature approaches that would provide stronger baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mUuKslaYyU", "reviewer": "Reviewer_gAQG", "summary": "This paper proposes a new framework for computing the sliced Wasserstein (SW) distance more efficiently by adaptively learning projection directions using Bayesian optimization (BO). While prior methods such as quasi–Monte Carlo (QSW) sampling select projection directions uniformly and independently of the data, this work leverages information from previously evaluated slices to focus on the most informative directions. The authors introduce four BO-based variants, BOSW, RBOSW, ABOSW, and ARBOSW, which differ in how often they refresh or refine projection sets and how they integrate with QSW initialization. These methods are designed as plug-and-play replacements requiring no change to downstream losses or gradients. Empirical results across synthetic benchmarks, point-cloud interpolation, image style transfer, and deep autoencoding show that the adaptive hybrids (especially ABOSW and ARBOSW) achieve state-of-the-art convergence with modest runtime overhead, outperforming or matching strong QSW baselines. Overall, the paper unites deterministic geometric designs with data-driven adaptivity to enhance efficiency and accuracy in large-scale optimal transport tasks.", "questions": "1. The proposed nonparametric regression framework seems to go well beyond simply constructing a good uniform set of projection directions. In my view, it effectively defines a data-dependent Radon transform (or equivalently, a new slicing distribution), which induces a novel form of metric as in [1]. In [1], the authors reweight directions according to their corresponding 1D costs. Your regression-based framework appears to be a  powerful way to achieve something similar, using in-sample weights (training directions) and out-of-sample weights (new proposed directions). Should a discussion be added?\n\n[1] Energy-based Sliced Wasserstein distance, Nguyen et al\n\n2.  Section 2 is quite restricted when discussing only empirical 1D Wasserstein. The proposed method in the paper does not depend on the form of the distance, so, it is better to  discuss general form (see Section 2.3 in [2]).\n\n[2]  An Introduction to Sliced Optimal Transport, Nguyen.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper introduces the first use of Bayesian optimization (BO) to adaptively select projection directions for sliced Wasserstein computation. The connection between nonparametric-regression and SW is interesting.\n\n2. The proposed methods (BOSW, RBOSW, ABOSW, ARBOSW) integrate seamlessly into existing SW pipelines without modifying loss functions or gradients.\n\n3. The paper benchmarked against baselines (MC, QSW, RQSW) across diverse tasks: synthetic landscapes, 3D point clouds, image style transfer, and deep autoencoding.\n\n4. Adaptive hybrids (especially ABOSW and ARBOSW) achieve state-of-the-art or comparable convergence with minimal runtime overhead.\n\n5. BO refinements can be combined with quasi-Monte Carlo (QSW) approaches, showing versatility rather than replacement.\n\n6. Code and experimental settings are carefully controlled for fair comparison, following prior benchmarks.", "weaknesses": "1.  BO introduces extra overhead and may not scale well to very high-dimensional problems or large projection budgets, despite some mitigation.\n\n2. Learned projection sets (especially BOSW) are not unbiased estimators of the true SW integral, reducing theoretical rigor for some applications.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)