SOL: Measuring Gaps between Text Distributions by Double Sliced Wasserstein Metrics

Gregor Kornhardt; Moritz Piening; Jannis Chemseddine; Gabriele Steidl

cs.CLOct 5, 2026

SOL: Measuring Gaps between Text Distributions by Double Sliced Wasserstein Metrics

Authors: Gregor Kornhardt, Moritz Piening, Jannis Chemseddine, Gabriele Steidl

Organizations: Institute of Mathematics, TU Berlin, Berlin, Germany · Department of Computer Science, University of Göttingen, Göttingen, Germany

Abstract

Evaluating text generation requires measuring how well the generated distribution matches the data distribution. For autoregressive models, this is done by the perplexity. Diffusion and flow-based language models can only provide a likelihood bound, whose tightness differs between model families. Sample-based substitutes such as generative perplexity with entropy do not consider the distribution fit. We propose SOL, a distance between text distributions. Each sequence is represented by the empirical measure of its hidden states under a fixed transformer and the distributions of these measures are compared by the double sliced Wasserstein distance. We prove that SOL is a metric if the transformer is injective. Experiments show that SOL detects distributional failures, recovers expected model trends, and provides stable sample-based estimates. We put forward SOL to fill the gap in the current evaluation protocol used for non auto-regressive models. As a first step we use SOL to re-evaluate a variety of models trained on OpenWebText.

Figures & tables

Figure 1: Illustration of one SOL slice for fixed directions

\xi

and

g

using a causal transformer. A sequence is represented by the point cloud

T^{\theta}(\mathbf{x})\in\mathcal{P}_{2}(\mathbb{R}^{d})

of its contextual hidden states, and a corpus becomes a collection of such clouds. The first slice projects each cloud along

\xi

; its one-dimensional empirical distribution is then represented by a quantile function and sliced again along

g

. This maps the corpus to

\mathcal{P}_{2}(\mathbb{R})

, where the final Wasserstein distance is computed. SOL integrates this quantity over

\xi

and

g

GPT-2	Gen. PPL ( $\downarrow$ )
Small	12.5
Medium	11.4
Large	7.5
XL	8.5

Table 1: Gen. PPL across GPT-2 sizes for unconditional generation prefers GPT-2 Large.

Figure 2: For a fixed diffusion language model and dataset, the reported MAUVE score depends on the number of evaluation samples $n$ and clusters $K$ . In particular, for the standard choice $K=n/10$ (red boxes), even after 5k samples it is not stable. Generation details are given in Appendix E.2.2 .

Figure 3: Replacing random tokens in the sequence beginning is not detected by MAUVE, but by SOL.

Figure 4: Mode collapse is invisible to Gen.PPL, while SOL rises sharply, see Subsection 5.1 .

Figure 5: Left: increasing the sequence length diminishes the generation quality as expected. Middle: behavior under mode-collapse , i.e. going from

Q_{2}

Q_{1}

(blue), and if we introduce additional modes , i.e. going from

Q_{2}

Q_{3}

(orange). Right: SOL under word shuffling for different embeddings.

	MAUVE			SOL
Criterion $\uparrow$	GPT-2 Large	OLMo2 1B	Dream 7B	GPT-2 Large	OLMo2 1B	Dream 7B
Human-like	0.857	0.881	0.881	0.881	0.929	0.952
Interesting	0.714	0.762	0.762	0.762	0.786	0.762
Sensible	0.762	0.786	0.786	0.786	0.881	0.905

Table 2: Human judgement comparison at sequence length 1024 for different embedding models. Worst-case Spearman rank correlation

\rho_{\min}

between metric-induced model rankings and the Bradley–Terry (BT) human ranking. Results are shown for MAUVE and SOL for a full setup see Section C.4 .

Figure 8

Model	Setting	$H$	Gen. ppl.	SOL $\downarrow$
Real OWT G	—	5.449	14.9	$2.066$
Real ELF OWT E	—	5.402	17.0	$2.649$
Autoregressive
AR G	Temperature $1.0$	5.582	34.9	$22.207$
Discrete diffusion
DUO-base G	Ancestral, $T=1024$	5.537	76.9	$32.642$

Table 3: OpenWebText baselines using 5,000 samples per distribution and a 1,024-token evaluation horizon. SOL uses Dream-7B as the reference encoder. The complete benchmark is reported in Table 24 .

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Dream-7B	All other encoders	Description
$L_{\xi}$	$8192$	$1024$	First-level projection directions
$L_{g}$	$8192$	$1024$	Second-level GP directions
$M$	$64$		Resolution of second-level functions
$\lambda$	$0.1$		RBF-GP length scale
$\alpha$	$d$		Dimension correction factor
Projection seeds	$3$		Independent $\xi$ draws

Appendix

Table 4: Default SOL estimator parameters by encoder. Here,

d

denotes the encoder’s hidden dimension. We evaluate all

3\times 3

combinations of projection and GP seeds. Experiment-specific deviations are stated in the corresponding sections.

$K=L$	8	16	32	64	128	256	512	1024	2048	4096	8192	16384
Time [s]	0.38	0.39	0.38	0.39	0.41	0.47	0.63	1.02	2.16	5.87	18.95	61.16

Appendix

Table 5: Median post-hidden DSW runtime across three repeats on a single NVIDIA GeForce RTX 5090, with 5,000 texts per arm and

M=64

. Hidden states remain on GPU across direction blocks, the initial CPU-to-GPU transfer is included.

Figure 12

Figure 10: Increasing

M

for the quantile resolution has a small impact beyond

M=32

Perturbation	mean-SW	SOL	$\Delta$
Selected-word shuffle	10.66	15.75	$0.390$
Adjacent word swaps	7.09	9.63	$0.305$
Short-window shuffle	5.63	7.61	$0.301$
Reverse short spans	7.77	10.81	$0.329$
Clause permutation	2.61	3.36	$0.252$
Sentence permutation	2.12	2.60	$0.203$

Appendix

Table 6: Clean-normalized ordinary-distance responses of mean-SW and SOL under structural perturbations. Response columns are arithmetic means of paired ratios.

\Delta

is the mean paired log-response contrast.

Encoder	Length	$t=0.7$	$t=0.9$	$t=1.0$	$t=1.1$	$t=1.3$
GPT-2-large	512	2.901	1.066	0.994	1.038	1.355
	1024	2.872	1.091	1.014	1.070	1.434
Qwen3-0.6B	512	6.755	5.085	5.204	5.440	6.332
	1024	6.779	5.179	5.257	5.565	6.538
OLMo-2-1B	512	7.719	4.051	4.864	5.947	8.211
	1024	8.562	4.103	4.758	6.051	8.561

Appendix

Table 7: SOL evaluating generated text by GPT-2 Large at different decoding temperatures using differnet embedding models.

Temp.	Metric	vMF	Soft-Masked + ReMDM	MDLM
$0.7$	SOL	$48.6981\pm 0.292$	$57.6332\pm 0.420$	$54.3128\pm 0.404$
	Gen. PPL ( $H$ )	$7.92\;(3.950)$	$3.24\;(3.490)$	$7.26\;(3.884)$
$0.8$	SOL	$37.9781\pm 0.160$	$41.3254\pm 0.303$	$42.0520\pm 0.244$
	Gen. PPL ( $H$ )	$15.28\;(4.656)$	$5.33\;(4.360)$	$14.18\;(4.657)$
$0.9$	SOL	$33.1134\pm 0.076$	$24.7670\pm 0.161$	$37.4379\pm 0.149$
	Gen. PPL ( $H$ )	$26.36\;(5.054)$	$9.18\;(4.982)$	$27.01\;(5.093)$

Appendix

Table 8: Temperature sweep on OpenWebText for vMF, Soft-Masked with ReMDM-loop, and MDLM. Panels (a) and (b) report SOL using Dream-7B and GPT-2 Large, respectively. Gen. PPL is evaluated with GPT-2 Large and

H

denotes mean per-sequence token-frequency entropy in nats; these quantities are independent of the SOL encoder. SOL uses 5,000 samples per side and a 1,024-token horizon, with mean

\pm

population SD over nine seed pairs. Lower SOL is better and bold marks its column minimum.

Figure 11: SOL under increasing [CLS] insertion (left) and word-position shuffling (right) for four encoders. For all four language-model encoders, SOL increases monotonically with the corruption rate, and the two corruptions produce responses of similar magnitude. In contrast to the text encoders in Figure 12 , no encoder is markedly less sensitive to shuffling than to insertion.

Corruption	[CLS] pollution	Word shuffling
0%	0.9702	0.9702
1%	0.9367	0.9701
2%	0.9009	0.9667
5%	0.7888	0.9527
10%	0.6559	0.8423
20%	0.4558	0.3555

Appendix

Table 9: MAUVE under [CLS] pollution and word shuffling using last hidden representation.

Encoder	Clean	Prefix	Suffix
OLMo-2-1B	0.870	2.048	0.913
Dream-7B	2.493	4.496	4.621
Qwen3-0.6B	0.914	1.937	0.946
GPT-2-large	0.224	0.340	0.226

Appendix

Table 10: Shuffle 20% of the tokens the first or last 5% of the text.

Metric	$Q_{2}\!\rightarrow Q_{1}$	$Q_{2}\!\rightarrow Q_{3}$
SOL	1.000	1.000
mean-SW	1.000	0.9995

Appendix

Table 11: Spearman correlations with the target share along the

Q_{2}\!\rightarrow Q_{1}

and

Q_{2}\!\rightarrow Q_{3}

interpolation paths.

Encoder	Length	$t=0.7$	$t=0.9$	$t=1.0$	$t=1.1$	$t=1.3$
GPT-2-large	512	2.141	1.299	1.138	1.035	1.032
	1024	–	–	–	–	–
Qwen3-0.6B	512	6.083	4.310	4.288	4.347	4.847
	1024	6.695	4.513	4.470	4.645	5.311
OLMo-2-1B	512	7.822	5.452	5.016	4.808	5.203
	1024	9.413	6.440	5.421	4.916	4.884

Appendix

Table 12: SOL scores using the

[\mathbf{x},\mathbf{x}]

second-pass representation across sampling temperatures.

Region	$[\mathbf{x}]$	$[\mathbf{x},\mathbf{x}]$	$[\mathbf{x},\texttt{<sep>},\mathbf{x}]$
Clean	0.910	0.732	0.853
Prefix	1.710	1.167	1.926
Middle	1.335	1.347	1.375
Suffix	0.942	1.599	0.958

Appendix

Table 13: OLMo regional corruption, cap 1,024. Mean

\mathrm{SOL}

for 10% replacement within a 5% region, taking the square root per seed before averaging, clean scores use uncorrupted candidate text.

Temp.	$[\mathbf{x}]$	$[\mathbf{x},\mathbf{x}]$	$[\mathbf{x},\texttt{<sep>},\mathbf{x}]$
$0.7$	8.562	9.413	8.152
$0.9$	4.103	6.440	4.060
$1.0$	4.758	5.421	4.703
$1.1$	6.051	4.916	5.931
$1.3$	8.561	4.884	8.205

Appendix

Table 14: SOL: GPT-2-large generations scored with OLMo at cap 1,024. Square root of the mean squared score across seeds. Bold marks the minimum within each method.

Figure 12: SOL under increasing shuffling of tokens (left) and inserting a CLS token (right). Compared to GPT2-Large, the text encoders are less sensitive to shuffling than GPT-2 Large.

Insertion rate	SOL $(\downarrow)$
$20\%$	1
$50\%$	0.942

Appendix

Table 15: NeoBERT SOL under increasing [CLS] insertion, normalized to the

20\%

value.

	MAUVE			SOL
Criterion $\uparrow$	GPT-2 Large	OLMo2 1B	Dream 7B	GPT-2 Large	OLMo2 1B	Dream 7B
Human-like	0.881	0.929	0.905	0.952	0.952	0.976
Interesting	0.643	0.786	0.690	0.762	0.762	0.857
Sensible	0.833	0.881	0.881	0.929	0.929	0.976

Appendix

Table 16: Human judgement comparison at sequence length 256. Worst-case Spearman rank correlation between metric-induced model rankings and the Bradley–Terry (BT) human ranking. Results are shown for MAUVE and SOL using GPT-2 Large, OLMo2 1B, and Dream 7B embeddings. Reported are the minimum correlations

\rho_{\min}

across the evaluated settings at sequence length 256.

Corruption	[CLS] pollution	Word shuffling
0%	0.9725	0.9725
1%	0.9590	0.9710
2%	0.9183	0.9697
5%	0.4709	0.9597
10%	0.0624	0.8236
20%	0.0181	0.1208

Appendix

Table 17: MAUVE under [CLS] pollution and word shuffling using mean pooling.

Figure 13: For two disjoint LM1B corpora, the reported MAUVE score depends on the number of evaluation samples $n$ and clusters $K$ . In comparison to generated vs true data the ratio $n/10=K$ is stable, but increasing $n$ converges to 1.

Model	Middle	Final
Real OWT	6.578	6.572
DUO ancestral	6.455	6.441
DUO greedy	6.317	5.011

Appendix

Table 18: Across-document token entropy

H_{\mathrm{pos}}

in nats (

N=5000

), measured at position 512 and the last valid token.

	MAUVE $(\uparrow)$		SOL $(\downarrow)$
Model	Base	Reweighted	Base	Reweighted
DUO ancestral	0.836	0.843	1.670	1.7418
DUO greedy	0.492	0.591	1.748	1.8538

Appendix

Table 19: Final-token reweighting on matched corrected inputs (

N=5000

). GPT-2-large SOL values average nine projection-seed combinations.

Model	Original	Remove final 8
ELF-M	$0.455\pm 0.011$	$0.823\pm 0.017$
ELF-L	$0.287\pm 0.006$	$0.798\pm 0.013$

Appendix

Table 20: MAUVE

(\uparrow)

for shared ELF sampling. Truncation removes eight tokens from both generated and reference texts. Mean

\pm

SD over three clustering seeds.

Rep.	$\Delta P$	$\Delta R$	$\Delta C$
1	+2.14	+6.70	+3.10
2	+3.22	+4.88	+1.80
3	+2.10	+8.32	+0.84
Mean	+2.49	+6.63	+1.91
SD	0.64	1.72	1.13

Appendix

Table 21: ReMDM-loop minus MDLM at

k=4

, in percentage points. SD is across three independent repetitions.

Sampler	MAUVE ( $\uparrow$ )	SOL ( $\downarrow$ )
Greedy	$0.0697\pm 0.0012$	$3.7755\pm 0.1190$
Ancestral	$0.9407\pm 0.0028$	$0.5083\pm 0.0113$
Nucleus ( $p=0.95$ )	$\mathbf{0.9607\pm 0.0017}$	$\mathbf{0.2712\pm 0.0065}$

Appendix

Table 22: GPT-2-XL conditional completion on

5{,}000

WebText documents, using shared

35

-token prefixes and a maximum total sequence length of

256

tokens. SOL uses GPT-2-large.

Sampler	MAUVE ( $\uparrow$ )	SOL ( $\downarrow$ )
Ancestral	$\mathbf{0.8832\pm 0.0071}$	$\mathbf{0.8358\pm 0.0245}$
Nucleus ( $p=0.95$ )	$0.8740\pm 0.0073$	$0.9072\pm 0.0279$

Appendix

Table 23: GPT-2-XL unconditional generation evaluated against

5{,}000

WebText documents. Each of the

5{,}000

samples starts from the end-of-text token and stops at EOS or

256

generated tokens, the initial seed token is excluded from evaluation. SOL uses GPT-2-large.

				SOL $(\downarrow)$
Model	Setting	$H@1024$	Gen. ppl.	GPT-2-large	OLMo-2-1B	Dream-7B
Real OWT G	—	5.449	14.9	$0.2002\pm 0.0116$	$0.7800\pm 0.0552$	$2.066\pm 0.109$
Real ELF OWT E	—	5.402	17.0	$0.2457\pm 0.0239$	$0.8857\pm 0.0636$	$2.649\pm 0.219$
Autoregressive model Sahoo et al. (2024)
AR G	Temperature 1.0	5.582	34.9	$1.064\pm 0.008$	$7.450\pm 0.132$	$22.207\pm 0.099$
Discrete diffusion

Appendix

Table 24: OpenWebText baselines using 5,000 samples per distribution and a 1,024-token horizon. The standard SOL hyperparameters are used; see Section C.3 . Superscripts G, E and B identify the evaluation reference family. SOL is reported as mean

\pm

population std over nine crossed direction/GP seed pairs.

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