CorrFlow: Exact and Permutation-Equivariant Flow Matching on the Correlation Elliptope

Gautier Marti

stat.MLOct 9, 2026

CorrFlow: Exact and Permutation-Equivariant Flow Matching on the Correlation Elliptope

Authors: Gautier Marti

Organizations: Abu Dhabi Investment Authority (ADIA), Abu Dhabi, United Arab Emirates

Abstract

Correlation matrices combine nonlinear positive-definiteness constraints with a simultaneous row-column permutation symmetry. Their generation therefore involves two distinct problems. First, every output must remain on the correlation elliptope, irrespective of how its variables are ordered. Second, the generator must learn the domain-specific distribution occupied by empirical data inside that valid space. Existing financial image generators mix these problems: they learn ordered matrix entries and repair invalid draws afterward. We introduce CorrFlow, which separates them. A generalized Fisher transform maps full-rank correlation matrices bijectively to unconstrained coordinates, guaranteeing valid outputs without projection. A graph flow-matching network learns the empirical distribution in those coordinates and is permutation equivariant: reordering the variables only reorders the output. We compare a conditional CorrGAN reimplementation, image diffusion, and CorrFlow on $80\times80$ Taiwan equity matrices using five seeds per neural method. CorrFlow produces 100% valid draws without repair and obtains feature sliced-Wasserstein distance $.712\pm.141$ , versus $4.339\pm.252$ for CorrGAN and $4.610\pm.191$ for image diffusion after their invalid outputs are projected back to the correlation space. It also improves this distance over historical lookup, while lookup retains higher local coverage. These results support the combined construction under the evaluated training budget; they do not establish superiority over fully tuned image models or isolate the contribution of each component.

Figures & tables

Figure 1: Geometry versus domain distribution for

3\times 3

matrices. Valid correlations lie inside the gray elliptope (

1+2\rho_{12}\rho_{13}\rho_{23}-\rho_{12}^{2}-\rho_{13}^{2}-\rho_{23}^{2}>0

). (a) Choosing entries independently produces invalid points. (b) Every unconstrained GFT vector maps inside the elliptope. (c) Nine hundred random three-stock submatrices from the Taiwan training windows occupy a much smaller, mostly positive region. GFT enforces the universal constraint; CorrFlow learns the domain-specific distribution within it.

Figure 2: What permutation equivariance guarantees. Colors track eight variable identities. The top path maps edge noise $X_{0}$ to a valid correlation matrix $C$ . The bottom path applies the same permutation $P$ to the input and uses the same CorrFlow weights and condition; its output is exactly the corresponding row–column permutation $PCP^{\top}$ . Thus relationships follow their variables and the model does not learn a privileged ordering. The two paths agree exactly in exact arithmetic; a numerical implementation may differ only through floating-point rounding.

Real-data references
Model	Valid before repair $\uparrow$	F-SWD $\downarrow$	Energy $\downarrow$	Precision $\uparrow$	Recall $\uparrow$	Classifier AUC
Real–real	–	.061 (.009)	.010 (.003)	.890 (.012)	.896 (.014)	.499 (.021)
Train–test temporal shift	–	–	–	–	–	.950 (.005)
Historical lookup (not generative)
Historical replay	1.000 (.000)	.886 (.010)	2.715 (.072)	.457 (.011)	.589 (.036)	.951 (.003)
Matched replay	1.000 (.000)	.899 (.011)	2.820 (.057)	.424 (.006)	.576 (.024)	.950 (.003)

Table 1: Taiwan test results. Distances (

\downarrow

) use repaired image outputs; validity before repair, precision, and recall (

\uparrow

) are better when higher. For classifier AUC, .5 means indistinguishable and 1 perfectly separable. Method rows give mean with sample s.d. in parentheses over five training or generation seeds; AUC is first averaged over 20 matched subsamples within each seed. Real-data references use 20 resamples. Bold marks the best fitted generator. Matched rows use the continuous performance score.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Group	Count	Definition	Interpretation
Correlation moments	2	Mean and standard deviation of $\{r_{ij}:i<j\}$	Overall correlation level and cross-sectional dispersion.
Correlation quantiles	7	Quantiles $.01,.05,.25,.50,.75,.95,.99$ of $\{r_{ij}:i<j\}$	Shape and tails of the pairwise-correlation distribution.
Eigenvalue quantiles	6	Quantiles $0,.05,.25,.50,.75,.95$ of $\{\lambda_{i}\}_{i=1}^{n}$	Shape of the bulk spectrum, including its smallest eigenvalue.
Leading eigenvalue mass	2	$\lambda_{n}$ and $\sum_{i=n-4}^{n}\lambda_{i}$	Variance concentrated in the largest common mode and in the five largest components.
Effective ranks	2	$\exp(-\sum_{i}p_{i}\log p_{i})$ and $1/\sum_{i}p_{i}^{2}$	Effective number of independent spectral dimensions; smaller values mean greater concentration.
Log determinant	1	$\log\det C=\sum_{i}\log\lambda_{i}$	Spectral volume, with strong sensitivity to small eigenvalues.

Appendix

Table 2: The 21 structural summaries used in distributional evaluation.

Dataset	Model	F-SWD $\downarrow$	Energy $\downarrow$	Precision $\uparrow$	Recall $\uparrow$
S&P	Dense GFT flow	.308	.430	.657	.597
S&P	Dense normalized-Cholesky flow	.335	.642	.167	.157
S&P	CorrFlow	.294	.393	.575	.420
Taiwan pilot	Dense GFT flow	1.073	4.465	.347	.217
Taiwan pilot	CorrFlow	.951	3.283	.497	.603

Appendix

Table 3: Exploratory coordinate and architecture diagnostics after the numerical inverse audit; one seed per model. Distances are comparable within each dataset only. All rows have 100% validity.

Objective	F-SWD $\downarrow$	Energy $\downarrow$	Precision $\uparrow$	Recall $\uparrow$
Rectified flow	.904	2.929	.335	.541
GFT diffusion, noise prediction	4.049	38.389	.000	.000
GFT diffusion, velocity prediction	1.151	5.070	.159	.429

Appendix

Table 4: Exploratory objective comparison; one matched training seed.

Method	F-SWD $\downarrow$	Precision	Recall	Portfolio SWD $\downarrow$	C2ST AUC
CorrFlow	$0.471\pm 0.062$	0.524	0.320	$0.624\pm 0.094$	0.999
DiffeoCFM	$0.751\pm 0.019$	0.304	0.355	$0.862\pm 0.026$	0.997
Transformed Gaussian	$0.871\pm 0.006$	0.301	0.504	$0.811\pm 0.013$	0.995
Training replay	$0.877\pm 0.008$	0.450	0.590	$1.107\pm 0.009$	0.955
cCorrGAN reconstruction	$4.039\pm 0.154$	0.000	0.401	$818.722\pm 824.701$	1.000

Appendix

Table 5: Matched categorical Taiwan comparison. Identical 1,500 test cases, 500 per regime. Means and sample standard deviations over five training/paired generation seeds for neural models; five draws from the same fitted training data for controls. Selected weights; CorrFlow 20 epochs, DiffeoCFM 200 and cCorrGAN 200, not equal compute.

Model	F-SWD $\downarrow$	Energy $\downarrow$	Precision $\uparrow$	Recall $\uparrow$
Historical replay	1.013	2.922	.550	.144
PCA-64 Gaussian	1.725	9.020	.006	.000
CorrFlow	1.374 (.114)	5.480 (.860)	.036	.017
GFT diffusion, velocity	1.641 (.116)	7.525 (.775)	.023	.008

Appendix

Table 6: Japan diagnostic. Neural distances are means (sample s.d.) over five training seeds; precision and recall show seed means. Controls use one generation seed. Different sample counts and market-specific feature scales preclude direct comparison of distances with the Taiwan table.

Assets	Training step (s)	Peak training (GiB)	ODE (s/matrix)	Inverse GFT (s/matrix)
80	.006	.097	.042	.009
160	.009	.192	.144	.008
200	.014	.260	.238	.012
500	.101	1.282	1.867	.028
1,000	.389	4.933	7.553	.082

Appendix

Table 7: Synthetic GB10 scaling at batch size one. Memory is peak PyTorch allocated memory during training, excluding other processes. The two sampling stages use different synthetic inputs and are not an end-to-end latency measurement.

Figure 3: The spectral change between the full, naturally weighted training and test splits. (a) Median cumulative variance explained by the leading eigenvalues; bands span the 10th–90th percentiles across matrices. (b) Test matrices shift toward a larger share in the five leading eigenvalues (median .382 versus .338). (c) Their entropy-based effective rank shifts lower (median 37.9 versus 42.9 out of 80). Dashed lines mark medians. The test period therefore contains fewer effective directions of standardized variation, consistent with stronger common movement and less diversification potential for broad portfolios.

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Flow-matching models generate samples through iterative velocity predictions, but typically do not revisit these predictions using the intermediate representations that produced them. We introduce CorrFlow, a self-correcting flow-matching framework that reuses this information through a lightweight refiner. Under rectified-flow interpolation, the error of the one-step clean prediction equals the realized velocity residual scaled by the noise level. Motivated by this identity, the refiner learns to predict residual corrections from the base model's clean prediction, conditioning, noise level and hidden states from the same forward pass. The correction is added to the base velocity before each sampling update, so subsequent predictions are made from the corrected state. This requires no additional base-model evaluations and only one lightweight refiner evaluation per step. Our theoretical analyses establish conditions under which feature reuse outperforms raw-input affine predictors and refiner training achieves greater residual-risk reduction than continued base-model training under a matched training compute budget. Experiments on class-conditional image generation, text-to-video generation and image-to-video generation show consistent improvements over the corresponding base models, including gains in visual quality and physical consistency. Project Page: https://fogradio.github.io/corrflow-project

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Generative models learn the statistical properties of their training data, so high-quality generation depends on clean and representative datasets. In scientific imaging, acquisition often yields noisy measurements, while collecting clean references can be costly, impractical or even unattainable. Training directly on these measurements results in a model that reproduces the corrupted data. This can be circumvented by learning the clean population distribution directly from the noisy data. Conditional flow matching (CFM) combines a simple regression objective with stable training, efficient sampling, and strong image-generation performance, making it a natural framework for this setting. We introduce Noise-Robust Conditional Flow Matching (NR-CFM), an unconditional generator that learns from one corrupted observation per image. NR-CFM provides a closed-form clean endpoint correction for additive white Gaussian noise and learns a data-driven correction for general Gaussian corruptions with more complex covariance structure. Across the evaluated corruption settings, NR-CFM outperforms NR-GAN in most cases and remains competitive with Ambient Diffusion in the high-noise regime. We further evaluate NR-CFM on scientific data at signal-to-noise ratios as low as

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Learning matrix-valued distributions from high-dimensional and possibly incomplete training data is challenging: ambient-space generative modeling is computationally expensive and statistically fragile when the matrix dimension is large but the sample size is limited. We propose CoreFlow, a geometry-preserving low-rank flow model that learns shared row/column subspaces across the matrix distribution, and then trains a continuous normalizing flow only on the induced low-dimensional core. CoreFlow is designed for settings where shared low-rank matrix geometry is present, especially in high-dimensional limited-sample regimes. This separates shared matrix geometry from sample-specific variation, preserves matrix structure, and substantially improves training efficiency. The same framework also handles incomplete training matrices through masked Riemannian updates and iterative completion. Across real and synthetic benchmarks, CoreFlow substantially improves spectral and moment-level generation quality in few-sample regimes while remaining competitive in data-rich settings, even under compression to 9% of the ambient dimension and with up to 40% missing training entries.

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Permutation-Equivariant Neural Networks

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