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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