MLVis: Machine Learning Methods in Visualisation for Big Data (2024)
Daniel Archambault, Ian Nabney, and Jaakko Peltonen (Editors)

Visualizing Riemannian data with Rie-SNE
A. Bergsson1 and S. Hauberg2
1 Marel, Reykjavík, Iceland
2 Technical University of Denmark, Kgs. Lyngby, Denmark

Abstract
Faithful visualizations of data residing on manifolds must take the underlying geometry into account when producing a flat planar view of the data. In this paper, we extend the stochastic neighbor embedding (SNE) algorithm to data on general Riemannian
manifolds. We replace standard Gaussian assumptions with Riemannian diffusion counterparts and propose an efficient approximation that only requires access to calculations of Riemannian distances and volumes. We demonstrate that the approach also
allows for mapping data from one manifold to another, e.g. from a high-dimensional sphere to a low-dimensional one.

1. Introduction
Visualizations are crucial to investigators trying to make senseof high-dimensional data. The most common visualization is a
two-dimensional plot (e.g. on a piece of paper or a computer
screen), so we rely on dimensionality reduction when handlinghigh-dimensional data. Most dimensionality reduction techniques
assume that data resides in an Euclidean domain, which presents a
problem when data is not quite that simple. Data residing on Riemannian manifolds, such as the sphere, appear in many domains
where either known constraints or other modeling assumptions impose a Riemannian structure. How should we visualize such data?
There are many concerns and questions when visualizing Riemannian data. The first is generic: all dimensionality reduction
tools amplify parts of the signal while reducing the remainder. This
inherent limitation should always be in mind when interpreting thevisualization. Since some loss of information is inevitable, should
we then loosen our grip on the data or its underlying Riemannian
structure? Gauss’s Theorema Egregium [GP05] informs us that if
the final plot is to be presented on a flat screen or piece of paper,then a distortion of the Riemannian structure is inevitable.
In practice, even if we accept the limitations of a visualization, actual algorithms for Riemannian data are missing. In thispaper, we develop an extension of Stochastic Neighbor Embedding (SNE) [HR03] to Riemannian data and thereby provide one
such tool. We call this Riemannian Stochastic Neighbor Embedding, or Rie-SNE for short. Our approach can embed data observedon one Riemannian manifold on another manifold. This allows formapping data from a Riemannian space to a two-dimensional Euclidean plane (for plotting), but also mapping to a two-dimensional
sphere, or similar when the Euclidean topology is inappropriate.
Rie-SNE does not claim to solve the above-mentioned limitationsof visualization-through-dimensionality-reduction, but it does provide a working tool with practical merit.
© 2024 The Authors.
Proceedings published by Eurographics - The European Association for Computer Graphics.
This is an open access article under the terms of the Creative Commons Attribution License, which
permits use, distribution and reproduction in any medium, provided the original work is properly
cited.

Figure 1: Rie-SNE visualizes high-dimensional Riemannian databy mapping to a low-dimensional Riemannian (or Euclidean) manifold. The figure shows high-dimensional spherical data mapped toeither a low-dimensional sphere or a low-dimensional plane. The
method also supports other manifolds, both as input and output.

2. Background and related work
Euclidean visualization. General data visualization is a vast
topic, and a complete review is beyond our scope [Van16]. We here
focus on the setting where data is represented as vectors (points) inan Euclidean space of high dimension. When data is two- or threedimensional a scatter plot can directly reveal its structure, and we
focus on the more difficult setting where data dimensions vastly
exceed the easily plottable. Here one may explore the data throughmultiple projections, such as pairwise scatter plots, which are usually manageable for data of up to around 10 dimensions. Eventually, the approach tends to become unwieldy and the greater picture
is lost. Alternatives include continuously interpolating between dimensions to provide an (interactive) animation [Asi85].
The approach we here explore is to find a non-linear mappingfrom the high-dimensional observation space into a two- or three-2 of 5

A. Bergsson & S. Hauberg / Visualizing Riemannian data with Rie-SNE

dimensional space, which is suitable for plotting. Principal component analysis (PCA) [Jol02] is arguably the most commonly usedapproach to dimensionality reduction. PCA is restricted to spanning a linear subspace of the observation space, which often implies that the low-dimensional view reveals little structure. Several
non-linear variants exist (e.g. [Law05, TSL00, RS00, HR03]), butthey tend to be brittle [FHH21]. One of the currently most popular
tools is t-SNE [vdMH08], which we review in Sec. 2.1.
Riemannian data. Data is often equipped with additional
knowledge such as constraints or given smooth structures thatgive the data a Riemannian structure. For example, if all observations have unit norm, they reside on the unit sphere, which has awell-studied Riemannian manifold. The a priori available knowledge giving rise to a Riemannian data interpretation differs between domains, and we here only name a few. The most prominent example is that of directional statistics [MJM00, KGH13,
Hau18] where data resides on the unit sphere. Other examples
include shape data [Ken84, FB12, SJML05, You12, KKG∗ 12],DTI images [LDF04, PFA06, WSCV14], image features [TPM06,
PTM06, FHB14], motion models [TVSC11, cV09], human poses[SCLBS07, HSP12, HSP10, HLP11], robotics [GKJH15, JBS∗ 22],social networks [MLLM∗ 19] and more.
Even if Riemannian data is becoming increasingly common,
tools for visualization have not followed. The most common approach for visualizing Riemannian data is to locate a point on
the manifold, e.g. the intrinsic mean [Pen06], and map the data
to its tangent space. That gives an Euclidean view of the data,
which can be visualized with one of the many methods for this domain. Applying PCA tangentially is the gold standard [FLPJ04].
Unless data concentrates around the point of tangency this approach gives a highly distorted view of the data, and in practice,most knowledge reflected in the geometry is lost by the linearization [SLHN10]. Several Riemannian PCA generalizations exist,
e.g. [HHM10, PPY14, JDM12]. These generalize the classic linear model, and less work has been done on extending nonlinear
methods. Two notable extensions are a ‘wrapped’ extension of the
GP-LVM [MHF18], and Riemannian principal curves [Hau15].
Measuring on Riemannian manifolds. A Riemannian manifold is a space that is locally Euclidean. This implies that in aneighborhood around a point µ on the manifold, we can get a Euclidean view of the manifold in the form of a tangent space, whichis equipped with an inner product, ⟨xi , x j ⟩µ = x⊤i Gµ x j , where Gµ is
a symmetric positive definite matrix that reflects the inner productat µ . This is allowed to change smoothly between tangent spaces,
to compensate for the approximation error induced by the linear
view of the manifold. This distortion can be characterized by thechangepin volume between the manifold and its tangent, which follows det Gµ [Pen06].
The inner product allows us to define local distances, which
can be integrated to provide a notion of curve length. That is,given
a curve c on a manifold, we may compute its length as
Rp⟨ċt , ċt ⟩ct dt, where ct and ċt to denote the position and velocity of the curve, respectively. From the notion of a curve length,
it is trivial to define the distance between two points as the lengthof the shortest curve, i.e dist = minc Length[c]. The shortest curvecommonly goes under the name geodesic.

2.1. Stochastic neighbor embedding
Stochastic neighbor embedding (SNE) [HR03] preserves similarity between neighboring points when mapped to a low-dimensional
representation. Assume that we have access to functions shigh andslow , which measure the similarity between observation pairs in thehigh-dimensional observation space and the low-dimensional representation space, respectively. Now define the conditional probability, p j|i that xi would pick x j as its neighbor [vdMH08]
p j|i =

shigh (x j |xi ),∑k̸=i shigh (xk |xi )(1)with pi|i = 0. We can renormalize this to form a distribution over
all n observations
p j|i + pi| j.
(2)
pi j =
2n
To learn a low-dimensional representation, the key idea is to repeat
the above over the low-dimensional space to form
q j|i =

slow (y j |yi ),∑k̸=i slow (yk |yi )qi j =

q j|i + qi| j
2n

.

(3)

We can now compare the similarity of our data and the representation with the Kullback-Leibler divergence between pi j and qi j ,nn

KL (P||Q) = ∑ ∑ pi j log
i=1 j=1

pi j
.
qi j

(4)

This can then be minimized using gradient descent with respect
to the low-dimensional representation {yi }ni=1 . In its classic form,
SNE picks the measures of similarity as Gaussian functions
!

−D/2
∥x j − xi ∥22shigh (x j |xi ) = 2πσi
exp −
,
2σ2i
(5)
!
∥y j − yi ∥2−d/2
slow (y j |yi ) = (2π)
exp −
.
2
With this choice, the normalization constants of Eq. 5 cancel out
when computing p j|i and q j|i . Note that this approach gives a perobservation variance σ2i , such that different points effectively canhave different sizes of neighborhoods. To determine σ2i , the user
specifies a perplexity parameter, which can be thought of as a measure of the effective number of neighbors [vdMH08],
perplexity = 2H(Pi ) ,(6)

where H(Pi ) is the Shannon entropy [Sha48] of Pi . For a specificuser-provided perplexity, we can perform a binary search over σ2isuch that Eq. 6 holds. In practice, the user experiments with different choices of perplexities to see which reveals a pattern.
The t-distributed stochastic neighbor embedding (t-SNE) isthe most popular variant of SNE [vdMH08]. This is motivatedby the ‘crowding problem’, where the low-dimensional representations significantly overlap without revealing much underlying
structure. The idea is to use a similarity in representation space withheavy tails. Specifically, slow is a t-distribution with one degree of
freedom centered around one representation,−1slow (y j |yi ) = π−1 1 + ∥y j − yi ∥2.
(7)© 2024 The Authors.
Proceedings published by Eurographics - The European Association for Computer Graphics.

3 of 5

A. Bergsson & S. Hauberg / Visualizing Riemannian data with Rie-SNE

Figure 2: Two-dimensional
Euclidean embeddings ofspherical MNIST. Rie-SNE(left) discovers structure,
which is lost on tangent
space PCA (right).

3. Method
Brownian motion on a Riemannian manifold. The key building
block for generalizing SNE to Riemannian manifolds is a suitable
generalization of the Gaussian distribution. Here we consider a density derived from a Brownian motion on a Riemannian manifold forhigh-dimensional probability computations.
The end-point x of Brownian motion in Euclidean space follows
a Gaussian distribution. Infinitessimal increments of the Brownian
motion can be projected onto the tangent space of corresponding
points on a Riemannian manifold without error. Then, a Brownian
motion starting at point λ running for some time t can be projected
onto a D-dimensional Riemannian manifold. The resulting randomvariable can be interpreted as the probability that a Brownian motion starting at λ will end in point x on the manifold. This approximately has density [HS02, KEAH20]:!
D
dist2 (x, λ )λ,t) ≈ (2πt)− 2 H0 exp −BM(x|λ(8)2twhere t > 0 is the duration of the Brownian motion (variance in
Euclidean space), λ is the starting point of the Brownian motion,and H0 is the ratio of Riemannian volume measures evaluated atpoints x and λ respectively. Superficially, Eq. 8 looks like the density of the normal distribution, with the Euclidean distance being
replaced by its Riemannian counterpart. The normalization factor
H0 , however, differs from the Euclidean distribution.

3.1. Rie-SNE
Rie-SNE works similarly as SNE and t-SNE, i.e. it will also produce two probability distributions P and Q from the data and aim
to make them as similar as possible and in the process capturesome underlying structure in the produced low dimensional embedding. However, computing the high-dimensional probability distribution P comes with an added cost. To preserve the Riemannian
nature of the data, a different density is used when computing highdimensional probabilities belonging to P, namely the approximate
density induced by the heat kernel of a Brownian motion on a Riemannian manifold given in Eq. 8, i.e. we pickshigh (x j |xi ) = BMx j |xi ,ti .

(9)The added computational cost is that the evaluation of BM(·|·)
is more demanding than the conventional Gaussian similarity (5).
Specifically, the normalization H0 and the geodesic distance may be
demanding, depending on the manifold on which the data resides.
© 2024 The Authors.
Proceedings published by Eurographics - The European Association for Computer Graphics.

With the Brownian motion model, we get
dist2 [i, j]
H0 [i, j] · exp − 2ti,
p j|i ≈
dist2 [i,k]∑ H0 [i, k] · exp − 2ti

(10)

k̸=i

where we use the notations dist2 [i, j] = dist2 (xi , x j ) and H0 [i, j] =p
det Gx /det Gx to emphasize that these quantities can be precomputed. As with SNE, we can optimize ti to match a prespecified perplexity using a binary search.
ij

Choice of representation. As mentioned in Sec. 1, Gauss’s
Theorema Egregium [GP05] inform us that we cannot isometricallyembed data from a curved space into a space of different curvature without introducing distortion. Specifically, if we embed datafrom a nonlinear manifold onto a flat two-dimensional representation (for plotting) then the curvature mismatch between spaces
induces a distortion. This is a fundamental limitation that any visualization of Riemannian data will face, but we may nonethelesstry to limit its impact. One approach is to embed the data onto amanifold of similar curvature as that of the manifold on which thedata resides. For example, if the data resides on a high-dimensional
sphere, it is perhaps more prudent to embed onto a two-dimensional
sphere for plotting, rather than a Euclidean space.
With this in mind, we choose different distributions over the lowdimensional representation, depending on user preference.
• Euclidean. If the user prefers a Euclidean low-dimensional representation, we use a student-t as in regular t-SNE.
• Spherical. If the data manifold has positive curvature it may bebeneficial to embed it on a sphere, in which case we opt to use avon Mises-Fisher distribution [MJM00]. Spherical embeddings
have been explored elsewhere, e.g. [LCY19].
• Other.
When the user wants to embed on another lowdimensional manifold, we suggest using the Riemannian Brownian over the low-dimensional representation.
Once we have defined both shigh and slow , we can estimate therepresentations using gradient descent just as regular SNE. Having performed T iterations (with a sufficiently large value for T ) of
the gradient descent, the two probability distributions P and Q will
have a minimal KL-divergence resulting in near-optimal positionsof the points in the low-dimensional embedding.
Implementation details. Our implementation of Rie-SNE relies on two approximation techniques that are traditionally also
used when performing regular t-SNE. First, we compute the highdimensional probabilities in P using a sparse nearest neighborbased approximation technique [vdM14]. Second, we use the4 of 5

A. Bergsson & S. Hauberg / Visualizing Riemannian data with Rie-SNE

Figure 3: Four different
views of embeddings ofspherical MNIST. The
top row shows sphericalembeddings obtained using Rie-SNE, while thebottom row shows threedimensional
embeddings
using the gold standard
tangent space PCA. Notethat the clustering structure
is significantly more evident
in Rie-SNE.

Barnes-Hut approximation [BH86], whenever we use a student’s tdistribution over the low-dimensional representation. This reduces
the complexity to O(n log n).
4. Results
The performance of Rie-SNE is shown by comparing it to the gold
standard of visualizing non-Euclidean data. This amounts to firstcomputing the intrinsic mean, mapping all data to the tangent spaceat this point, and performing PCA over the tangential data.
Spherical MNIST. We start with the classic MNIST dataset
consisting of 24 × 24 dimensional gray-scale images. We consider
digits 0–5 to reduce clutter. To induce a non-Euclidean data geometry, we project the data onto the unit sphere of R24×24 and denote
the resulting data spherical MNIST. We visualize the resulting datausing both Rie-SNE and tangent space PCA. First, we embed thedata onto the plane, R2 , and show the resulting plots in Fig. 2. Hereit can be seen that Rie-SNE captures well the underlying relation-

ship between the data points (the same digits are grouped), while
tangent space PCA produces a cluttered view that does not reveal
the underlying structure. Since the data has a spherical geometry, it
may be beneficial to embed onto a low-dimensional sphere to preserve topology and curvature better. Figure 3 shows the Rie-SNE
embedding onto S 2 , where the clustering is again evident. As abaseline, the figure also shows a tangent space PCA embedding on
R3 . Although some structure can be captured with tangent space
PCA here, Rie-SNE still gives better separation of the digit classes.
Crypto-tensors. Following Mallasto et al. [MHF18] we consider the price of 10 popular crypto-currencies over the period
2.12.2014 — 15.5.2018. As is common in economy [WG10], therelationship between prices is captured by a 10 × 10 covariancematrix constructed from the past 20 days. This gives rise to a timeseries of covariance matrices, each of which resides on the coneof symmetric positive definite matrices. We provide visualizationsof the data in Fig. 4. Rie-SNE is used to produce visualizations on
both the plane R2 and the sphere S 2 , showing a one-dimensional
structure capturing the time evolution behind the data. In contrast,
tangent space PCA produces R2 and R3 visualizations showing little to no structure in the embeddings.

5. Conclusions
We presented a new type of visualization technique, Rie-SNE, thatis aimed at data residing on Riemannian manifolds, such as spheres.
It is an SNE-based technique that can additionally produce different kinds of low-dimensional embeddings depending on user preference and the curvature of the original data manifold. We achievefavorable results using Rie-SNE compared to the gold standard tangential PCA. We hope that the present work also paves the way forother visualization tools for Riemannian data in order to support
investigators relying on geometric models.
Figure 4: Embeddings of symmetric positive definite matrices using Rie-SNE (left) and tangent space PCA (right). The top row
shows two-dimensional Euclidean embeddings, while the bottom
row shows spherical and R3 embeddings, respectively. In bothcases, Rie-SNE recovers a one-dimensional signal matching the underlying time series, while tangent space PCA does not.

Acknowledgements.
SH was supported by research grants(15334, 42062) from VILLUM FONDEN. This project has also
received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement no 757360). This work was funded inpart by the Novo Nordisk Foundation through the Center for Basic
Machine Learning Research in Life Science (NNF20OC0062606).
© 2024 The Authors.
Proceedings published by Eurographics - The European Association for Computer Graphics.

A. Bergsson & S. Hauberg / Visualizing Riemannian data with Rie-SNE

References[Asi85] A SIMOV D.: The grand tour: a tool for viewing multidimensional
data. SIAM journal on scientific and statistical computing 6, 1 (1985). 1[BH86] BARNES J. E., H UT P.: A hierarchical O(n-log-n) force calculation algorithm. Nature 324 (1986), 446. 4[cV09] Ç ETINGUL H., V IDAL R.: Intrinsic mean shift for clustering on
stiefel and grassmann manifolds. In IEEE CVPR (2009). 2[FB12] F REIFELD O., B LACK M.: Lie bodies: A manifold representationof 3D human shape. In ECCV (2012). 2[FHB14] F REIFELD O., H AUBERG S., B LACK M. J.: Model transport:
Towards scalable transfer learning on manifolds. In CVPR (2014). 2[FHH21] F ELDAGER C. W., H AUBERG S., H ANSEN L. K.: Spontaneous symmetry breaking in data visualization. In ICANN (2021). 2[FLPJ04] F LETCHER P. T., L U C., P IZER S. M., J OSHI S.: Principal
Geodesic Analysis for the study of Nonlinear Statistics of Shape. IEEE
Transactions on Medical Imaging (TMI) 23, 8 (2004), 995–1005. 2[GKJH15] G ILITSCHENSKI I., K URZ G., J ULIER S. J., H ANEBECKU. D.: Unscented orientation estimation based on the bingham distribution. IEEE Transactions on Automatic Control (2015). 2

5 of 5[LCY19] L U Y., C ORANDER J., YANG Z.: Doubly stochastic neighborembedding on spheres. Pattern Recognition Letters (2019). 3[LDF04] L ENGLET C., D ERICHE R., FAUGERAS O.: Inferring whitematter geometry from diffusion tensor MRI: Application to connectivity
mapping. In ECCV (2004). 2[MHF18] M ALLASTO A., H AUBERG S., F ERAGEN A.: Probabilistic
riemannian submanifold learning with wrapped gaussian process latentvariable models. In AISTATS (2018). 2, 4
[MJM00] M ARDIA K. V., J UPP P. E., M ARDIA K.: Directional statistics, vol. 2. Wiley Online Library, 2000. 2, 3
[MLLM∗ 19] M ATHIEU E., L E L AN C., M ADDISON C. J., T OMIOKA
R., T EH Y. W.: Continuous hierarchical representations with poincaré
variational auto-encoders. NeurIPS 32 (2019). 2[Pen06] P ENNEC X.: Intrinsic Statistics on Riemannian Manifolds: Basic
Tools for Geometric Measurements. JMIV (2006). 2[PFA06] P ENNEC X., F ILLARD P., AYACHE N.: A Riemannian framework for tensor computing. IJCV (2006). 2[PPY14] PANARETOS V. M., P HAM T., YAO Z.: Principal flows. Journalof the American Statistical Association (JASA) (2014). 2[GP05] G AUSS K. F., P ESIC P.: General investigations of curved surfaces. Courier Corporation, 2005. 1, 3

[PTM06] P ORIKLI F., T UZEL O., M EER P.: Covariance tracking using
model update based on Lie algebra. In CVPR (2006). 2[Hau15] H AUBERG S.: Principal curves on riemannian manifolds. IEEE
Transactions on Pattern Analysis and Machine Intelligence (2015). 2[RS00] ROWEIS S. T., S AUL L. K.: Nonlinear dimensionality reduction
by locally linear embedding. science 290, 5500 (2000), 2323–2326. 2[Hau18] H AUBERG S.: Directional statistics with the spherical normal
distribution. In Proceedings of FUSION 2018 (2018). 2[SCLBS07] S AID S., C OURTY N., L E B IHAN N., S ANGWINE S. J.: Exact principal geodesic analysis for data on SO(3). In The 15th European
Signal Processing Conference (2007). 2[HHM10] H UCKEMANN S., H OTZ T., M UNK A.: Intrinsic shape analysis: geodesic PCA for Riemannian manifolds modulo isometric Liegroup actions. Statistica Sinica 20, 1 (2010), 1–58. 2[Sha48] S HANNON C. E.: A mathematical theory of communication.
Bell Syst. Tech. J. 27, 3 (1948), 379–423. 2[HLP11] H AUBERG S., L AUZE F., P EDERSEN K. S.: Unscented Kalman
Filtering on Riemannian Manifolds. JMIV (2011). 2[SJML05] S RIVASTAVA A., J OSHI S. H., M IO W., L IU X.: Statisticalshape analysis: Clustering, learning, and testing. TPAMI (2005). 2[HR03] H INTON G. E., ROWEIS S.: Stochastic neighbor embedding. In
NeurIPS (2003), vol. 15, MIT Press. 1, 2

[SLHN10] S OMMER S., L AUZE F., H AUBERG S., N IELSEN M.: Manifold Valued Statistics, Exact Principal Geodesic Analysis and the Effectof Linear Approximations. In ECCV (2010). 2[HS02] H SU E., S OCIETY A. M.: Stochastic Analysis on Manifolds.
American Mathematical Society, 2002. 3[HSP10] H AUBERG S., S OMMER S., P EDERSEN K. S.: Gaussian-like
Spatial Priors for Articulated Tracking. In ECCV (2010), Springer. 2[HSP12] H AUBERG S., S OMMER S., P EDERSEN K. S.: Natural metrics
and least-committed priors for articulated tracking. Image and Vision
Computing 30, 6-7 (2012), 453–461. 2[JBS∗ 22] JAQUIER N., B OROVITSKIY V., S MOLENSKY A., T ERENIN
A., A SFOUR T., ROZO L.: Geometry-aware bayesian optimization in
robotics using riemannian matérn kernels. In CoRL (2022). 2[JDM12] J UNG S., D RYDEN I. L., M ARRON J.: Analysis of principal
nested spheres. Biometrika 99, 3 (2012), 551–568. 2[Jol02] J OLLIFFE I. T.: Principal component analysis for special typesof data. Springer, 2002. 2[KEAH20] K ALATZIS D., E KLUND D., A RVANITIDIS G., H AUBERG
S.: Variational autoencoders with riemannian brownian motion priors,2020. arXiv:2002.05227. 3[Ken84] K ENDALL D.: Shape manifolds, Procrustean metrics, and complex projective spaces. Bulletin of the London Mathematical Society 16,
2 (1984), 81–121. 2[KGH13] K URZ G., G ILITSCHENSKI I., H ANEBECK U. D.: Recursive
nonlinear filtering for angular data based on circular distributions. In
2013 American Control Conference (2013), IEEE. 2[KKG∗ 12] K URTEK S., K LASSEN E., G ORE J. C., D ING Z., S RIVAS TAVA A.: Elastic geodesic paths in shape space of parameterized surfaces. IEEE TPAMI (2012). 2[Law05] L AWRENCE N.: Probabilistic non-linear principal component
analysis with gaussian process latent variable models. The Journal of
Machine Learning Research 6 (2005), 1783–1816. 2© 2024 The Authors.
Proceedings published by Eurographics - The European Association for Computer Graphics.

[TPM06] T UZEL O., P ORIKLI F., M EER P.: Region covariance: A fast
descriptor for detection and classification. In ECCV (2006), Springer. 2[TSL00] T ENENBAUM J. B., S ILVA V. D ., L ANGFORD J. C.: A global
geometric framework for nonlinear dimensionality reduction. science
290, 5500 (2000), 2319–2323. 2[TVSC11] T URAGA P. K., V EERARAGHAVAN A., S RIVASTAVA A.,
C HELLAPPA R.: Statistical computations on Grassmann and Stiefel
manifolds for image and video-based recognition. TPAMI (2011). 2[Van16] VANDER P LAS J.: Python data science handbook: Essential
tools for working with data. " O’Reilly Media, Inc.", 2016. 1[vdM14] VAN DER M AATEN L.: Accelerating t-sne using tree-based algorithms. J. Mach. Learn. Res. 15, 1 (Jan. 2014), 3221–3245. 3[vdMH08] VAN DER M AATEN L., H INTON G.: Visualizing data using
t-SNE. Journal of Machine Learning Research 9 (2008). 2[WG10] W ILSON A. G., G HAHRAMANI Z.: Generalised wishart processes. arXiv preprint arXiv:1101.0240 (2010). 4[WSCV14] WANG Y., S ALEHIAN H., C HENG G., V EMURI B. C.:
Tracking on the product manifold of shape and orientation for tractography from diffusion MRI. In IEEE Conference on Computer Vision and
Pattern Recognition (CVPR) (2014), pp. 3051–3056. 2[You12] YOUNES L.: Spaces and manifolds of shapes in computer vision:
An overview. Image and Vision Computing 30, 6 (2012), 389–397. 2