Harnessing Data Asymmetry: Manifold Learning in the Finsler World

Thomas Dagès; Simon Weber; Daniel Cremers; Ron Kimmel

cs.LGMar 12, 2026

Harnessing Data Asymmetry: Manifold Learning in the Finsler World

Authors: Thomas Dagès, Simon Weber, Daniel Cremers, Ron Kimmel

Organizations: Technion – Israel Institute of Technology · Technical University of Munich · Munich Center for Machine Learning · University of Oxford

Abstract

Manifold learning is a fundamental task at the core of data analysis and visualisation. It aims to capture the simple underlying structure of complex high-dimensional data by preserving pairwise dissimilarities in low-dimensional embeddings. Traditional methods rely on symmetric Riemannian geometry, thus forcing symmetric dissimilarities and embedding spaces, e.g. Euclidean. However, this discards in practice valuable asymmetric information inherent to the non-uniformity of data samples. We suggest to harness this asymmetry by switching to Finsler geometry, an asymmetric generalisation of Riemannian geometry, and propose a Finsler manifold learning pipeline that constructs asymmetric dissimilarities and embeds in a Finsler space. This greatly broadens the applicability of existing asymmetric embedders beyond traditionally directed data to any data. We also modernise asymmetric embedders by generalising current reference methods to asymmetry, like Finsler t-SNE and Finsler Umap. On controlled synthetic and large real datasets, we show that our asymmetric pipeline reveals valuable information lost in the traditional pipeline, e.g. density hierarchies, and consistently provides superior quality embeddings than their Euclidean counterparts.

Figures & tables

Figure 1: Motivation: how asymmetry can arise and why preserving it matters. We aim to recover the underlying smooth US manifold from US cities (latitude-longitude). Hidden external factors (e.g. mountain ranges) bias the sampling density: fewer cities lie in high-altitude regions. Reweighing distances by local density yields asymmetric dissimilarities that encode differences in geographical setting. Symmetrising and embedding in symmetric spaces (e.g. Isomap, t-SNE, Umap, Poincaré maps) discards this information. Our asymmetric dissimilarity construction can be fed to existing traditional asymmetric embedders (e.g. slide-vector, radius-distance), but these are heuristic and non-metric. Finsler geometry provides a principled metric framework, enabling the existing Finsler MDS [ 37 ] and our novel and more scalable Finsler t-SNE and Umap to better embed and reveal the hidden terrain.

Figure 2: The traditional manifold learning pipeline leads to asymmetric data dissimilarities on sampled data due to directed proximity graphs and local distance transforms. As this violates the Riemannian manifold assumption and is incompatible with symmetric Euclidean space embeddings, heuristic symmetrisation of data dissimilarities is required yet theoretically unjustified. We propose to equip the data manifold with a Finsler metric allowing asymmetric dissimilarities. Embeddings are then performed in a canonical Finsler space, enabling us not only to accurately capture the structure of the data but also harness and reveal the natural asymmetry of the sampling.

Figure 4: Metrics define distances in tangent spaces (left) via their convex unit tangent ball. Riemannian metrics – whether isotropic or anisotropic – remain symmetric due to symmetric unit tangent balls. Finsler metrics allow asymmetry, so geodesic paths and distances need not be symmetric (right). Courtesy of [ 121 , 37 , 36 ] .

Data

Isomap

t-SNE

Umap

Finsler MDS

Finsler t-SNE

Finsler Umap

Figure 6: Swiss roll embeddings. Euclidean baselines aim only to preserve the manifold, with debatable success. Our asymmetric Finsler embeddings additionally reveal density asymmetry: dense points (blue) are embedded lower than sparse ones (red). For Finsler MDS and t-SNE we show, top to bottom, the embedding with equal axes, a rotated side view, and an unequal-axes view that magnifies height differences.

Figure 7: Our Finsler approach clusters and reveals cluster hierarchy, as sparser clusters are embedded higher. This is shown either locally (Finsler t-SNE, default Finsler Umap) or globally (Finsler MDS, extended Finsler Umap), and is absent in Euclidean methods.

Figure 9: Percentage difference in mean performance between our Finsler methods and their traditional Euclidean baselines. Positive (resp. negative) differences, in green (resp. red), means we get better (resp. worse) scores comparing kMeans clusters with groundtruth class labels. See section F.3.1 for full results on all tested datasets, with visual summaries ( fig. 14 ), raw performance values, and standard deviations ( tables 2 and 3 ).

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 10 — Figure 10: Embeddings of the toy planar data with non-uniform density. Each pair of rows corresponds to a different number of datapoints $n$ , with $n=300$ (top), $n=1000$ (middle), and $n=3000$ (bottom). Within each pair of plots, the top (resp. bottom) row corresponds to embedding results in the recommended (resp. unrecommended) embedding dimension, namely $\mathbb{R}^{2}$ (resp. $\mathbb{R}^{3}$ ) for symmetric methods and $\mathbb{R}^{3}$ (resp. $\mathbb{R}^{2}$ ) for asymmetric Finsler embeddings to the canonical Randers space. The last dimension of Finsler embeddings, i.e. the $z$ -axis (resp. $y$ -axis) in three-dimensional (resp. two-dimensional) embeddings fully encodes the asymmetry, providing more information on the data than vanilla methods like Isomap. On the other hand, traditional symmetric embedding methods do not reveal the density difference between points, unlike the Finsler embedding methods, even those that use fewer dimensions than recommended for a two-dimensional data manifold.

Figure 11: Embeddings of simulated mutation data with exponentially decreasing samples per groundtruth Gaussian. While traditional methods cluster, with more or less success, our Finsler approach both clusters correctly and encodes a cluster hierarchy via height, as sparser low-count clusters are embedded higher. This is shown either locally, when handling sparse dissimilarities (Finsler t-SNE, default Finsler Umap), or globally when using dense dissimilarities due to geodesic extensions of graph distances (Finsler MDS, extended Finsler Umap). This semantic hierarchy is absent in Euclidean methods. This figure is an enlarged version of fig. 7 .

Figure 12: Imagenette: Finsler t-SNE vs. Poincaré maps, a non-Euclidean hyperbolic baseline. Right: percentage difference in kMeans scores (positive/green means Finsler t-SNE is better). The gains reveal that modelling asymmetry with Finsler geometry may capture structure better than other non-Euclidean embeddings.

		Label-related scores							Label-unrelated scores
Dataset	Method	AMI (+)	ARI (+)	NMI (+)	HOM (+)	COM (+)	V-M (+)	FMI (+)	SIL (+)	DBI (–)	CHI (+)
Iris	Umap	0,724 ( $\pm$ 0,248)	0,688 ( $\pm$ 0,241)	0,728 ( $\pm$ 0,245)	0,719 ( $\pm$ 0,243)	0,737 ( $\pm$ 0,247)	0,728 ( $\pm$ 0,245)	0,794 ( $\pm$ 0,159)	0,721 ( $\pm$ 0,019)	0,389 ( $\pm$ 0,025)	2526 ( $\pm$ 1202)
Iris	Finsler Umap (Ours)	0,794 ( $\pm$ 0,024)	0,746 ( $\pm$ 0,032)	0,796 ( $\pm$ 0,023)	0,785 ( $\pm$ 0,026)	0,808 ( $\pm$ 0,020)	0,796 ( $\pm$ 0,023)	0,832 ( $\pm$ 0,020)	0,705 ( $\pm$ 0,036)	0,412 ( $\pm$ 0,047)	1471 ( $\pm$ 331)
MNIST	Umap	0,847 ( $\pm$ 0,017)	0,782 ( $\pm$ 0,040)	0,847 ( $\pm$ 0,017)	0,839 ( $\pm$ 0,019)	0,855 ( $\pm$ 0,015)	0,847 ( $\pm$ 0,017)	0,805 ( $\pm$ 0,035)	0,580 ( $\pm$ 0,018)	0,606 ( $\pm$ 0,033)	194060 ( $\pm$ 19078)
MNIST	Finsler Umap (Ours)	0,858 ( $\pm$ 0,012)	0,793 ( $\pm$ 0,032)	0,858 ( $\pm$ 0,012)	0,848 ( $\pm$ 0,014)	0,868 ( $\pm$ 0,011)	0,858 ( $\pm$ 0,012)	0,816 ( $\pm$ 0,028)	0,611 ( $\pm$ 0,029)	0,585 ( $\pm$ 0,060)	119299 ( $\pm$ 10201)
Fashion-MNIST	Umap	0,616 ( $\pm$ 0,009)	0,445 ( $\pm$ 0,028)	0,617 ( $\pm$ 0,009)	0,610 ( $\pm$ 0,011)	0,623 ( $\pm$ 0,008)	0,617 ( $\pm$ 0,009)	0,503 ( $\pm$ 0,024)	0,491 ( $\pm$ 0,017)	0,738 ( $\pm$ 0,039)	173803 ( $\pm$ 3963)

Appendix

Table 2: Mean clustering performance using kMeans on Umap and Finsler Umap data embeddings. The mean and standard deviation over 10 runs is presented. (+)/(-) signifies that higher/lower is better.

Dataset	Method	AMI (+)	ARI (+)	NMI (+)	HOM (+)	COM (+)	V-M (+)	FMI (+)	SIL (+)	DBI (–)	CHI (+)
		Label-related scores							Label-unrelated scores
Iris	t-SNE	0,829	0,851	0,831	0,831	0,831	0,831	0,900	0,665	0,453	3028
Iris	Finsler t-SNE (Ours)	0,845	0,868	0,846	0,846	0,847	0,846	0,911	0,664	0,465	4822
DTD	t-SNE	0,498	0,288	0,581	0,578	0,584	0,581	0,303	0,402	0,804	2492
DTD	Finsler t-SNE (Ours)	0,504	0,294	0,585	0,580	0,590	0,585	0,310	0,408	0,859	1186
Caltech101	t-SNE	0,804	0,377	0,830	0,865	0,798	0,830	0,428	0,547	0,610	23151

Appendix

Table 3: Clustering performance using kMeans on t-SNE and Finsler t-SNE embeddings across the smaller datasets. (+)/(-) signifies that higher/lower is better.

Supervised Evaluator	Embedding	DTD	Caltech101	OxfordFlowers102	OxfordIIITPet	Imagenette
5-NN classifier	UMAP	0.51	0.87	0.55	0.92	0.98
	Finsler UMAP	0.56	0.88	0.62	0.93	0.98
	t-SNE	0.57	0.88	0.63	0.92	0.98
	Finsler t-SNE	0.58	0.88	0.62	0.92	0.98
Linear classifier	UMAP	0.34	0.53	0.37	0.80	0.97
	Finsler UMAP	0.43	0.66	0.48	0.87	0.98

Appendix

Table 4: Supervised accuracy on image benchmarks for two classifiers. We report mean test accuracy over 2-stratified 5-fold CV (10 runs). Best/ties are bold/underlined. Finsler variants consistently outperform Euclidean ones, as suggested already by the unsupervised scores.

Figure 13: Percentage difference in mean performance between our Finsler methods and their traditional Euclidean baselines. Positive (resp. negative) differences, in green (resp. red) signifies we get better (resp. worse) scores evaluating the quality of kMeans clusters, either by comparing with groundtruth class labels or not. See fig. 15 . Raw performance values and standard deviations are pushed to tables 2 and 3 . This figure is the same as fig. 9 , except that we added to the label-related scores, in white background, the label-unrelated scores, in grey background. Having better label-unrelated score with worse label-related scores is particular bad, as it means that the method is artificially creating “nicely looking” clusters that do not correspond to the reality of the data manifold. This never happens for our Finsler methods, as we systematically have better label-related measures.

Figure 14: Complete version of fig. 9 for all tested datasets, presenting the percentage difference in mean performance between our Finsler methods and their traditional Euclidean baselines. Positive (resp. negative) differences, in green (resp. red) signifies we get better (resp. worse) scores comparing kMeans clustering with groundtruth class labels. Raw performance values are in tables 2 and 3 .

Figure 15: Complete version of fig. 13 for all datasets, presenting the percentage difference in mean performance between our Finsler methods and their traditional Euclidean baselines. Positive (resp. negative) differences, in green (resp. red) signifies we get better (resp. worse) evaluating the quality of kMeans clusters, either by comparing with groundtruth class labels or not. Raw performance values are in tables 2 and 3 . This figure is the same as fig. 14 , except that we added to the label-related scores, in white background, the label-unrelated scores, in grey background. Having better label-unrelated score with worse label-related scores is particular bad, as it means that the method is artificially creating “nicely looking” clusters that do not correspond to the reality of the data manifold. This never happens for our Finsler methods, as we systematically have better label-related measures.

Figure 16: Embeddings results, using recommended embedding dimensions, on all our tested reference classification datasets using either traditional Euclidean methods or our Finsler pipelines.

Figure 17: Embeddings results, using unrecommended embedding dimensions, on all our tested reference classification datasets using either traditional Euclidean methods or our Finsler pipelines.

			Label-related scores							Label-unrelated scores
Dataset	Method	Dimensionality $d$	AMI (+)	ARI (+)	NMI (+)	HOM (+)	COM (+)	V-M (+)	FMI (+)	SIL (+)	DBI (–)	CHI (+)
CIFAR10	Umap	2	0,654 ( $\pm$ 0,327)	0,619 ( $\pm$ 0,310)	0,654 ( $\pm$ 0,327)	0,651 ( $\pm$ 0,325)	0,656 ( $\pm$ 0,328)	0,654 ( $\pm$ 0,327)	0,658 ( $\pm$ 0,279)	0,585 ( $\pm$ 0,115)	0,570 ( $\pm$ 0,128)	239743 ( $\pm$ 98339)
	Finsler Umap (Ours)	2	0,817 ( $\pm$ 0,009)	0,767 ( $\pm$ 0,029)	0,817 ( $\pm$ 0,009)	0,812 ( $\pm$ 0,013)	0,822 ( $\pm$ 0,005)	0,817 ( $\pm$ 0,009)	0,791 ( $\pm$ 0,024)	0,647 ( $\pm$ 0,016)	0,516 ( $\pm$ 0,033)	187999 ( $\pm$ 11072)
	Umap	5	0,814 ( $\pm$ 0,011)	0,763 ( $\pm$ 0,023)	0,814 ( $\pm$ 0,011)	0,810 ( $\pm$ 0,012)	0,819 ( $\pm$ 0,009)	0,814 ( $\pm$ 0,011)	0,788 ( $\pm$ 0,020)	0,634 ( $\pm$ 0,013)	0,546 ( $\pm$ 0,038)	228691 ( $\pm$ 12209)
	Finsler Umap (Ours)	5	0,815 ( $\pm$ 0,011)	0,767 ( $\pm$ 0,025)	0,815 ( $\pm$ 0,011)	0,811 ( $\pm$ 0,013)	0,820 ( $\pm$ 0,009)	0,815 ( $\pm$ 0,011)	0,791 ( $\pm$ 0,021)	0,638 ( $\pm$ 0,015)	0,541 ( $\pm$ 0,045)	173549 ( $\pm$ 11579)
	Umap	10	0,816 ( $\pm$ 0,010)	0,770 ( $\pm$ 0,014)	0,816 ( $\pm$ 0,010)	0,812 ( $\pm$ 0,010)	0,820 ( $\pm$ 0,009)	0,816 ( $\pm$ 0,010)	0,794 ( $\pm$ 0,012)	0,637 ( $\pm$ 0,012)	0,538 ( $\pm$ 0,034)	233395 ( $\pm$ 9628)

Appendix

Table 5: Mean clustering performance using kMeans on Umap and Finsler Umap data embeddings with varying dimensionality

d

. The mean and standard deviation over 10 runs is presented. (+)/(-) signifies that higher/lower is better.

Dataset	Method	$\lVert\omega\rVert_{2}$	AMI (+)	ARI (+)	NMI (+)	HOM (+)	COM (+)	V-M (+)	FMI (+)	SIL (+)	DBI (–)	CHI (+)
			Label-related scores							Label-unrelated scores
Iris	Umap	—	0,724 ( $\pm$ 0,248)	0,688 ( $\pm$ 0,241)	0,728 ( $\pm$ 0,245)	0,719 ( $\pm$ 0,243)	0,737 ( $\pm$ 0,247)	0,728 ( $\pm$ 0,245)	0,794 ( $\pm$ 0,159)	0,721 ( $\pm$ 0,019)	0,389 ( $\pm$ 0,025)	2526 ( $\pm$ 1202)
	Finsler Umap (Ours)	0,5	0,626 ( $\pm$ 0,313)	0,584 ( $\pm$ 0,293)	0,630 ( $\pm$ 0,309)	0,620 ( $\pm$ 0,304)	0,641 ( $\pm$ 0,314)	0,630 ( $\pm$ 0,309)	0,726 ( $\pm$ 0,194)	0,742 ( $\pm$ 0,063)	0,418 ( $\pm$ 0,113)	912 ( $\pm$ 778)
	Finsler Umap (Ours)	0,1	0,719 ( $\pm$ 0,242)	0,677 ( $\pm$ 0,228)	0,723 ( $\pm$ 0,239)	0,713 ( $\pm$ 0,236)	0,732 ( $\pm$ 0,242)	0,723 ( $\pm$ 0,239)	0,786 ( $\pm$ 0,151)	0,731 ( $\pm$ 0,021)	0,382 ( $\pm$ 0,027)	1760 ( $\pm$ 493)
	Finsler Umap (Ours)	0,01	0,722 ( $\pm$ 0,243)	0,683 ( $\pm$ 0,230)	0,726 ( $\pm$ 0,240)	0,717 ( $\pm$ 0,237)	0,735 ( $\pm$ 0,243)	0,726 ( $\pm$ 0,240)	0,790 ( $\pm$ 0,152)	0,717 ( $\pm$ 0,018)	0,402 ( $\pm$ 0,031)	1485 ( $\pm$ 275)
	Finsler Umap (Ours)	0,001	0,794 ( $\pm$ 0,024)	0,746 ( $\pm$ 0,032)	0,796 ( $\pm$ 0,023)	0,785 ( $\pm$ 0,026)	0,808 ( $\pm$ 0,020)	0,796 ( $\pm$ 0,023)	0,832 ( $\pm$ 0,020)	0,705 ( $\pm$ 0,036)	0,412 ( $\pm$ 0,047)	1471 ( $\pm$ 331)

Appendix

Table 6: Mean clustering performance using kMeans on Umap and Finsler Umap data embeddings. We use various levels of emphasis for metric asymmetry, via

\lVert\omega\rVert_{2}\in\{0.001,0.01,0.1,0.5\}

, in the embedding space of the Finsler methods. The mean and standard deviation over 10 runs is presented. (+)/(-) signifies that higher/lower is better.

Dataset	Method	$\lVert\omega\rVert_{2}$	AMI (+)	ARI (+)	NMI (+)	HOM (+)	COM (+)	V-M (+)	FMI (+)	SIL (+)	DBI (–)	CHI (+)
			Label-related scores							Label-unrelated scores
Iris	t-SNE	—	0,829	0,851	0,831	0,831	0,831	0,831	0,900	0,665	0,453	3028
	Finsler t-SNE (Ours)	0,5	0,845	0,868	0,846	0,846	0,847	0,846	0,911	0,664	0,465	4822
	Finsler t-SNE (Ours)	0,1	0,835	0,851	0,837	0,836	0,837	0,837	0,900	0,659	0,465	5092
	Finsler t-SNE (Ours)	0,01	0,835	0,851	0,837	0,836	0,837	0,837	0,900	0,670	0,442	5063
	Finsler t-SNE (Ours)	0,001	0,835	0,851	0,837	0,836	0,837	0,837	0,900	0,663	0,450	5263

Appendix

Table 7: Mean clustering performance using kMeans on t-SNE and Finsler t-SNE data embeddings across the smaller datasets. We use various levels of emphasis for metric asymmetry, via

\lVert\omega\rVert_{2}\in\{0.001,0.01,0.1,0.5\}

, in the embedding space of the Finsler methods. (+)/(-) signifies that higher/lower is better.

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Finsler Geometry Graph Embedding Manifold Learning Spectral Methods

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Many datasets carry an intrinsic directionality: citations point backward in time, cells differentiate along lineages, and traffic follows preferred routes. Spectral embedding methods, including most of their extensions to directed graphs, discard this information: they symmetrize the data and map it into a Euclidean space where asymmetry cannot be represented. We instead model directed data as sampled from a Finsler manifold, whose distance depends on the direction of travel, and study the kernel operator built from this asymmetric distance. Through a moment expansion of this operator, we show that its symmetric and antisymmetric parts separate geometry from direction. As the bandwidth of the kernel vanishes, the symmetric part converges to a weighted Laplacian, recovering diffusion maps in the Riemannian case, while the antisymmetric part converges to a first-order transport operator that encodes the directionality. We prove that the corresponding graph operators, built from finitely many samples, converge uniformly and almost surely to these limits. For Randers metrics, this vector field is explicit and yields an embedding algorithm recovering both the manifold structure, from the spectrum of the symmetric part, and the underlying drift. We illustrate the approach on synthetic directed graphs and point-clouds.

Abstract

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