On Riemannian manifolds, generative-model evaluation is geometry-dependent since curvature and dimension jointly determine distances and volume growth. Our framework systematically compares geodesic, ambient, and tangent-space versions of Wasserstein and MMD across different geometries. Using controlled-noise perturbations and Riemannian Flow Matching samples, we study metric agreement, noise sensitivity, and their relation to geometry-induced volume scaling. We find that no metric is uniformly reliable: evaluation distance can affect both measured discrepancies and noise sensitivity, varying across curvature and dimension. Our analysis leads to practical recommendations for reporting evaluation distances, comparing metric types, and calibrating metrics to the geometry of the samples.
Figures & tables
Figure 1: Metric discrepancies depend on evaluation geometry. Cells show log2(Dgeodesic/Dother) over (κ,d) for GtG, comparing geodesic evaluation with ambient (top) and tangent-space (bottom) distances. Positive values mean geodesic eval. yields a larger discrepancy. W1 is mostly larger under geodesic, whereas MMD decreases in high d .
Figure 2: Evaluation distance changes metric trends. Curves show log2(Dgeodesic/Dother) across dimensions for four curvatures. W1 and MMD2 respond differently to the same change from intrinsic to non-intrinsic evaluation.
Table 3
Appendix figures & tables18 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 5: The three evaluation distances, illustrated for spherical and hyperbolic space. Samples are compared intrinsically by geodesic distance, extrinsically by ambient Euclidean distance, or locally after log-mapping to the tangent space at o . The same pair of points can thus produce different error.
Figure 6: Illustration of the evaluation metrics. The same predicted samples can be assessed by transport cost ( W1 ), kernel similarity ( MMD2 ), and dispersion via Fréchet variance.
Figure 7: Jκ versus r for different κ .
Figure 8: Overview of the target distributions. Columns show GtG, SGR, and MoG targets; rows show Euclidean, spherical, and hyperbolic geometries. The same tangent-space distributions lead to different sample geometries after mapping to manifolds with different curvature.
Parameter
Value
Curvatures ( κ )
-10, -5, -3, -2, -1, 0, 1, 2, 3, 5, 10
Dimensions ( d )
2, 3, 4, 8, 16, 32, 64, 256, 512
Seeds
34,42,50
Appendix
Table 3: General hyperparameters.
Parameter
Value
Base standard deviation ( σ0 )
0.3
Target standard deviation ( σ1 )
0.3
Separation distance ( δ )
0.5
Appendix
Table 4: Additional hyperparameters for the GtG experiments.
Parameter
Value
Radius ( rSGR )
0.35 , 0.60 , 0.85
Radial standard deviation ( σSGR )
0.05
Appendix
Table 5: Additional hyperparameters for the SGR experiments.
Parameter
Value
Number of target modes ( n )
2, 3, 5
Base standard deviation ( σ0 )
0.3
Target standard deviation ( σ1 )
0.3
Balanced weights ( wn )
[1/n,…,1/n]
Unbalanced weights ( wn )
w2=31[1,2],w3=41[1,1,2],w5=81[1,2,3,1,1]
Appendix
Table 6: Additional hyperparameters for the MoG experiments.
Experiment
Hidden layers
Batch size
Learning rate
d
∣κ∣
Seeds
GtG, SGR, MoG
8
512
1×10−4
2, 3, 4, 8, 16, 32, 64, 256, 512
0, 1, 2, 3, 5, 10
34, 42, 50
Appendix
Table 7: Model hyperparameters and sweep configurations.
Figure 9: Metric-type agreement is target-dependent. Spearman correlations are computed across all (κ,d,seed) -configurations for each target. For Fréchet variance we use ∣logρF∣ so lower values indicate better quality. Distance representations are strongly correlated within a metric type, but agreement across W1 , MMD2 , and Fréchet variance is weaker and depends on target structure.
Curvature
d=3
16
64
256
512
Hyperbolic
1.00
0.99
0.77
0.18
0.11
Spherical
1.00
1.01
0.95
1.01
1.01
Appendix
Table 8: Ambient Fréchet variance underestimates high- d hyperbolic dispersion. Entries show ρFdambient/ρFdgeodesic for RFM SGR samples. Values below 1 mean ambient evaluation reports a more small dispersion error than geodesic evaluation. The gap grows with dimension in hyperbolic space but stays near 1 on spheres.
Figure 10: Dispersion errors depend on target structure and representation. Ratio of predicted-to-target Fréchet variance across signed curvature κ and dimension d , using geodesic, tangent-space, and ambient distances. GtG remains close to 1 across the sweep, while SGR develops large high- d overdispersion under geodesic and tangent-space evaluation. Ambient hyperbolic evaluation compresses this magnitude, consistent with Table 8 .
d
ρW1
ρMMD
2
1.0000
0.9903
3
1.0000
0.9894
4
1.0000
0.9940
8
1.0000
0.9949
16
0.9997
0.9940
64
0.9959
0.9760
Appendix
Table 9: Geodesic W1 remains monotone, while geodesic MMD2 weakens at high d . Mean Spearman correlations between noise level σ and metric value on GtG, averaged over curvatures and seeds. Values near 1 indicate that the metric increases consistently with controlled noise. Spherical configurations at d∈{256,512} are excluded (see Appendix D.3 ).
Figure 11: Noise responses depend on dimension and evaluation distance. At d=2 and d=64 , both W1 and MMD2 generally increase with controlled noise. However, the responses differ per evaluation distance, especially in hyperbolic space.
Figure 12: Noise-detection reliability varies by metric and target structure. SNR is shown as a function of dimension for GtG, SGR, and MoG, using geodesic, tangent-space, and ambient distances. This supports the main-text claim that reliability is not a property of the metric alone.
Figure 13: SGR radius alone does not determine geodesic SNR. Mean geodesic SNR is shown against the target radius rp1 in spherical and hyperbolic geometries, coloured by dimension. SNR remains high across radii, while dimension explains much of the variation.
Figure 14: Volume scaling explains some, but not all, SNR gaps. Each point is a (κ,d) -configuration on GtG. The x-axis shows target volume scaling, logμJκ(p1) , and the y-axis shows the geodesic SNR advantage over the compared distance. Positive values mean geodesic evaluation detects controlled noise more reliably. Volume scaling correlates with the geodesic advantage over ambient W1 and over both MMD alternatives, with the relationship weaker for tangent space W1 .
Sample validity
Relative tangency violation
∥vt(x)∥gκ (p95)
Target
# runs
finite (%)
failed (%)
median
max
median
max
GtG
330
100.0
0.0
4.5×10−6
4.6×10−5
0.45
20.40
SGR
1350
100.0
0.0
2.3×10−6
5.3×10−4
0.71
5.20
MoG
1980
100.0
0.0
1.0×10−5
2.9×10−4
0.45
12.76
All
3660
100.0
0.0
7.1×10−6
5.3×10−4
0.52
20.40
Appendix
Table 10: Numerical stability for RFM sweeps. All completed runs produce finite samples, and relative tangency violations remain very small. This supports the interpretation that the metric differences in the text are not caused by invalid samples or off-manifold vector fields. The p95 vector-field norm is included as a secondary scale check.
We propose the Monge Inception Distance (MIND), a metric for evaluating generative models that addresses key limitations of the widely adopted Fréchet Inception Distance (FID). The MIND metric leverages the sliced Wasserstein distance to compare distributions by averaging one-dimensional optimal transport distances, efficiently computed via sorting. This approach circumvents the estimation of high-dimensional means and covariance matrices, which underlie FID's poor sample complexity and vulnerability to adversarial attacks. We empirically demonstrate three primary advantages: (i) it is more sample-efficient by one order of magnitude, (ii) it is faster to compute by two orders of magnitude, (iii) it is more robust to adversarial attacks such as moment-matching. We show that MIND with 5k samples can replace the evaluation performance of FID with 50k samples, providing high correlation with this standard benchmark and superior discriminative performance. We further demonstrate that even smaller sample sizes (e.g., 1k or 2k) remain highly informative for rapid model iteration.
Many scientific datasets, such as molecular conformational ensembles or single-cell tissue measurements, are naturally modeled as meta-distributions: distributions over probability measures on non-Euclidean domains. Existing generative methods largely assume Euclidean geometry and fail to capture this structure. We introduce Riemannian Wasserstein Entropic Flow Matching (RWEFM), a generative framework on the Wasserstein space P2(M) of a Riemannian manifold (M,g). RWEFM is trained by regressing a neural vector field onto Riemannian optimal transport velocities, using McCann displacement interpolations as conditional paths. We confirm theoretically that this construction leads to a valid flow matching approach on P2(M) and introduce the Riemannian Entropic Map, a GPU-efficient approximation of the optimal transport map on manifolds. Our experiments show that by respecting the intrinsic geometry of the data, RWEFM can generate whole single-cell samples in hyperspherical latent spaces and protein conformational ensembles on the torus. As RWEFM requires only a geodesic distance and a projection operator, it is not restricted to manifolds with closed-form geometry, which we demonstrate by generating distributions on a general triangulated mesh.
Doron Haviv, Edward De Brouwer, Rishabh Anand +4
Genentech Inc., South San Francisco, CA. · Yale University, New Haven, CT.
The geometry of generative models serves as the basis for interpolation, model inspection, and more. Unfortunately, most generative models lack a principal notion of geometry without restrictive assumptions on either the model or the data dimension. In this paper, we construct a general geometry compatible with different metrics and probability distributions to analyze generative models that do not require additional training. We consider curves analogous to geodesics constrained to a suitable data distribution aimed at targeting high-density regions learned by generative models. We formulate this as a (pseudo)-metric and prove that this corresponds to a Newtonian system on a Riemannian manifold. We show that shortest paths in our framework can be characterized by a system of ordinary differential equations, which locally corresponds to geodesics under a suitable Riemannian metric. Numerically, we derive a novel algorithm to efficiently compute shortest paths and generalized Fréchet means. Quantitatively, we show that curves using our metric traverse regions of higher density than baselines across a range of models and datasets.
Frederik Möbius Rygaard, Shen Zhu, Yinzhu Jin +2
DTU Compute, Technical University of Denmark, Kongens Lyngby, Denmark · Department of Computer Science, University of Virginia, Charlottesville (VA), USA · Department of Electrical and Computer Engineering, University of Virginia, Charlottesville (VA), USA