Toscani-Fourier Distance on Probability Measures: Wasserstein Control, Topological Equivalence on Model Classes, and Duality

Mehrdad Mohammadi

stat.MLSep 19, 2026

Toscani-Fourier Distance on Probability Measures: Wasserstein Control, Topological Equivalence on Model Classes, and Duality

Authors: Mehrdad Mohammadi

Organizations: Department of Statistics University of Illinois Urbana-Champaign Champaign, IL 61820, USA

Abstract

Comparing probability measures in machine learning trades transport geometry against computational cost: Wasserstein distances encode the geometry of $\mathbb R^d$ but require solving a transport problem, while kernel discrepancies are cheap to evaluate yet depend delicately on their test class. We study the Toscani--Fourier family $\mathrm T_{s,p}$ , the weighted $L^p$ norm of the difference of two characteristic functions, as a continuous Fourier-side discrepancy on $\mathbb R^d$ . For $1\le p<\infty$ we show that $d/p<s<1+d/p$ is exactly the window in which $\mathrm T_{s,p}$ is finite on $\mathcal P_p(\mathbb R^d)$ , both endpoints already failing for a pair of Dirac measures, and we establish the metric, embedding, and compactness structure of the resulting space, which we prove to be complete. Duality identifies $\mathrm T_{s,p}$ as an integral probability metric over a homogeneous Fourier--Lebesgue ball, with an explicit extremizer when $1<p<\infty$ . We prove the global bound $\mathrm T_{s,p}\lesssim W_p^{\,s-d/p}$ , whose exponent is sharp, show that no global converse of any form can hold, and recover topological equivalence with $W_p$ on bounded-support and uniform-tail classes, together with explicit reverse moduli on bounded-support classes that improve the imported energy-kernel exponent at $p=2$ . There, $\mathrm T_{s,2}$ is a constant multiple of the classical energy distance, which yields an exact finite-sample identity for the mean of the empirical discrepancy; the numerical experiments are otherwise diagnostic.

Figures & tables

Panel of Figure 1 — Figure 1: Main metric relations in the paper.

Panel of Figure 2 — Figure 2: Sensitivity of $\widehat{\mathrm{T}}_{s,p}$ to the truncation cutoffs. The origin cutoff $r_{\min}$ (left column) and the infinity cutoff $r_{\max}$ (right column), for several values of $s$ relative to the admissible window. Instability concentrates at the endpoint predicted by Proposition 8 . The lower cutoff matters for $s$ at or above $1+d/p$ , the upper cutoff for $s$ at or below $d/p$ , while interior values of $s$ are stable against both.

$d$	$p$	$s$	$\alpha$	slope vs. ${}^{\text{(sliced)}}\mathrm{W}_{p}$	RMSE	slope vs. control	RMSE
2	2	1.6	0.6	0.994	0.124	1.443	0.164
2	5	1.0	0.6	0.993	0.347	1.390	0.374
16	2	8.6	0.6	0.987	0.153	1.645	0.153
16	5	3.8	0.6	1.013	0.327	1.687	0.327

Table 1: Translation-family comparison of

\widehat{\mathrm{T}}_{s,p}

with sliced Wasserstein.

Panel of Figure 3 — Figure 3: Translation-family comparison of $\widehat{\mathrm{T}}_{s,p}$ with sliced Wasserstein. Each row corresponds to one $(d,p)$ setting. The first column plots $\widehat{\mathrm{T}}_{s,p}$ against $\max\{({}^{\mathrm{(sliced)}}W_{p})^{\alpha},{}^{\mathrm{(sliced)}}W_{p}\}$ , the second plots it directly against ${}^{\mathrm{(sliced)}}W_{p}$ , and the third shows the corresponding normalized ratio.

$d$	$s$	$\alpha=2s-d$	theory $c_{\alpha}$	fitted $c$	rel. gap in $c$	median / 95% rel. err.
2	1.6	1.2	5.6579	5.6145	0.0077	0.187 / 0.739
16	8.6	1.2	0.8831	0.9291	0.0521	0.342 / 1.081

Table 2: Numerical check of the

p=2

identity

\widehat{\mathrm{T}}_{s,2}^{2}\approx c_{\alpha}\,\mathrm{MMD}_{k_{\alpha}}^{2}

Panel of Figure 4 — Figure 4: Scatter of $\widehat{\mathrm{T}}_{s,2}^{2}$ versus $\mathrm{MMD}_{k_{\alpha}}^{2}$ for $\alpha=1.2$ , together with the fitted linear relation through the origin. The fitted and theoretical constants are closer in $d=2$ than in $d=16$ , and the point cloud is also substantially tighter in $d=2$ .

Figure 5: DOTmark 1.0 data set at resolution $128\times 128$ . Each row corresponds to a class. The eight classes shown are those retained in the experiments.

Figure 6: DOTmark scatter comparison of $\mathrm{T}_{s,2}$ (with $s=1.66$ , a value from the tabulated grid of Table 3 ) against sliced $W_{2}$ , one panel per resolution. Axes are on the pixel-coordinate scale of the footnote in Section 7.2 and are not comparable across resolutions.

Figure 7: As in Figure 6 , for $\mathrm{T}_{s,5}$ with $s=1.39$ versus sliced $W_{5}$ .

Figure 8: DOTmark ROC curves for distinguishing same-class from different-class image pairs by thresholding the distance (smaller distance indicates same class), for $\mathrm{T}_{s,2}$ with $s=1.66$ and sliced $W_{2}$ , one panel per resolution, with the area under each curve in the legends.

Figure 9: As in Figure 8 , for $\mathrm{T}_{s,5}$ with $s=1.39$ and sliced $W_{5}$ .

$s$ N-RMSE Spearman Ordinal agr. 32 $\times$ 32 Pixels $p=2$ 1.01 0.3176 0.8782 0.8403 1.34 0.3043 0.8886 0.8461 1.66 0.3000 0.8919 0.8476 1.99 0.2995 0.8918 0.8467 $p=5$ 0.41 0.3057 0.8668 0.8229 0.74 0.2870 0.8861 0.8335 1.06 0.2838 0.8892 0.8358 1.39 0.2847 0.8879 0.8343	$s$ N-RMSE Spearman Ordinal agr. 128 $\times$ 128 Pixels $p=2$ 1.01 0.3359 0.8930 0.8469 1.34 0.3172 0.9055 0.8546 1.66 0.3099 0.9116 0.8580 1.99 0.3077 0.9136 0.8585 $p=5$ 0.41 0.3325 0.8830 0.8291 0.74 0.3092 0.9014 0.8394 1.06 0.3026 0.9073 0.8429 1.39 0.3019 0.9086 0.8435
$s$ N-RMSE Spearman Ordinal agr. 64 $\times$ 64 Pixels $p=2$ 1.01 0.3495 0.8797 0.8365 1.34 0.3403 0.8901 0.8437 1.66 0.3377 0.8938 0.8460 1.99 0.3376 0.8948 0.8461 $p=5$ 0.41 0.3565 0.8681 0.8212 0.74 0.3378 0.8901 0.8341 1.06 0.3323 0.8955 0.8379 1.39 0.3309 0.8966 0.8383	$s$ N-RMSE Spearman Ordinal agr. 256 $\times$ 256 Pixels $p=2$ 1.01 0.3661 0.8413 0.8252 1.34 0.3551 0.8529 0.8320 1.66 0.3511 0.8573 0.8346 1.99 0.3498 0.8590 0.8351 $p=5$ 0.41 0.3604 0.8165 0.8071 0.74 0.3421 0.8421 0.8189 1.06 0.3360 0.8491 0.8232 1.39 0.3337 0.8513 0.8236

Table 3: DOTmark. Agreement of

\mathrm{T}_{s,p}

with sliced

W_{p}

across resolutions and

s

grids for

p\in\{2,5\}

. N-RMSE after affine calibration, Spearman rank correlation, and ordinal (anchored triplet) agreement, as defined in the text, over all

3{,}160

image pairs.

	Method	Rejected pairs	Reject rate	Agreement
$\mathbf{p=2}$	Sliced Wasserstein	13/28	46.4%	–
	Toscani ( $s=1.01$ )	18/28	64.3%	21/28 (75.0%)
	Toscani ( $s=1.34$ )	18/28	64.3%	21/28 (75.0%)
	Toscani ( $s=1.66$ )	17/28	60.7%	22/28 (78.6%)
	Toscani ( $s=1.99$ )	17/28	60.7%	22/28 (78.6%)
$\mathbf{p=5}$	Sliced Wasserstein	19/28	67.9%	–

Table 4: Gaussian-kernel MMD two-sample testing on DOTmark at resolution

256\times 256

. There are

28

class pairs, each with

n_{1}=n_{2}=10

images and

2000

permutations. Decisions are at level

\alpha=0.05

. The counts are derived from the pair-level records in Appendix B of the online appendix.

Class pair	Sliced $W_{2}$	$\mathbf{s=1.01}$	$\mathbf{s=1.34}$	$\mathbf{s=1.66}$	$\mathbf{s=1.99}$
ClassicImages vs GRFsmooth	Reject	Non-reject	Non-reject	Non-reject	Non-reject
GRFmoderate vs Shapes	Non-reject	Reject	Reject	Reject	Reject
GRFsmooth vs MicroscopyImages	Non-reject	Reject	Reject	Reject	Reject
GRFsmooth vs Shapes	Non-reject	Reject	Reject	Non-reject	Non-reject
LogGRF vs MicroscopyImages	Non-reject	Reject	Reject	Reject	Reject
LogGRF vs Shapes	Non-reject	Reject	Reject	Reject	Reject

Table 5: Class pairs on which Gaussian-kernel MMD tests based on Toscani distances and sliced Wasserstein give different decisions at level

\alpha=0.05

, for

p=2

at resolution

256\times 256

. Pairs on which all five tests agree are omitted.

Class pair	Sliced $W_{5}$	$\mathbf{s=0.41}$	$\mathbf{s=0.74}$	$\mathbf{s=1.06}$	$\mathbf{s=1.39}$
ClassicImages vs GRFsmooth	Reject	Non-reject	Non-reject	Non-reject	Non-reject
ClassicImages vs LogitGRF	Reject	Non-reject	Non-reject	Non-reject	Non-reject
GRFsmooth vs Shapes	Reject	Reject	Non-reject	Non-reject	Non-reject
LogitGRF vs MicroscopyImages	Non-reject	Non-reject	Non-reject	Non-reject	Reject
MicroscopyImages vs Shapes	Reject	Non-reject	Non-reject	Non-reject	Non-reject

Table 6: Class pairs on which Gaussian-kernel MMD tests based on Toscani distances and sliced Wasserstein give different decisions at level

\alpha=0.05

, for

p=5

at resolution

256\times 256

. Pairs on which all five tests agree are omitted.

Metric	1D cost	Cost for $d\geq 2$	Optimization
$W_{p}$ (exact, discrete)	$\mathcal{O}(n\log n)$	LP on $\Theta(n^{2})$ variables	Yes
Sinkhorn / entropic OT	$\mathcal{O}(n^{2})$ per iter	$\mathcal{O}(n^{2})$ per iter	Yes (iterative)
Sliced $W_{p}$ with $L$ projections	$\mathcal{O}(Ln\log n)$	$\mathcal{O}(Ldn+Ln\log n)$	No (projection sorting)
$\mathrm{T}_{s,p}$ (sample-based approx.)	$\mathcal{O}(M(n+m))$	$\mathcal{O}(Md(n+m))$	No

Table 7: Comparison of computational scaling and algorithmic structure. The frequency budget

M

controls accuracy in a distribution-dependent way (see the discussion below). The table compares algorithmic structure, not error-matched cost. On grids, convolutional Sinkhorn variants reduce the per-iteration cost to

O(N\log N)

( Peyré and Cuturi, 2019 ) .

$p$	Resolution	$s$	Sliced W	Sinkhorn	$\mathrm{T}_{s,p}$
2	32	1.99	0.0118	0.0593	0.0002
	64	1.99	0.0593	1.5615	0.0003
	128	1.99	0.2525	21.2620	0.0012
	256	1.99	1.5013	$\spadesuit$	0.0037
5	32	1.39	0.0126	0.0466	0.0001
	64	1.39	0.0561	0.7048	0.0002

Table 8: Per-pair empirical mean runtime (seconds) of

\mathrm{T}_{s,p}

versus Wasserstein-type baselines.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Class pair	Metric	$\mathbf{s}$	Bandwidth	$\mathbf{\widehat{\mathrm{MMD}}_{u}^{2}}$	$\mathbf{p}$ -value	Reject
ClassicImages vs GRFmoderate	Sliced W		11.259	-0.001	0.442	0
ClassicImages vs GRFmoderate	Toscani	1.01	0.310	-0.011	0.641	0
ClassicImages vs GRFmoderate	Toscani	1.34	0.985	-0.013	0.625	0
ClassicImages vs GRFmoderate	Toscani	1.66	3.035	-0.014	0.633	0
ClassicImages vs GRFmoderate	Toscani	1.99	9.883	-0.015	0.628	0
ClassicImages vs GRFrough	Sliced W		6.828	0.243	4.998e-04	1

Appendix

Table B.1: Detailed Gaussian-kernel MMD two-sample testing results on DOTmark at resolution

256\times 256

with

p=2

, for the sliced-Wasserstein baseline and for

\mathrm{T}_{s,2}

at every tabulated

s

. All tests use

n_{1}=n_{2}=10

images and

2000

permutations. For Sliced W, the

s

column is left blank.

Class pair	Metric	$\mathbf{s}$	Bandwidth	$\mathbf{\widehat{\mathrm{MMD}}_{u}^{2}}$	$\mathbf{p}$ -value	Reject
ClassicImages vs GRFmoderate	Sliced W		14.727	-0.006	0.501	0
ClassicImages vs GRFmoderate	Toscani	0.41	0.188	-0.027	0.798	0
ClassicImages vs GRFmoderate	Toscani	0.74	0.615	-0.027	0.788	0
ClassicImages vs GRFmoderate	Toscani	1.06	1.940	-0.027	0.776	0
ClassicImages vs GRFmoderate	Toscani	1.39	6.448	-0.028	0.791	0
ClassicImages vs GRFrough	Sliced W		9.430	0.240	4.998e-04	1

Appendix

Table B.2: Detailed Gaussian-kernel MMD two-sample testing results on DOTmark at resolution

256\times 256

with

p=5

, for the sliced-Wasserstein baseline and for

\mathrm{T}_{s,5}

at every tabulated

s

. All tests use

n_{1}=n_{2}=10

images and

2000

permutations. For Sliced W, the

s

column is left blank.

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Optimal Transport Wasserstein Distance

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We study the approximation of probability measures in the Wasserstein-

p

distance by structured classes of approximators, motivated by applications in imaging, machine learning, and physical measurement under sensor constraints. We obtain three sets of results. First, for measures with densities bounded away from zero on a bounded Lipschitz domain

Ω

, we prove that any approximation scheme for functions in

\mathrm{L}_p(Ω)

transfers, with linear rate, to a corresponding approximation scheme for measures in

\mathrm{W}_p(Ω)

. The argument applies a theorem of Bogovskii on regularity of solutions to the continuity equation in the Benamou-Brenier formulation of optimal transport. We exhibit concrete approximation schemes (polynomials, shift-invariant spaces, cardinal interpolation with radial basis functions, kernel density estimators, and piecewise approximations on nonuniform Voronoi partitions) that fit the framework. As a matter of independent interest, we prove a negative Sobolev lower bound that generalizes existing bounds from

p=2

to all

p\in(1,\infty)

. We also consider deterministic bounds for discrete approximations to arbitrary measures in terms of the mesh norm of a quasi-uniform set of points. We specialize these bounds to show that compactly supported measures admit a deterministic

N

-term approximation

μ_N

such that

\mathrm{W}_p(μ,μ_N) = O(N^{-\frac{1}{d}})

for all

d\geq 1

, which matches the asymptotic optimal quantizer rate. We also extend these results to non-compactly supported measures with appropriate tail decay.

Abstract

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