Published in Transactions on Machine Learning Research (10/2023)

Identifying latent distances with Finslerian geometry
Alison Pouplin

alpu@dtu.dk

Technical University of Denmark

David Eklund

david.eklund@ri.se

Research Institutes of Sweden

Carl Henrik Ek

che29@cam.ac.uk

University of Cambridge

Søren Hauberg

sohau@dtu.dk

Technical University of Denmark
Reviewed on OpenReview: https: // openreview. net/ forum? id= Q2Gi0TUAdS

Abstract
Riemannian geometry provides us with powerful tools to explore the latent space of generative models while preserving the underlying structure of the data. The latent space canbe equipped it with a Riemannian metric, pulled back from the data manifold. With thismetric, we can systematically navigate the space relying on geodesics defined as the shortest
curves between two points.
Generative models are often stochastic, causing the data space, the Riemannian metric,
and the geodesics, to be stochastic as well. Stochastic objects are at best impractical, andat worst impossible, to manipulate. A common solution is to approximate the stochastic
pullback metric by its expectation. But the geodesics derived from this expected Riemannian
metric do not correspond to the expected length-minimising curves.
In this work, we propose another metric whose geodesics explicitly minimise the expected
length of the pullback metric. We show this metric defines a Finsler metric, and we compare
it with the expected Riemannian metric.
 In high dimensions, we prove that both metrics
1
converge to each other at a rate of O D. This convergence implies that the established expected Riemannian metric is an accurate approximation of the theoretically more grounded
Finsler metric. This provides justification for using the expected Riemannian metric forpractical implementations.

1

Introduction

Generative models provide a convenient way to learn low-dimensional latent variables z corresponding todata observations x through a smooth function f : Z ⊂ Rq → X ⊂ RD , such that x = f (z). Throughthis learnt manifold, one can generate new data or compare observations by interpolating or computing
distances. However, doing so by using the Euclidean distance in the latent space is misleading (Hauberg,
2018a), because the latent variables are not statistically identifiable. If our observations are lying near amanifold (Fefferman et al., 2016), we want to equip our latent space with a metric that preserves distancemeasures on it. Figure 1 (left panel) illustrates the need for defining geometric-aware distances on manifolds.
Distances on a manifold can be precisely defined using a norm, which is a mathematical function that exhibits
several desirable properties such as non-negativity, homogeneity, and the triangle inequality. In particular,a norm can be induced by an inner product (i.e., a quadratic function) that associates each pair of points
on the manifold with a scalar value.
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Published in Transactions on Machine Learning Research (10/2023)

To derive the standard Riemannian interpretation of the latent space, we first compute the infinitesimal
2
Euclidean norm according to the data space. Using the Taylor expansion, we have: f (z + ∆z) − f (z) 2 ≈2f (z) + J(z)∆z − f (z) 2 = ∆z > J(z)> J(z)∆z. As a first approximation, the norm defined in the latent
space locally preserves the Euclidean norm defined in the data space. The curvature of our data manifold
is condensed in the Riemannian metric tensor Gz = J > (z)J(z), which serves as a proxy to define the
Riemannian metric: gz : (u, v) → u> Gz v. In mathematical jargon, we say that the Riemannian manifold(Z, g) is obtained by pulling back the Euclidean metric through the map f .
Riemannian geometry enables the exploration of the latent space in precise geometric terms, and quantities
of interest such as the length, the energy or the volume can be directly derived from the pullback metric.
These geometric quantities are, by construction, known to be invariant to reparametrizations of the latent
space Z, and are thus statistically identifiable (Hauberg, 2018a). While this geometric framework exclusively
handles deterministic objects, generative models are often stochastic. The learnt map f , that mathematically describes those models, is stochastic too. For example, in the celebrated Gaussian Process Latent
Variable Model (GPLVM) (Lawrence, 2003), this function is a Gaussian process. The pullback metric isthen stochastic, and standard differential geometric constructions no longer apply. Navigating the latent
manifolds through geodesics (length-minimising curves) becomes practically infeasible.
Previous research tried to circumvent this problem by approximating the stochastic pullback metric withthe expected value of the Riemannian metric tensor, but the derived length is an unintuitive quantities that
do not correspond to the expected length:
L(γ) |E[G] :=Z
γ(t) E[G] dt

6=

E L(γ) |G :=ZE γ(t) G dt.

Instead of taking the expectation of the metric tensor, which serves as a surrogate to define a norm, we
propose to take the expectation of the norm directly.
In this paper, we compare our expected norm with the norm induced by the the commonly used expected
metric tensor. The main findings are:
1.
2.
3.

The expected norm defines a Finsler metric. Finsler geometry is a generalisation of Riemannian
geometry.
For Gaussian processes, the stochastic norm obtained through the pullback metric follows a noncentral Nakagami distribution, so our Finsler metric has a closed-form expression.
In high dimensions, for Gaussian processes, our Finsler
metric and the previously studied Riemannian1metric converge to each other at a rate of O D, with D the dimension of the data space.

We conclude that the Riemannian metric serves as a good approximation of the Finsler metric, which istheoretically more grounded. It further justifies the use of the Riemannian metric in practice when exploring
stochastic manifolds.
1.1

Outline of the paper

The paper explores the geometry of latent spaces learned by generative models, which encode a latent
low-dimensional manifold that represents observed high-dimensional data. The latent manifold is denoted
Z ⊂ Rq and the data manifold is denoted X ⊂ RD .
Assuming the manifold hypothesis holds, we need to define an infinitesimal norm in the latent manifold
to compute distances that respect the underlying geometry of the data. Such a norm can be constructedby pulling back the Euclidean distance through the smooth function that maps the latent manifold to thedata manifold. This map, f : Z → X , mathematically describes the decoder of a trained generative model.
Those models being often stochastic, we consider f being a stochastic
process. It means that the pullback√
metric tensor, G = J > J, and its induced norm, k·kG : u → u> Gu, are also stochastic. In section 2, wemathematically define the notion of stochastic pullback metric and stochastic manifolds.
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Published in Transactions on Machine Learning Research (10/2023)

Figure 1: In this illustration, we show the 2-dimensional representation (θ, ϕ) of the sphere parametrised in R3 withf : (θ, ϕ) → cos(θ) sin(ϕ), sin(θ) cos(ϕ), sin(ϕ). In practice, we don’t have access to well-parametrised manifolds,
but instead those manifolds are being shaped by data points in RD . The data points are depicted as tiny dots forillustrative purposes. On the left figure, the map f is deterministic, and on the right figure, we imagine that f isstochastic. When we pullback, through f , a circle from the sphere to the plane, the circle is deformed to an ellipse.
Those ellipses, called indicatrices, are the fingerprints of the metric and they showcase the deformation of the plane.
Left figure: In blue, the Euclidean distance, is represented by the straight line in the plane. It is not identifiable andit does not represent a geodesic on the sphere. In orange, the length-minimising curve obtained through the pullback
metric of the mapping f leads to a great circle on the sphere. Following the great circle is the fastest way to go fromone point to another. Effectively, the pullback metric leads to a geodesic, contrary to the Euclidean metric. Notice
that, in the plane, the orange geodesic also follows the direction of the indicatrices, since it is length-minimising.
Right figure: We imagine that a generative model maps latent space to data space to the data space using astochastic process f = {f1 , f2 , . . . }. This leads to a stochastic pullback metric, stochastic indicatrices, and stochastic
paths in the latent space.

To circumvent all the challenges posed by this stochastic component, a deterministic approximation of thenorm is needed. It can be defined by taking the expectation of the metric tensor. This norm, that we willnote k·kR : u → kukE[G] , has been studied before by Tosi et al. (2014), and is explained in section 2.2. In
this paper, we propose
to directly take the expectation of the stochastic norm. This expected norm, instead
noted k·kF : u → E kukG , is introduced in section 2.4. The norm k·kR is defined by a Riemannian metric,
and we show that the norm k·kF defines a Finsler metric. We explain the general difference between Finsler
and Riemannian geometry in section 3.
The aim of this paper is to compare those two norms. We first draw absolute bounds in section 3.2, andthen relative bounds in 3.3. We also investigate the relative difference of the norms when the dimension of
the data space increases, in section 3.4. Finally, we perform some experiments in Section 4, that illustrate
the Riemannian and Finsler norm in the same latent space.

1.2

Related works

Riemannian geometry for machine learning
When navigating a learnt latent space, the geodesics obtained through the pullback metric not only follow
geometrically coherent paths, they also remain invariant under different representations. While two runs of
the same model will produce distinct latent manifolds, the geodesics connecting the same chosen points shouldhave the same length. We say that the pullback metric effectively solved the identifiability problem (Hauberg,
2018a). This has led to the growing adoption of Riemannian geometry in machine learning applications. In
robotics, Beik-Mohammadi et al. (2021) used the pullback metric from a variational autoencoder to safely
navigate the space of motion patterns, and Scannell et al. (2021) use geodesics under the expected metric in
a GPLVM to control quadrotor robot. In proteins modelling, Detlefsen et al. (2022) showed that the derived
geodesics on the space of beta-lactamase follow their phylogenic tree structure. In game content generation,
González-Duque et al. (2022) generated game levels more coherent and reliable by interpolating data along
the geodesics. Jørgensen & Hauberg (2021) used the expected Riemannian metric from observations ofpairwised distances through a GPLVM to build a probabilistic dimensionality reduction model.
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Published in Transactions on Machine Learning Research (10/2023)

Finsler geometry in machine learning
Our work crucially relies on Finslerian geometry, which has been well-studied mathematically, but has only
seen very limited use in machine learning and statistics. We point to two notable exceptions, which are quitedistinct from our work. Lopez et al. (2021) use symmetric spaces to represent graphs and endow these witha Finsler metric to capture dissimilarity structure in the observational data. Ratliff et al. (2021) discuss therole of differential geometry in motion planning for robotics. Along the way, they touch upon Finsleriangeometry, but mostly as a neat tool to allow for generalizations. To the best of our knowledge, no prior work
has investigated the links between stochastic and Finslerian geometry.
Strategies to deal with stochastic Riemannian geometry
Tosi et al. (2014) and Arvanitidis et al. (2018) introduced approximation of the pullback metric by taking the
expectation of the metric tensor. In those two cases, the map f is respectively a trained Gaussian process, orthe decoder of a VAE. In this paper, the derivations only hold if f is a smooth stochastic process (Definition
2.2), which is not the case1 of the VAEs, and hence, our results are not directly applicable to those models.
In addition to the work of Tosi et al. (2014), a solution to circumvent the randomness of the metric tensor is
to consider that the data follows a specific probability distribution. Instead of looking at the shortest pathon the data manifold, Arvanitidis et al. (2021) borrow tools from information geometry and consider thestraightest paths on the manifold whose elements are probability distributions.

2

Expectation on random manifolds

The metric pulled back by a stochastic mapping is, de facto, stochastic and endows a random manifold.
Unfortunately, we are not yet equipped to derive geometric objects on a random manifold. Instead, we
dodge this problem by seeking a deterministic approximation of this stochastic metric. As mentioned above,
a common solution is to approximate such a metric by its expectation. In section 2.2, we study the expected
Riemannian metric.
The solution suggested by this paper is to approximate the expectation of the lengths instead of the random
metric itself. In section 2.4, we show that this new metric is not Riemannian but Finslerian (Proposition
2.2), and it has a closed-form expression when the map f is a Gaussian process (Proposition 2.3).
2.1

Random Riemannian geometry

The pullback metric is defined as a Riemannian metric if and only if the mapping f is an immersion,
which is a differentiable function whose derivatives are injective everywhere on the manifold (Lee, 2013,
Proposition 13.9). A manifold equipped with a Riemannian metric is called a Riemannian manifold.
Definition 2.1. The pullback of the Euclidean metric through the immersion f : Z → X is a Riemannian metric. It is defined as the inner product gz : (Tz Z, Tz Z) → R+ : (u, v) → u> Gv, at a specificpoint z in the manifold Z. u and v are vectors lying in the tangent plane Tz Z (ie: the set of all tangent
vectors) of the manifold. G = J > J, with J the Jacobian of f .
Since a Riemannian metric is an inner product, it induces a norm: k·kG . We can then define the curve
R1
R1
2
length and curve energy on a manifold: LG (γ) = 0 γ̇(t) G dt and EG (γ) = 12 0 γ̇(t) G dt, with γ acurve defined on Z, and γ̇ its derivative. A locally length-minimising curve between two connecting points
is a geodesic. To obtain a geodesic, we can minimise the curve length, but in practice minimising the
1 the decoder of a VAE, while it decodes to a Gaussian, cannot be considered as a differentiable stochastic process. One
Qn
reason is because the independence of the probability of the data: p(x|z) =p(xi |zi ). Let us assume the opposite: thei=1
decoder is a Gaussian process. The covariance of the Gaussian process would be a diagonal matrix because of the independenceof the probability of the data. The covariance would correspond to a dirac distribution: cov(xi , xj ) = δij . However, a stochastic
process is differentiable only if the covariance is differentiable, which is not the case of the dirac distribution.

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Published in Transactions on Machine Learning Research (10/2023)

curve energy is more efficient. OnR the manifold, Rwe also define a volume measure
in order to integrate√
probability functions: for U ⊂ Z, f (U ) h(x) dx = U h(f (z))VR dz, with VR (z) = Gz the volume measure.
In addition, we are considering the case where the immersion f is a stochastic process. The outputs of ourtrained model, x ∈ X , which represent our data, are random variables.
Definition 2.2. A stochastic process is a collection of random variables {X(t, ω), t ∈ T } indexedby an index set T defined on a sample space Ω, which represents the set of all possible outcomes. An
outcome in Ω is denoted by ω, and a realisation of the stochastic process is the sequence of X(·, ω) that
depends on the outcome ω.
In this framework, our index set is our latent manifold T = Z, and our sample space Ω is defined as the setof the model evaluations. For every point z ∈ Z, every time we execute our model, the output x = f (z)is a random variable following a specific distribution. When the data x follow a Gaussian distribution, thestochastic process is called a Gaussian process. A GP-LVM (Lawrence, 2003) is a model that learns how
to map the data from a latent space to a data space through a Gaussian process.
When f is a stochastic immersion, the metric tensor becomes a random matrix. In this paper, we call amanifold equipped with the stochastic pullback metric a random manifold, noted (Z, g). As a consequenceof the stochastic aspect of the metric, all the functionals are stochastic themselves, and they are no longer
trivial to manipulate.
Definition 2.3. A random Riemannian metric tensor is a matrix-valued random field (ie: acollection of matrix-valued random variables {G(z, ω), z ∈ Z}), whose realisation for a specific evaluationω ∈ Ω is a Riemannian metric tensor. A random Riemannian metric is a metric induced by arandom Riemannian metric tensor: gz : (Tz Z, Tz Z) → R+ : (u, v) → u> Gv. For the rest of the paper,
the associated stochastic norm is noted:√pk·kG : Tz Z → R+ : u → gz (u, u) := u> Gu
If this stochastic norm is induced by f defined as a Gaussian process, then k·kG follows a non-central
Nakagami distribution. This is explained in the proof of Proposition 2.3.
2.2

Norm induced by the expected metric tensor

One way to approximate a random metric tensor is to take its expectation with respect to the collection of
random metrics induced by the stochastic process. This has been introduced before by Tosi et al. (2014)
GP-LVMs.
Definition 2.4. Let G be a stochastic Riemannian metric tensor on the manifold Z. We refer to E[G]as the expected metric tensor. It induces a Riemannian metric and a norm on Z. We will note thenorm induced by the expected metric tensor as:qk·kR : Tz Z → R+ : u → kukE[G] := u> E[G]uLike any Riemannian metric, we can define the following functionals: LR (γ) =p
R1ER (γ) = 0 γ̇(t)> E[G]γ̇(t) dt = E[ER (γ)], and VR (z) = det E[G].
2.3

R1p
0

γ̇(t)> E[G]γ̇(t) dt,

Expected paths on random manifolds

Approximating the stochastic metric by its expectation seems a natural but also ad-hoc solution. If wewant to explore a manifold, we might prefer to use a representative quantity, such as the lengths between
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Published in Transactions on Machine Learning Research (10/2023)

data points. The expectation of the lengths can give us an idea about how, on average, two points areconnected on a random manifold. The expected curve length,
and its corresponding
curve energy oni
R 1 hp>the random manifold (Z, g) are defined as: LF (γ) = 0 E
γ̇(t) Gγ̇(t) dt = E[LG (γ)], and EF (γ) =i2
R 1 hpEγ̇(t)> Gγ̇(t) dt.
0
One observation made by Eklund & Hauberg (2019) is that the length (LR ) derived from the expected
Riemannian metric is not equal to the expected curve length (LF ), and their respective energy curves differ
by a variance term:
ER (γ) − EF (γ) =Z 1γ̇(t)> E[G]γ̇(t) − Eqγ̇(t)> Gγ̇(t)

2
dt

0

=

Z 1

h
E

γ̇(t)

0

2
G

i

−E

h

γ̇(t) G

i2

dt =

Z 1
0

Var

h

i
γ̇(t) G dt

This term can be regarded as a regularisation term for the Riemannian energy curve: the curve energy
ER might be penalised when the curve goes through regions with high-variance. In practice, for a Gaussian
process with a stationary kernel, this variance term is upper bounded by the posterior variance that is
relatively low next to the training points and is high outside of the support of the data. Later, we will also
see that the functionals agree in high dimensions, leading to the same geodesics (Section 3).
Eklund & Hauberg (2019) also noted that these quantities are bounded by the number of dimensions:
Proposition 2.1. (Eklund & Hauberg, 2019) Let f : Rq → RD be a stochastic process such that the0sequence: {f10 , f20 , . . . , fD
} has uniformly bounded moments. There is then a constant C such that:0≤

2.4

LR − LF
C≤
LR
8D

Expected norm and Finsler geometry

Our work builds on Eklund & Hauberg (2019)’s research. We are interested in approximating the stochastic
norm instead of the metric tensor, and by doing so, the derived curve length and curve energy are the sameones studied by Eklund & Hauberg (2019). We go further as we not only compare curve lengths, but thedeterministic norms obtained with the stochastic metric.
Definition 2.5. Let G be a stochastic Riemannian metric tensor on the manifold Z. It induces astochastic norm, k·kG on Z. We will note the expected norm as:h√ik·kF : Tz Z → R+ : u → E[kukG ] := E u> Gu
While it cannot be induced by an inner-product, it is sufficiently convex to be defined as a Finsler metric.
Definition 2.6. Let F : T Z → R+ be a continuous non-negative function defined on the tangent
bundle T Z of a differentiable manifold Z.
We say that F is a Finsler metric if, for each point z of Z and v on Tz Z, we have (1) Positive
homogeneity: ∀λ ∈ R+ , F (λv) = λF (v). (2) Smoothness: F is a C ∞ function on the slit tangent∂2F 2
bundle T Z \ {0}. (3) Strong convexity criterion: the Hessian matrix gij (v) = 12 ∂v
i v j (v) is positive
definite for non-zero v.
A differentiable manifold equipped with a Finsler metric is called a Finsler manifold. Finsler geometry can beseen as an extension of Riemannian geometry, since the requirements for defining a metric are less restrictive.
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Proposition 2.2. Let G be a stochastic Riemannian metric tensor. Then, the function Fz : Tz Z → R :u → kukF defines a Finsler metric, but it is not induced by a Riemannian metric.
Proof.
If F was induced by apRiemannian metric, then this metric would be defined as:
fz : Rq × Rq → R+ : (v1 , v2 ) → E[ v1> Gv2 ]2 . Since a Riemannian metric is an inner product, it
should be symmetric, positive, definite and bilinear. Here, we can see that√ fx is not bilinear, so fz is not a
Riemannian metric. However, we can prove that Fz : Rq → R : v → E[ v > Gv] is positive, homogeneous,smooth and strongly convex, and so Fz is a Finsler metric (Shen & Shen, 2016, Definition 2.1). For the full
proof, see Section B.1.
So far, we have assumed that f is an immersion and a stochastic process. If we consider f to be a GaussianProcess in particular, the Finsler norm can be rewritten in a closed form expression.
Proposition 2.3. Let f be a Gaussian process and J its Jacobian, with J ∼ N (E[J], Σ). The Finsler
norm can be written as:


1
√ √
Γ( D
1 D ω
2 + 2)>
Fz : Tz Z → R+ : kvkF := v → 2 v Σv,−,
1 F1 − ,
2 2
2
Γ( D
2)
with 1 F1 as the confluent hypergeometric function of the first kind and ω = (v > Σv)−1 (v > E[J]> E[J]v).
Proof. We suppose that f is a Gaussian process, and so is its Jacobian. G follows a non-central Wishart
distribution: G = J > J ∼ Wq (D, Σ, Σ−1 E[J]> E[J]). v > Gv is a scalar and also follows a non-central Wishart
distribution: v > Gv ∼ W1 (D, σ, ω), with σ = v > Σv and ω = (v > Σv)−1 (v > E[J]> E[J]v) (Kent & Muirhead, 1984, Definition 10.3.1). The square-root of a non-central Wishart distribution follows a non-central
Nakagami distribution (Hauberg, 2018b). Then, by construction, the stochastic norm k·kG follows a noncentral Nakagami distribution. The expectation of this distribution is known, and it has a closed-form
expression.
The confluent hypergeometric function of the first kind, also known as the Kummer function, is a special
function that is defined as the solution of a specific second-order linear differential equation. The term ωappears from the non-central Wishart distribution. When ω is non-zero, the distribution of the Jacobian
shifts away from the origin, and ω represents the magnitude and the direction of this shift, balanced by thecorrelation between the variables. In Section 3.4, to prove our results in high-dimensions, we will assume
that our manifold Z is bounded, and so is ω.

3Comparison of Riemannian and Finsler metrics

3.1

Theoretical comparison

In geometry, we need to define a norm to compute functionals, and the Riemannian metric is conveniently
obtained by constructing an inner product. Because of its bilinearity, it greatly simplifies subsequent computations, but it is also restrictive. Relaxing this assumption and defining a metric as a more general2 norm
has been studied by Finsler (1918), who gave his name to this discipline.
Finsler geometry is similar to Riemannian geometry without the bilinear assumption, most of the functionals(curve length and curve energy) are defined similarly to those obtained in Riemannian geometry. However,the volume measure is different, and there are at least two definitions of volume measure used in Finsler
geometry: the Busemann-Hausdorff volume and the Holmes-Thomson volume measure (Wu, 2011). In this
2 The norm actually needs to be strongly convex, but it is not necessary symmetric. This means that, for a vector v, we
can have a non reversible Finsler metric: Fx (v) 6= Fx (−v). Intuitively, this means that the path used to connect two points
would be different depending on the starting point. This asymmetric property becomes valuable when studying the geometryof anisotropic media (Markvorsen, 2016), for example. In our case, our Finsler metric is reversible.

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paper, we decided to focus on the Busemann-Hausdorff definition (Definition 3.1), which is more intuitiveand easier to derive. If the Finsler metric is a Riemannian metric, the definition of volume naturally
coincides with the Riemannian volume measure.
Definition 3.1. For a given point z on the manifold, we define the Finsler indicatrix as the setof vectors in the tangent space such that the Finsler metric is equal to: {v ∈ Tz Z|Fz (v) = 1}). We
call Bn (1) the Euclidean unit ball, and vol(·) the standard Euclidean volume. In local coordinates(e1 , · · · , ed ) on a Finsler manifold M, the Busemann-Hausdorff volume form is defined as dVF =VF (z)e1 ∧ · · · ∧ ed , with:
vol(Bn (1))
VF (z) =.
vol({v ∈ Tz Z|Fz (v) < 1})
In the definition above, we introduce the notion of indicatrix. An indicatrix is a way to represent the
distortion induced by the metric on a unit circle. If our metric is Euclidean, we will only have a linear
transformation between the latent and the observational spaces, and the indicatrix would still be a circle.
Because the Riemannian metric is quadratic, it will always generate an ellipse in the latent space. The
Finsler indicatrix, however, would have a convex, even asymmetrical, shape. This difference can be observedin the indicatrix-field represented in Figure 2: The Finsler indicatrices in purple can have almost rectangularshape, while the Riemannian indicatrices, in orange, are ellipses.

A

B

Figure 2: Indicatrice field over the latent space of the pinwheel data (in grey) representing the Riemannian (in orange)
and Finslerian (in purple) metrics (See Section C). (A) The indicatrices are computed over a grid in the latent space.
(B) The indicatrices are computed along a geodesic: the Riemannian and Finslerian metrics coincide.

There are also a few observations to note in Figure 2. First, in the area of low predictive variance (wheredata points lie in the latent space), the Finsler and Riemannian indicatrices are alike. This follows fromthe preceding comment that the metrics diverge by a variance term. If our mapping f was deterministic,
both metrics would agree. Second, for every point, the Riemannian indicatrices are always contained by theFinslerian ones, illustrating Proposition 3.1 on our absolute bounds in the following section.
3.2

Absolute bounds on the Finsler metric

The Finsler norm is upper bounded with the Riemannian norm obtained from the expected metric tensor.
It is also lower bounded:8

Published in Transactions on Machine Learning Research (10/2023)

Proposition 3.1. We define α = 2



1
Γ( D
2 +2)
Γ( D
)
2

2

. The Finsler norm: k·kF is bounded by two norms,k·kαΣ and k·kR , induced by the two respective Riemannian metric tensors: the covariance tensor αΣz
and the expected metric tensor E[Gz ].
∀(z, v) ∈ Z × Tz Z : kvkαΣ ≤ kvkF ≤ kvkR
Proof. The full proof is detailed in Section√ B.2.1, andp it can be summarised the following way. The upperbound kvkF ≤ kvkR , also rewritten √as: E[ v > Gv] ≤ v > E[G]v, is obtained by applying Jensen’s inequality,
knowingthat the√√ square root x → x is a concave function. The lower bound kvkαΣ ≤ kvkF , rewritten asv > αΣv ≤ E[ v > Gv], is obtained using the closed form expression of the Finsler function.
The result is illustrated in Figure 4 (lower right). Four metric tensors (G1 , G2 , G3 , G4 ), each following a
non-central Wishart distribution with a specific mean and covariance matrix, have been computed. For eachof them, we have drawn the indicatrices ({v ∈ Tz Z | kvk = 1}) induced by the norms: k·kF , k·kR andk·kαΣ . As expected, we can notice that the αΣ-indicatrix contains the Finsler indicatrix, itself containing
R-indicatrix.
By bounding the Finsler metric, we are able to bound their respective functionals:
Corollary 3.1. The length, the energy and the Busemann-Hausdorff volume of the Finsler metric arebounded respectively by the Riemannian length, energy and volume of the covariance tensor αΣ (noted
LαΣ , EαΣ , VαΣ ) and the expected metric E[G] (noted LR , ER , VR ):∀z ∈ Z, LαΣ (z) ≤ LF (z) ≤ LR (z)EαΣ (z) ≤ EF (z) ≤ ER (z)VαΣ (z) ≤ VF (z) ≤ VR (z)Proof. The full proof is detailed in Section B.2.1. From Proposition 3.1, we need to integrate each term of
the inequality to obtain the length and the energy. The volume is less trivial, since we use the BusemannHausdorff definition for measuring VF . We recast the problem in hyperspherical coordinates, and show that
the Finsler indicatrix is still bounded.
3.3 Relative bounds on the Finsler metric
Proposition 3.2. Let f be a stochastic immersion. f induces the stochastic norm k·kG , defined in
Section 2. The relative difference between the Finsler norm k·kF and the Riemmanian norm k·kR is:
h
i
2
Var
kvk
G
kvkR − kvkF
0≤
≤
h
i2 .
kvkR
2
2E kvkG

Proof. This proposition is a direct application of the Sharpened Jensen’s inequality (Liao & Berg, 2019).
The previous proposition is valid for any stochastic immersion. We can see that the metrics become equal
when the ratio of the variance over the expectation shrinks to zero. This happens in two cases: when thevariance converges to zero, which is similar to having a deterministic immersion, and when the number ofdimensions increases. The latter case is investigated below for a Gaussian process 3 .
3 Interestingly, the term E[ν]2 /Var[ν], when ν follows a central Nakagami distribution, is called a shape parameter. It has
been pintroduced by Nakagami himself to study the intensity of fading in radio wave propagation (Nakagami, 1960). Whenν :=
ξ with ξ ∼ W1 (Σ, D), then m = D/2. This is a particular result obtained from Proposition 3.3, when E[J] = 0.

9

Published in Transactions on Machine Learning Research (10/2023)

Proposition 3.3. Let f be a Gaussian process. We note ω = (v > Σv)−1 (v > E[J]> E[J]v), with J thejacobian of f , and Σ the covariance matrix of J.
The relative ratio between the Finsler norm k·kF and the Riemmanian norm k·kR is:0≤

kvkR − kvkF
1
ω
≤
+
.
kvkR
D + ω (D + ω)2

Proof.
v > Gv follows a one-dimension non-central Wishart distribution: v > Gz v ∼ W1 (D, σ, ω), with>
σ = v Σv and ω = (v > Σv)−1 (v > E[J]> E[J]v). We use the theorem of the moments to obtain both the
expectation and the variance, which leads us to the result.
As we have seen that the metrics are bounded, it is easy to show that the functionals derived from thosemetrics are also bounded:
Corollary 3.2. When f is a Gaussian Process, the relative ratio between the length, the energy andthe volume of the Finsler norm (noted LF , EF , VF ) and the Riemannian norm (noted LR , ER , VR ) is:
LR (z) − LF (z)
1
ω
0≤
≤ max
+
v∈Tz ZLR (z)
D + ω (D + ω)2()
ER (z) − EF (z)2
1 + 2ω
2ω
ω2
0≤
≤ max
+
+
+
v∈Tz ZER (z)
D + ω (D + ω)2(D + ω)3(D + ω)4
!q
VR (z) − VF (z)1ω0≤≤ 1 − 1 − max+v∈Tz ZVR (z)
D + ω (D + ω)2
Proof. We directly use Proposition 3.3. To obtain the inequalities with the lengths and the energies, wefirst multiply all the terms by the Riemannian metric, and we integrate every term. To obtain the inequality
with the volume, similarly to Corollary 3.1, we place ourselves in hyperspherical coordinates and bound theradius of the Finsler indicatrix. The full proof is in Section B.2.2.
A

0.6

B

C

0.6

0.4

0.4

0.2

0.2

0.0

0.0

0.0

−0.5

D

−0.5
−1.0
−1.5

−1.0

−2.0
−1.5

−0.2

−2.5

−0.2
−2.0

−0.4

−0.6

−3.0

−0.4
−2.5

−3.5

−3.0

−4.0

−0.6

Figure 3: Difference of volume for data embedded in the latent space. (A) Riemannian volume measure, (B) Finslerian(Busemann-Hausdorff) volume measure, (C) Variance of the Gaussian process, (D) Ratio between the Riemannianand Finslerian volume: (VR (z) − VF (z))/VR (z). All heatmaps are computed in logarithm scale.

In Figure 3, we can compare the volume measures obtained from the Riemannian and Finsler metrics, andin particular, their ratio in the top right image. When the metrics are computed next to the data points
in area where the variance is very low, we can see that the ratio of the volume measure is at the orderof magnitude 10−4 . Further away from the data points, the variance increases and so does the difference
between the Riemannian and Finsler volume measures.
3.4

Results in high dimensions

Proposition 3.3 and Corrollary 3.2 indicate that the metrics become similar when the dimension (D) of theobservational space increases. Ifwe assume that the latent space is a bounded manifold, the metrics converge
1
to each other at a rate of O D, as do their functionals.
10Published in Transactions on Machine Learning Research (10/2023)

We assume that the latent manifold is bounded. Then, we can deduce that (1) the term ω, which represents
the non-centrality of the data, does not grow faster than the number of dimensions (See lemma B.2.2, in
Section B.2.3) and (2) we that the metrics are finite.
Corollary 3.3. Let f be a Gaussian Process. In high dimensions, we have:LR (z) − LF (z)=O
LR (z)



1
D


,

ER (z) − EF (z)=O
ER (z)



1
D


,

and

VR (z) − VF (z)=O
VR (z)



q
D


.

When D converges toward infinity: LR ∼ LF , ER ∼ EF and VR ∼ VF .
+∞

+∞

+∞

Proof. This result follows from Corollary 3.2, assuming the latent manifold is bounded. The full proof canbe found in Setion B.2.3.
Corollary 3.4. Let f be a Gaussian Process. In high dimensions, the relative ratio between the Finsler
norm k·kF and the Riemmanian norm k·kR is: 
kvkR − kvkF
1
=O
kvkR
D
And, when D converges toward infinity: ∀v ∈ Tz Z, kvkR ∼ kvkF .
+∞Proof. Similarly, from Proposition 3.3, in a bounded manifold, both metrics converge to each other in highdimensions.

dim: 3

dim: 5

dim: 10

dim: 50

Lower bound metric tensor indicatrix
Finslerian indicatrix
Riemannian indicatrix

Figure 4: Left: Ratio of volumes (VR − VF )/VR decreasing with respect to the number of dimensions. The results
were obtained from using a collection of matrices {Gi } following a non-central Wishart distribution. Upper right:
The Finsler and Riemannian indicatrices converge towards each other when increasing the number of dimensions.
Lower right: Illustration of the absolute bounds in Proposition 3.1 with the αΣ-indicatrices, Riemannian indicatrices
and Finsler indicatrices.

4

Experiments

We want to illustrate cases where these metrics differ in practice. For this, we use two synthetic datasets,consisting of pinwheel and concentric circles mapped to a sphere, and four real-world datasets: a font11

Published in Transactions on Machine Learning Research (10/2023)

dataset (Campbell & Kautz, 2014), a dataset representing single-cells (Guo et al., 2010), MNIST (LeCun,
1998) and fashionMNIST (Xiao et al., 2017). We trained a GPLVM with and without using stochastic active
sets (Moreno-Muñoz et al., 2022) to learn a latent manifold. From the learnt model, we can access the
Riemannian and Finsler metrics, and minimise their respective curve energies to obtain the corresponding
geodesics. All the code has been built using Stochman (Detlefsen et al., 2021), a python library to efficiently
compute geodesics on manifolds.

4.1

Experiments with synthetic data showing high variance

The synthetic data correspond to simple patterns – a pinwheel and concentric circles – that have been
projected onto a sphere. In those examples, we are plotting curves that does not follow the data points
and so, go through regions of high variance. Notably, the background of the latent manifold represents thevariance of the posterior distribution in logarithmic scale. For both cases, we have the 3-dimensional data
space and the corresponding learnt latent space. The curves are mapped from the latent space to the data
space using the forward pass of the GPLVM.

Figure 5: The Riemannian (purple), Finslerian (orange) and Euclidean (dotted gray) geodesics obtained by pullingback the metric through the Gaussian processes of a trained GPLVM. The models, trained on the 3-dimensional
synthetic data (Figures A.2, B.2) learnt their latent representations (Figures A.1, B.1)

We can see that the Riemannian geodesics tend to avoid area of high variance in both cases: they are
attracted to the data, at the detriment of following longer paths in R3 . The Finslerian geodesics, on the
opposite, are not perturbed by the variance term, and will explore regions without any data, following shorter
paths in R3 . The geodesics have been plotted by computing a discretized manifold, obtained from a 10x10
grid with Detlefsen et al. (2021). This approach proved to be more effective than minimizing the energy
along a spline, which was prone to getting trapped in local minima. The GPLVM has been coded in Pyro
(Bingham et al., 2019). All the implementation details can be found in the Appendix C.

4.2

Experiments with a font dataset and qPCR dataset

The font dataset (Campbell & Kautz, 2014) represents the contour of letters in various font, and the qPCR
dataset (Guo et al., 2010) represents single-cells stages. We trained a GPLVM model, also using Pyro
(Bingham et al., 2019), to learn the latent space. From the optimised Gaussian process, we can access the
Riemannian and Finsler metric, and minimise their respective curve energies to obtain geodesics.
Similar to the previous experiment, the background colour represents the variance of the posterior distribution
in logarithmic scale. We can notice that the Riemannian and the Finsler geodesics agrees with each other.
This experiment also agrees with our finding that in high dimensions (the dimensions of the single-cells
data is 48, the font data is 256), both metrics converge to each other. This concludes that, in practice, theRiemannian metric is a good approximation of the Finsler metric, and Finsler geometry doesn’t need to be
used in high dimensions.
12Published in Transactions on Machine Learning Research (10/2023)

Figure 6: The Riemannian (plain line) and Finslerian (dotted line) geodesics obtained by pulling back the metricthrough the GPLVM. The models trained on single-cells data (which cannot be represented) and font data (FigureB.2) learnt their respective 2-dimensional latent representation (Figures A, B.1).

4.3

Experiments with MNIST and FashionMNISTGPLVMs are often hard to train, but they can be scaled effectively using variational inference and inducing
points. Using stochastic active sets has been shown to give reliable uncertainty estimates, and the model also
learn more meaningful latent representations, as shown in the original paper by Moreno-Muñoz et al. (2022).
For those experiments, we used those models on two well-known benchmarks: MNIST and FashionMNIST.
In both experiments, because the difference of the variance of the posterior learnt by the GPLVM acrossthe latent manifold is low, as we can notice in the background of the plots on the right, and because thedata is high dimensional, we cannot see any difference between the Finslerian and the Riemannian geodesics.
Again, in practice, the Riemannian metric is a good approximation of the Finslerian metric.

5

Discussion

Generative models are often used to reduce data dimension in order to better understand the mechanisms
behind the data generating process. We consider the general setting where the mapping from latent variables to observations is driven by a smooth stochastic process, and the sample mappings span Riemannian
manifolds. The Riemannian geometry machinery has already been used in the past to explore the latent
space.
In this paper, we have shown how curves and volumes can be identified by defining the length of a latent
curve as its expected length measured in the observation space. This is a natural extension of classicaldifferential geometric constructions to the stochastic realm. Surprisingly, we have shown that this does notgive rise to a Riemannian metric over the latent space, even if sample mappings do. Rather, the latent
representation naturally becomes equipped with a Finsler metric, implying that stochastic manifolds, suchas those spanned by Latent Variable Models (LVMs), are inherently more complex than their deterministic
counterparts.
The Finslerian view of the latent representation gives us a suitable general solution to explore a random
manifold, but it does not immediately translate into a practical computational tool. As Riemannian manifolds are better understood computationally than Finsler manifolds, we have raised the question: How good
an approximation of the Finsler metric can be achieved by a Riemannian metric? The answer turns out tobe: quite good. We have shown that as data dimension increases, the Finsler metric becomes increasingly
Riemannian. Since LVMs are most commonly applied to high-dimensional data (as this is where dimensionality reduction carries value), we have justification for approximating the Finsler metric with a Riemannian
metric such that computational tools become more easily available. In practice we find that geodesics under
the Finsler and the Riemannian metric are near identical, except in regions of high uncertainty.
13

Published in Transactions on Machine Learning Research (10/2023)

Figure 7: The Riemannian (purple) and the Finslerian (orange) geodesics are plotted. Upper figure: learnt latent
space with the images (left) and data points (right) of Fashion MNIST. Lower figure: learnt latent space with theimages (left) and data points (right) of MNIST.

Acknowledgments
The authors would like to thank Professor Steen Markovsen for the initial discussions on Finsler geometry,and Dr. Cilie Feldager for her invaluable help in taming GP-LVMs.
This work was funded in part by the Novo Nordisk Foundation through the Center for Basic Machine Learning
Research in Life Science (NNF20OC0062606). It also received funding from the European Research Council(ERC) under the European Union Horizon 2020 research, innovation programme (757360). SH was supported
in part by research grants (15334, 42062) from VILLUM FONDEN.
Code availability
The code used for the experiments in this study is available on GitHub at https://github.com/a-pouplin/latent_distances_finsler.
14

Published in Transactions on Machine Learning Research (10/2023)

Notations
Z, X
f
J
G
Tz Z
Σ
k·kG
k·kR
k·kF
LR , ER , VR
LF , EF , VF

Smooth differentiable latent (Z) and data (X ) manifold,
A stochastic immersion f : Z ⊂ Rq → X ⊂ RD ,Jacobian of the stochastic function f ,Stochastic metric tensor defined as the pullback metric through f : G = J > J,
Tangent space of the manifold Z at a point z,
QD
IF f is a Gaussian process, then J √∼ i=1 N (µi , Σ),
Stochastic induced norm: kvkG := pv > Gv,p>
Riemannian induced norm:√ kvkR := v E[G]v := g(v, v),
Finsler norm: kvkF := E[ v > Gv] = F (v),Length, energy and volume obatined from the Riemannian induced norm k·kR ,Length, energy and Busemann Hausdorff volume obtained from the Finsler norm k·kF .

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A

A primer on Geometry

The main purpose of the paper is to define and compare two legitimate metrics to compute the average length
between random points. Before going further, it’s important to formally define the two metrics (Riemannianand Finsler metrics, respectively) which we do in sections A.2 and A.3. They are both constructed on
topological manifolds, the definition of which is recalled in section A.1. We finally introduce the notion of
random manifold in section A.4, which is the last notion needed to frame our problem of interest: which
metric should we use to compute the average distance on a random manifold?
A.1

Topological and differentiable manifolds

This section aims to define core concepts in differential geometry that will be used later to define Riemannianand Finsler manifolds. Recall that two topological spaces are called homeomorphic if there is a continuous
bijection between them with continuous inverse.
Definition A.1. A d-dimensional topological manifold M is a second-countable Hausdorff topological space such that every point has an open neighbourhood homeomorphic to an open subset of
Rd .
Let M be a topological manifold. This means that for any x ∈ M there is an open neighbourhood Ux ofx and a homeomorphism φUx : Ux → Rd onto an open subset of Rd . Suppose that x, y ∈ M are such thatUx ∩ Uy 6= ∅, let U = Ux , V = Uy and consider the so-called coordinate change mapd:φV ◦ φ−1
U |φU (U ∩V ) φU (U ∩ V ) → R .

We call M together with an open cover {Ux }x∈M as above a differentiable or smooth manifold if thecoordinate maps are infinitely differentiable.
Beyond these technical definitions, one can imagine a differentiable manifold as a well-behaved smoothsurface that possesses locally all the topological properties of a Euclidean space. All the manifolds in thispaper are assumed to be differentiable and connected manifolds.
Definition A.2. We also define, for a differentiable manifold M, the tangent space Tx M as the setof all the tangent vectors at x ∈ M, and the tangent bundle T M the disjoint union of all the tangent
spaces: T M = ∪ Tx M .
x∈M

So far, we have only defined topological and differential properties of manifolds. In order to compute
geometric quantities, we need to equip those with a metric that helps us derive useful quantities such aslengths, energies and volumes. A metric is a scalar valued function that is defined for each point on the
topological manifold and takes as inputs one or two vectors (depending on the type of metric) from thetangent space at the specific point. Such a function can either be defined as a scalar product between two
vectors, this is the case of a Riemannian metric or, in the case of a Finsler metric, it is defined similarly tothe norm of a vector. We will formally define these metrics and highlight their differences in the following
sections.
A.2 Riemannian manifolds
Definition A.3. Let M be a manifold. A Riemannian metric is a map assigning at each pointx ∈ M a scalar product G(·, ·) : Tx M × Tx M → R, with G a positive definite bilinear map, whichis smooth with respect to x. A smooth manifold equipped with a Riemannian metric is called a
Riemannian manifold. We usually express the metric as a symmetric positive definite matrix G,>
where we have for two vectors
p u, v ∈ Tx M : G(u, v) = hu, viG = u Gv. We further define the induced
norm: v ∈ Tx M , kvkG = G(v, v).
The Riemannian metric here can either refer to the scalar product G itself, or the associated metric tensor
G.
18

Published in Transactions on Machine Learning Research (10/2023)

Tx M
gx (u, v) = 1

df (v)f

df (u)vuM
Figure 8: f is an immersion that maps a low dimensional manifold to a high dimensional manifold M. On M, atangent plane Tx M is draw at x. The indicatrice of the Euclidean metric is plotted in blue. When this metric ispulled-back through f , the low dimensional space is now equipped with the pullback metric g, which is a Riemannian
metric by definition. The vectors df (u) and df (v) are called the push-forwards of the vectors u and v through f .

Definition A.4. We consider a curve γ(t) and its derivative γ̇(t) on a Riemannian manifold M equippedwith the metric g. Then, we define the length of the curve:Z
Z p
LG (γ) =γ̇(t)) G dt =gt (γ̇(t), γ̇(t)) dt,
where gt = gγ(t) . Locally length-minimising curves between two connecting points are called Geodesics.
Definition A.5. The curve energy is defined as:
ZZ
2
EG (γ) =γ̇(t)) G dt = gt (γ̇(t), γ̇(t)) dt.

There are two interesting properties to note about the length of a curve and the curve energy. First, thelength is parametrisation invariant: for any bijective smooth function η on the domain of γ we have that
LG (γ ◦ η) = LG (γ). We also say the Riemannian metric gives us intrinsic coordinates to compute the length.
Secondly, for a given curve γ, we have: LG (γ)2 ≤ 2EG (γ). Because of the invariance of the curve, when we
aim to minimise it, a solver can find an infinite number of solutions. On the other hand, the curve energy isconvex and will lead to a unique solution. Thus, to obtain a geodesic, instead of solving the corresponding
ODE equations, or directly minimising lengths, it is easier in practice to minimise the curve energy, as aminimal energy gives a minimal length.
The Riemannian metric also provides us with an infinitesimal volume element that relates our metric G toan orthonormal basis, the same way the Jacobian determinant accommodate for a change of coordinates in
the change of variables theorem.
Definition A.6. In local coordinates (e1 , · · · , ed ), the volume form of the Riemannian manifold M,
equipped with the metric tensor G, is defined as: dVG = VG (x)e1 ∧ · · · ∧ ed , with:p
VG (x) = det(G).
Remark. The symbol ∧ represents the wedge product and it is used to manipulate differential k-forms. Here,the basis vectors (e1 , · · · , ed ) form a d-dimensional parallelepiped (e1 ∧ · · · ∧ ed ) with unit volume.

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Published in Transactions on Machine Learning Research (10/2023)

VG (x)
f
f (γ)

γ

M

Figure 9: Once the low dimensional manifold is equipped with a metric that captures the inherent structure of thehigh dimensional manifold, we can compute a geodesic γ, by minimising the energy functional between two points.
The geodesic f (γ) will be the shortest path between two points on the manifold M. The volume measure VG canalso be used to integrate functions over regions of the manifold, as we would do in the Euclidean space. It can also
be linked to the density of the data: if the data points are uniformly distributed over the high-dimensional manifold,in the low-dimensional manifold, a low volume would correspond to a high density of data. It is a useful way to give
more information about the distribution of the data.

A.3

Finsler manifolds

Finsler geometry is often described as an extension of Riemannian geometry, since the metric is defined ina more general way, lifting the quadratic constraint. In particular, the norm of a Riemmanian metric is a
Finsler metric, but the converse is not true.
Definition A.7. Let F : T M → R+ be a continuous non-negative function defined on the tangent
bundle T M of a differentiable manifold M . We say that F is a Finsler metric if, for each point x of
M and v on Tx M , we have:
1. Positive homogeneity: ∀λ ∈ R+ , F (λv) = λF (v).
2. Smoothness: F is a C ∞ function on the slit tangent bundle T M \ {0}.
2

2

∂ F
3. Strong convexity criterion: the Hessian matrix gij (v) = 12 ∂v
i v j (v) is positive definite for nonzero v.

A differentiable manifold M equipped with a Finsler metric is called a Finsler manifold.
Here, it is worth noting that, for a given point in the manifold, the Finsler metric is defined with only one
vector in the tangent space, while the Riemannian metric is defined with two vectors. Moreover, from theprevious definition, we can deduce that the metric is:
1. Positive definite: for all x ∈ M and v ∈ Tx M , F (v) ≥ 0 and F (v) = 0 if and only if v = 0.
2. Subadditive: F (v + w) ≤ F (v) + F (w) for all x ∈ M and v, w ∈ Tx M .
We say that F is a Minkowski norm on each tangent space Tx M . Furthermore, if F satisfies the reversibility
property: F (v) = F (−v), it defines a norm on Tx M in the usual sense.
Similarly to Riemannian geometry, lengths, energies and volumes can be defined directly from the Finsler
metric:Definition A.8. We consider a curve γ and its derivative γ̇ on a Finsler manifold M equipped withthe metric F . We define the length of the curve as follows:Z
LF (γ) = F (γ̇(t))dt.

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Tx M

Fx (u) = 1
fuMFigure 10: f is an immersion that maps a low dimensional manifold to a high dimensional manifold M. On M, atangent plane Tx M is draw at x. Compared to the Riemannian manifold, the Finsler indicatrix, which represents
the all the vectors u ∈ Tx M such that Fx (u) = 1, is not necessarily an ellipse. It can be asymmetric if the metric is
asymmetric itself. It is always convex.

Definition A.9. The curve energy is defined as: EF (γ) =RF (γ̇(t))2 dt.

Not only are the definitions strikingly similar, they also share the same properties. The curve length isalso invariant under reparametrisation, and upper bounded by the curve energy. Computing geodesics on
a manifold is reduced to a variational optimisation problem. These propositions are proved in detail in
Lemmas B.1.4 and B.1.5, in the appendix.
In Riemannian geometry, the volume measure defined by the metric is unique. In Finsler geometry, different
definitions of the volume exist, and they all coincide with the Riemannian volume element when the metricis Riemannian. The most common choices of volume forms are the Busemann-Hausdorff measure and theHolmes-Thompson measure. According Wu (2011), depending on the Finsler metric and the topologicalmanifold, some choices seem more legitimate than others. In this paper, we decided to only focus on the
Busemann-Hausdorff volume, as its definition is the most commonly used and leads to easier derivations.
We will later show that in high dimensions, our Finsler metric converges to a Riemannian metric, and thus,
the results obtained for the Busemann-Hausdorff volume measure are also valid for the Holmes-Thomson
volume measure.
Definition A.10. For a given point x on the manifold, we define the Finsler indicatrix as the set ofvectors in the tangent space such that the Finsler metric is equal to one: {v ∈ Tx M |F (v) = 1}). We
denote the Euclidean unit ball in Rd by Bd (1) and for measurable subsets S ⊆ Rd we use vol(S) to
denote the standard Eulcidean volume of S. In local coordinates (e1 , · · · , ed ) on a Finsler manifold M,
the Busemann-Hausdorff volume form is defined as dVF = VF (x)e1 ∧ · · · ∧ ed , with:VF (x) =vol(Bd (1)).
vol({v ∈ Tx M |F (v) < 1})

We can interpret the volume as the ratio between the euclidean ball, and a convex ball whose radius isdefined as a unit Finsler metric. If the Finsler metric is replaced by a Riemannian metric, the volume of
the indicatrix will be an ellipsoid whose semi-axis are equal to the inverse of the squareroot of the metric’s
eigenvalues. The Finsler volume then reduces to the definition of the Riemannian volume.
A.4

Random manifolds

So far, we have only considered deterministic data points lying on a manifold. If we consider our data to berandom variables, we will need to define the associated random metric and manifold.

21

Published in Transactions on Machine Learning Research (10/2023){ft } = {f1 , f2 , f3 , · · · }
G1
G2
G3
M
Figure 11: Usually the immersion f would be deterministic. In the case of most generative models, where f is
described by a GP-LVM, or the decoder of a VAE, the immersion is stochastic. The pullback metric is stochastic defacto.

As said previously, if we have a function f : Rq → RD that parametrises a manifold, then we can construct
a Riemannian metric G = Jf> Jf , with Jf the Jacobian of the function f . In the previous cases, we assumed
f to be a deterministic function, and so is the metric. We construct a stochastic Riemannian metric in thesame way, with f being a stochastic process. A stochastic process f : Rq → RD is a random map in thesense that samples of the process are maps from Rq to RD (the so-called sample paths of the process).
Definition A.11. A stochastic process f : Rq → RD is smooth if the sample paths of f are smooth.
We call a smooth process f a stochastic immersion if the Jacobian matrix of its sample paths has fullrank everywhere. We can then define the stochastic Riemannian metric G = Jf> Jf .

The terms stochastic and random are used interchangeably. The definition of the stochastic immersion isfairly important, as it means that its Jacobian is full rank. Since the Jacobian is full rank, the random
metric G is positive definite, a necessary condition to define a Riemannian metric. Another definition of astochastic Riemannian metric would be the following:
Definition A.12. A stochastic Riemannian metric on Rq is a matrix-valued random field on Rq
whose sample paths are Riemannian metrics. A stochastic manifold is a differentiable manifold
equipped with a stochastic Riemannian metric.

Any matrice drawn from this stochastic metric would be a proper Riemannian metric. When using therandom Riemannian metric on two vectors u, v ∈ Tx M , G(u, v) = u> Gv is a random variable, but both u, v
are deterministic vectors. From this definition, it follows that the length, the energy and the also volume
are random variables.
ObjectRiemann

Finsler

metricg : Tx M × T x M → R

F : T M → R+

length structure

LG (γ) =Rp
gt (γ̇(t), γ̇(t)) dt

LF (γ) =RF (γ̇(t)) dt

energy structure

EG (γ) =R

gt (γ̇(t), γ̇(t)) dt

EF (γ) =RF (γ̇(t))2 dt

volume element

VG (x) =p| det{G}|VF (x) = vol(Bn (1))/vol({v ∈ Tx M |F (x, v) < 1})
Busemann-Hausdorff volume measure

Table 1: Comparison of Riemannian and Finsler metrics.

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Published in Transactions on Machine Learning Research (10/2023)

B

ProofsOne of the main challenges of this paper is to find coherent notations while respecting the tradition of twogeometric fields. In Riemannian geometry, we principally use a metric, noted gp : Tp M × Tp M → R+ ,that is defined as an inner product and thus can induce a norm, but is not a norm. In Finsler geometry,
we call intercheangeably Finsler function, Finsler metric or Finsler norm, the norm traditionally noted
F : T M → R+ , with Fp (u) := F (p, u) defined at a point p ∈ M for a vector u ∈ Tp M. We will assume that
all our metric are always defined for a specific point z (or p) on our manifold Z (or M), and so we will justdrop this index. The following notations will be used:
Stochastic pullback metric tensor
Stochastic pullback metric
Expected Riemannian metric
Stochastic pullback induced norm
Expected Riemannian induced norm
Finsler metric
B.1

G = Jf> Jfg̃ : (u, v) → u> Gu
g : (u, v) →√u> E[G]v
k·kG : u → pu> Gupk·kR : u → √u> E[G]u := g(u, u)
k·kF : u → E[ u> Gu] := F (u)Finslerian geometry of the expected length

In this section, we will always let f : Rq → RD be a stochastic immersion, Jf its Jacobian, and G = Jf> Jfh√ia metric tensor. We will first prove that the function F : T M → R : v → E v > Gv is a Finsler metric.
Then, for the specific case where Jf follows a non-central normal distribution, the Finsler metric F defined as the expected length follows a non-central Nakagami distribution and can be expressed in closed form.
To prove that the function F is indeed a Finsler metric, we will need to verify the criteria above, among themthe strong convexity criterion is less trivial to prove than the others. It will be detailed in Lemma B.1.3.
∂2F 2
Strong convexity means that the Hessian matrix 12 Hess(F (v)2 ) = 12 ∂v
i v j (v) is strictly positive definite for
non-negative v. This matrix, when F is a Finsler function, is also called the fundamental form and plays animportant role in Finsler geometry. To prove the strong convexity criterion, we will need the full expressionof the fundamental form, detailed in Lemma B.1.1.
h√iLemma B.1.1. The Hessian matrix 12 Hess(F (v)2 ) of the function F (v) = E v > Gv is given by
h
i h
i
h
i2
1
1
3
1
1
Hess(F (v)2 ) = E (v > Gv) 2 E (v > Gv)− 2 G − (v > Gv)− 2 Gvv > G + E (v > Gv)− 2 G vv > .
2√
Proof.
Let G be a random positive definite symmetric matrix and define g : Rq → R : v 7→ v > Gv,
where v is considered a column vector. We would like to know the different derivatives of g with respect
to v. We name by default Jg and Hg , its Jacobian and Hessian matrix. Using the chain rule, we have:1
1
3
Jg = (v > Gv)− 2 v > G and Hg = (v > Gv)− 2 G − (v > Gv)− 2 (Gvv > G).
For the rest of the proof, we need to show that derivatives and expectation values commute.
Using the Fubini theorem, we can show that tha derivatives and the expectation values commute.
√
For F : Rq → R : v 7→ E[ v > Gv],
hi
1
3
Hess(F ) = E[Hg ] = E (v > Gv)− 2 G − (v > Gv)− 2 Gvv > G
1∇F = E[Jg ] = E[(v > Gv)− 2 Gv].
√
We now consider the function h : Rq → R : v 7→ E[ v > Gv]2 = F (v)2 . Using the chain rule and changing
the order of expectation and derivatives, we have its Hessian
Hh = 2F · Hess[F ] + 2∇F > ∇F = 2E[g]E[Hg ] + 2E[Jg ]> E[Jg ].
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Finally, replacing Jg and Hg previously obtained in this expression, we conclude:
h
i h
i
h
i2
1
1
3
1
1
Hh (x, v) = E (v > Gv) 2 E (v > Gv)− 2 G − (v > Gv)− 2 Gvv > G + E (v > Gv)− 2 G vv > .
2
Remark. Before going further, it’s important to note that G = Jf> Jf is a random matrix that is positive
definite: it is symmetric by definition and has full rank. The later statement is justified by the assumptionthat the stochastic process f : Rq → RD is an immersion, then Jf is full rank.
h√iLemma B.1.2. The function F (v) = E v > Gv is:
1. positive homogeneous: ∀λ ∈ R+ , F (λv) = λF (v)2. smooth: F (v) is a C ∞ function on the slit tangent bundle T M \ {0}Proof. 1) Let λ ∈ R, then we have: F (λv) = E

h√

i
h√
i
λ2 v > Gv = |λ| v > Gv .

2) The multivariate function: Rq \{0} → R∗+ : v → v > Gv is C ∞ and strictly positive, since G = Jf> Jf is√positive definite. The function R∗+ → R∗+ : x → x is also C ∞ . Finally, R∗+ → R∗+ : x → E[x] is by definition∞differentiable. By composition, F (v) is a C function on the slit tangent bundle T M \ {0}.
h√iLemma B.1.3. The function F (v) = E v > Gv satisfies the strong convexity criterion.
Proof. Proving that F satisfies the strong convexity criterion is equivalent to show that the Hessian matrix
H = 12 Hess(F (v)2 ) is strictly positive definite. Thus, we need to prove that ∀w ∈ Rq \{0}, w> Hw > 0.
According to Lemma B.1.1, because the expectation is a positive function, it’s straightforward to see that∀w ∈ Rq \{0}, w> Hw ≥ 0. The tricky part of this proof is to show that w> Hw > 0. This can be obtained if
one of the terms (F · Hess(F ) or ∇F > ∇F ) is strictly positive.
First, let’s decompose H as the sum of matrices: H = F Hess(F ) + ∇F > ∇F (Lemma B.1.1), with:
h
i 


1
>
>
− 32
>
>
2
F · Hess(F ) = E (v Gv) E (v Gv)(v Gv)G − Gv(Gv),hi2
1∇F > ∇F = E (v > Gv)− 2 G vv > .
We will study two cases: when w ∈ span(v), and when w ∈/ span(v). We will always assume that v 6= 0, andso by definition: F (v) > 0.
Let w ∈ span(v). We will show that w> ∇F > ∇F w > 0. We have w = αv, α ∈ R. Because F is 1homogeneous and using Euler theorem, we have: ∇F (v)v = F (v). Then (αv)> ∇F > ∇F (αv) = α2 F 2 , and
α2 F (v)2 > 0.
Let w ∈
/ span(v). F being a scalar function, we have: w> F Hess[F ]w = F w> Hess[F ]w. We would liketo show that: w> Hess[F ]w > 0. The strategy is the following: if we prove that the kernel of Hess[F ]is equal to the span(v), then w ∈/ span(v) is equivalent to say that w ∈/ ker(Hess[F ]) and we can conclude that: w> Hess[F ]w > 0. Let’s prove span(v) ∈ ker(Hess(F )). We know that Hess(F )v = 0,
since F is 1-homogeneous, so we have span(v) ∈ ker(Hess(F )). To obtain the equality, we just needto prove that the dimension of the kernel is equal to 1. Let z ∈ span(v > G)> , which is (Gv)T z = 0.
>>>>
We havedim(span(vhi M )) = 1, and thus: dim(span(v G) ) = q − 1. Furthermore, z Hess[F ]z =1

z > E M (v > M v)− 2 z > 0, so we can deduce that dim(im(Hess[F ])) = q − 1. Using the Rank-Nullity
theorem, we conclude that dim(ker(Hess(F ))) = q − dim(im(Hess[F ])) = 1, which concludes the proof.
In conclusion, ∀w ∈ Rq \{0}, w> 12 Hess(F (v)2 )w > 0. The function F satisfies the strong convexity criterion.

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Proposition 2.2. Let G be a stochastic Riemannian metric tensor. Then, the function Fz : Tz Z → R :u → kukF defines a Finsler metric, but it is not induced by a Riemannian metric.
hpi
Proof. Let’s define F as a Riemannian metric: F : Rq × Rq → R : (v1 , v2 ) → E
v1> Gv2 . If F were a
Riemannian metric, then it would be bilinear, which is clearly not the case. Thus, F is not a Riemannian
metric. According to Lemma B.1.2 and Lemma B.1.2, F is a Finsler metric.
Proposition 2.3. Let f be a Gaussian process and J its Jacobian, with J ∼ N (E[J], Σ). The Finsler
norm can be written as:


1
√ √
Γ( D
1 D ω
2 + 2)>
Fz : Tz Z → R+ : kvkF := v → 2 v Σv,−,
1 F1 − ,
2 2
2
Γ( D
2)
with 1 F1 as the confluent hypergeometric function of the first kind and ω = (v > Σv)−1 (v > E[J]> E[J]v).
Proof.
The objective of the proof is to show that, if the Jacobian Jf follows a non-central normal
distribution, then, ∀v ∈ Rq , the expectation E[v > Jf> Jf v] will follow a non-central Nakagami distribution.
This is a particular case of the derivation of moments of non-central Wishart distributions, previously shownand studied by Kent & Muirhead (1984); Hauberg (2018b).
By hypothesis, Jf follows a non-central normal distribution: Jf ∼ N (E[J], ID ⊗ Σ). Then, G = Jf> Jf
follows a non-central Wishart distribution: G ∼ Wd (D, Σ, Σ−1 E[J]> E[J]). According to (Kent & Muirhead,
1984, Theorem 10.3.5.), v > Gv will also follow a non-central Wishart distribution: v > Gv ∼ W1 (D, v > Σv, ω),with: ω = (v > Σv)−1 (v > E[J]> E[J]v).
√
To compute E[ v > Gv], we shall look at the derivation of moments.
Theorem 10.3.7.)
states that: if X ∼ Wq (D, Σ, Ω0 ), with q ≤(Kent & Muirhead, 1984,
D, then E[(det(X))k ] =Γ ( D +k)1 0>(det{Σ})k 2qk qΓ (2 D ) 1 F1 (−k, D
2 , − 2 Ω ). We directly apply the theorem to our case, knowing that v Gv
q 2
is a scalar term, so det v > Gv = v > Gv, q = 1, and k = 12 :√√ √Γ( D + 1 )1 D 1
kvkF := E[ v > Gv] = 2 v > Σv 2 D 2 1 F1 (− , , − ω)
2 22
Γ( 2 )
Lemma B.1.4. The length of a curve using a Finsler metric is invariant by reparametrisation.
Proof. The proof is similar to the one obtained on a Riemannian manifold (Lee (2013), Proposition 13.25),
where we make use of the homogeneity property of the Finsler metric.
Let (M, F ) be a Finsler manifold and γ : [a, b] → M a piecewise smooth curve segment. We call γ̃ areparametrisation of γ, such that γ̃ = γ ◦ φ with φ : [c, d] → [a, b] a diffeomorphism. We want to show that
LF (γ) = LF (γ̃).
Z d

Z d

d
(γ ◦ φ(t))) dtdtcc
Z φ−1 (b)Z b=|φ̇(t)|F (γ̇ ◦ φ(t)) dt =F (γ̇(t)) dt = LF (γ)

LF (γ̃) =˙F (γ̃(t))
dt =F(φ−1 (a)aLemma B.1.5. If a curve globally minimizes its energy on a Finsler manifold, then it also globally minimizes
its length and the Finsler function F of the velocity vector along the curve is constant.
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Published in Transactions on Machine Learning Research (10/2023)

R1
Proof.
The curve energy and the curve length are defined as: EF (γ) = 0 F 2 (γ̇(t))dt and LF (γ) =R1F (γ̇(t))dt, with γ : [0, 1] → Rd . Let’s define f and g two real-valued functions such that: f : R → R : t 7→0
F (γ̇(t)) and g : R → R : t 7→ 1. Applying Cauchy-Schwartz inequality, we directly obtain:
Z 1
0

!2
F (γ̇(t))dt

Z 1≤

F (γ̇(t)) dt ·
2

Z 1

0

12 dt,

which means:

LF (γ)2 ≤ EF (γ).

0The equality is obtained exactly when the functions f and g are proportional, hence, when the Finsler
function is constant.
B.2

Comparison of Riemannian and Finsler metrics√
We have defined both a Riemannian (g : (v1 , v2 ) → v1> E[G]v2 ) and a Finsler (F : (x, v) → E[ v > Gv])
metric, in the hope to compute the average length between two points on a random manifold created by
the random field f : G = Jf> Jf . The main idea of this section is to better compare those two metrics andin what extend they differ in terms of length, energy and volume. From now on, f : Rq → RD will always
be defined as a stochastic non-central gaussian process. Its Jacobian Jf also follows √a non-central gaussian>
distribution, G = Jf Jf a non-central Wishart distribution, and F : (x, v) = E[ v > Gv] a non-central
Nakagami distribution (Proposition 2.3). The Finsler metric can be written in closed form.
In section B.2.1, we will see that the Finsler metric is upper and lower bounded by two Riemannian tensors(Proposition 3.1), and we can deduce an upper and lower bound for the length, the energy and the volume(Corollary 3.1). Then, in section B.2.2, we will show that the relative difference between the Finsler norm andthe Riemannian induced norm is always positive and upper bounded a term that is inversely proportional to
the number of dimensions D (Proposition 3.3). Similarly, we will deduce the same for the length, the energy
and the volume (Corollary 3.2). From this last results, we can directly conclude in section B.2.3 that both
metrics are equal in high dimensions (Corollary 3.4). A possible interpretation is that in high dimensions
the data distribution obtained on those manifolds becomes more and more concentrated around the mean,
reducing the variance term to zero. The manifold becoming deterministic, both metrics become equal.
Remark. Most of the following proofs will be a bit technical, as they rely on the closed form expressionof the non-central Nakagami distribution. Once proving the main propositions, obtaining the corollaries isstraightforward. While we do not have closed form expression of the indicatrix, we will show that it’s amonotoneous function which can upper and lower bounded.
B.2.1

Bounds on the Finsler metric

Proposition 3.1. We define α = 2



1
Γ( D
2 +2)
Γ( D
)
2

2

. The Finsler norm: k·kF is bounded by two norms,k·kαΣ and k·kR , induced by the two respective Riemannian metric tensors: the covariance tensor αΣz
and the expected metric tensor E[Gz ].
∀(z, v) ∈ Z × Tz Z : kvkαΣ ≤ kvkF ≤ kvkR
Proof. Let’s first recall that the Finsler function can be written as:
kvkF := F (v) =√ √Γ( D + 1 )1 D 1
2 v > Σv 2 D 2 · 1 F1 (− , , − ω).
2 22
Γ( 2 )P∞kk z
The confluent hypergeometric function is defined as: 1 F1 (a, b, z) = k=0 (a)(b)k k! , with (a)k and (b)k being
the Pochhammer symbols. Note that, despite their confusing notation, they are defined as rising factorials.
Γ(a+k) Γ(b)k
By definition, we have: (a)(b)k = Γ(b+k) Γ(a) . We can use the Kummer transformation to obtain:
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Published in Transactions on Machine Learning Research (10/2023)

1 F1 (a, b, −z) = e

−z

1
D
1
1 F1 (b − a, b, z). Replacing a = − 2 , b = 2 and z = 2 ω, we finally get:
∞
1
k
X
√ √
Γ( D
−z
2 + 2 + k) z>
F (v) = 2 v Σv · e.
DΓ( 2 + k) k!
k=0

1) Let’s show that: ∀v ∈ Tx M :√v > αΣv ≤ F (v), with α = 2



1
Γ( D
2 +2)
Γ( D
2 )

2

.

∗
The Pochhammer symbole is defined as (x)k = x(x + 1) . . . (x + k − 1) = Γ(x+k)Γ(x) . For x ∈ R+ , we have:(x)k ≤ (x + 12 )k . Thus,

Γ( D
Γ( D + 1 +k)
2 +k)≤ Γ(2 D +2 1 ) . The Gamma function being strictly positive on R+ , we obtain:
Γ( D
2 )
2
2

1
1
Γ( D
Γ( D
2 + 2)
2 + 2 + k)
≤
Γ( D
Γ( D
2)
2 + k)
∞
∞
D
1
1
k
k
X
X
√
√
√
√
Γ( 2 + 2 ) −z
Γ( Dz
2 + 2 + k) z
> Σv · e−z
2 v > Σv
·
e
≤
2
v
k!
k!
Γ( D
Γ( D
2)
2 + k)
k=0

k=0

∞
1
k
X
√ Γ( D
√ √
+ 1)√
Γ( D
2 + 2 + k) z
2 2 D 2 v > Σv ≤ 2 v > Σv · e−zDΓ( 2 )
Γ( 2 + k) k!
k=0√v > αΣv ≤ F (v).

2) Let’s show that: ∀v ∈ Tx M : F (v) ≤pv > E[G]v.

1
Wendel (1948) proved: Γ(x+y)≤ xy , for x > 0 and y ∈ [0, 1]. With x = D
Γ(x)2 + k, y = 2 , we obtained
q
q
√
D
1
P∞
Γ( 2 + 2 +k)zk≤ D
2v > Σv · e−z k=0 D
2 + k, which leads to: F (v) ≤2 + k k! .
Γ( D
2 +k)
q
P∞ −z zk
D
Furthermore,
e
=
1
and
the
functionx→
Then by Jensen’s ink=0k!
2 + x is concave.
q
q
P
P
P
k
k
∞
∞ z
∞ zk
z
D
−z
equality: e−z k=0 D≤= zez , we have:
k=0 k! k. Knowing thatk=0 k!
2 + k k!
2 +e
q
q
P∞
zk
D
e−z k=0 D
2 + k k! ≤2 + z.
pΩ
And with z = 2 , we obtain: F (v) ≤ v > Σ(D + Ω)v.

From (Kent & Muirhead, 1984, p. 442), the expectation of a non-central Wishart distribution (G ∼Wq (D, Σ, Ω)) is: E[G] = DΣ + ΣΩ. This finally leads to:
F (v) ≤

qv > E[G]v.

Remark. As a side note, the second part of the inequality F (v) ≤Proposition 3.2.

pv > E[G]v can be obtained using directly

Corollary 3.1. The length, the energy and the Busemann-Hausdorff volume of the Finsler metric arebounded respectively by the Riemannian length, energy and volume of the covariance tensor αΣ (noted
LαΣ , EαΣ , VαΣ ) and the expected metric E[G] (noted LR , ER , VR ):∀z ∈ Z, LαΣ (z) ≤ LF (z) ≤ LR (z)EαΣ (z) ≤ EF (z) ≤ ER (z)VαΣ (z) ≤ VF (z) ≤ VR (z)27Published in Transactions on Machine Learning Research (10/2023)pp
Proof. From Proposition 3.1, we have ∀(x, v) ∈ M × Tx M :h(v) ≤ F (v) ≤ g(v), with h : v → v > αΣv
and g : v → v > E[G]v Riemannian metrics. We also define the parametric curve: ∀t ∈ R, γ(t) = x andγ̇(t) = v.
1) Let’s show Rthat
Because of the monotonicity of the Lebesgue integrals, we
p LΣ (x) ≤ LRF (x) ≤ LR (x).R p
directly have:h(γ̇(t))dt ≤ F (γ̇(t))dt ≤g(γ̇(t))dt.
2) Let’s show that EΣ (x) ≤ EF (x) ≤ ER (x). Since all the functions are positive, we can raise
themto the R power two, and again,
with the monotonicity of the Lebesgue integrals, we have:RRph(γ̇(t))dt ≤ F 2 (γ(t), γ̇(t))dt ≤g(γ̇(t))dt.
3) Let’s show that VΣ (x) ≤ VF (x) ≤ VR (x). We write the vectors v ∈ Tx M in hyperspherical coordinates: v = re, with r = kvk the radial distance and e the angular coordinates. With v = re, we have:
p
p
p
−1
−1
r · h(e) ≤ r · F (e) ≤ r · g(e) ⇐⇒h(e) ≥ F (e)−1 ≥ g(e) .
We want to identify an inequality between
noted vol(Ih ), vol(Ig ), vol(IF ), formed by the
p the indicatrices,pfunctions h, g and F . Let’s define: rg h(e) = rh g(e) = rF F (e) = 1. For every angular coordinatee, we obtain: rh ≥ rF ≥ rg . Intuitively, this means that the finsler indicatrix will always be boundedby the indicatrices formed by h and g. The Busemann-Hausdorf volume of a function f is defined as:σB (f ) = vol(Bn (1))/vol(If ), with vol(Bn (1)) the volume of the unit ball and vol(If ) the volume of theindicatrix formed by f . The previous inequality and the definition of the Busemann-Hausdorff volume
implies that: vol(Ih ) ≥ vol(IF )p≥ vol(Ig ) ⇒ σB (h) ≤pσB (F ) ≤ σB (g). The functions g and h being
Riemannian, we have: σB (h) = det(αΣ) and σB (g) = det(E[G]), which concludes the proof.

B.2.2 Relative bounds between the Finsler and the Riemannian metric
Proposition 3.2. Let f be a stochastic immersion. f induces the stochastic norm k·kG , defined in
Section 2. The relative difference between the Finsler norm k·kF and the Riemmanian norm k·kR is:
h
i
2
Var
kvk
G
kvkR − kvkF
0≤
≤
h
i2 .
kvkR
2
2E kvkG
Proof. We will directly use a sharpen version of Jensen’s inequality obtained by Liao & Berg (2019): Let
X be a one-dimensional random variable with mean µ and P (X ∈ (a, b)) = 1, where −∞ ≤ a ≤ b ≤ +∞.
φ0 (µ)Let φ a twice derivable function on (a, b). We further define: h(x, µ) = φ(x)−φ(µ)(x−µ)2 − x−µ . Then:inf {h(x, µ)}Var[X] ≤ E[φ(x)] − φ(E[x]) ≤ sup {h(x, µ)}Var[X].

x∈(a,b)x∈(a,b)√In our case, we will chose φ : z → √z with z a one-dimensional random variable defined as z = v > Gv. a = 0,√√b = +∞ and µ = E[z]. h(z, µ) = ( z − µ)(z − µ)−2 − (2(z − µ) µ)−1 . Because its first derivative φ0 isconvex, the function x → h(x, µ) is monotonically increasing. Thus:√infz∈(0,+∞){h(x, µ)} = lim = −z→0

It finally gives:

µ
2µ2

and

√

−

sup

{h(x, µ)} = lim = 0.

z∈(0,+∞)p√µVar[z] ≤ E[ z] − E[z] ≤ 0.
2
2µ
28

z→+∞

Published in Transactions on Machine Learning Research (10/2023)pp√√
Replacing kvkF := F (v) = E[ z] and kvkR := g(v) = E[z] = µ concludes the proof.
Lemma B.2.1. Let z ∼ W1 (D, σ, Ω) following a one-dimensional non-central Wishart distribution. Then:
Var[z]
1Ω=+
2E[z]2
D + Ω (D + Ω)2
Proof.

(Kent & Muirhead, 1984, Theorem 10.3.7.)

Γ( D +k)1
σ k 2k Γ(2 D ) 1 F1 (−k, D
2 , − 2 Ω).
2states that if z ∼ W1 (D, σ, ω) then E[z k ] =In particular, for k = 1 and k = 2, we have 1 F1 (−1, b, c) = 1 − cb and


Γ( D
Γ( D
2c
c2
D
D D
2 +1)
2 +2)
F
(−2,
b,
c)
=
1
−
+
.
We
also
have
=
and
=
+
1
, which leads to:
1 1bb(b+1)
2
2
2
Γ( D )
Γ( D )
2

2

E[z] = σ(D + Ω) and E[z 2 ] = σ 2 (2ω + 2(D + ω) + (D + ω)2 ). Finally, we conclude:
Var[z]
E[z 2 ]
2ω
2
=
−1=
+
.
2
E[z]
E[z]2
(D + ω)2
D+ω
Proposition 3.3. Let f be a Gaussian process. We note ω = (v > Σv)−1 (v > E[J]> E[J]v), with J thejacobian of f , and Σ the covariance matrix of J.
The relative ratio between the Finsler norm k·kF and the Riemmanian norm k·kR is:0≤

kvkR − kvkF
1
ω
≤
+
.
kvkR
D + ω (D + ω)2

Proof. The result is directly obtained using Proposition 3.2 and Lemma B.2.1.
Corollary 3.2. When f is a Gaussian Process, the relative ratio between the length, the energy andthe volume of the Finsler norm (noted LF , EF , VF ) and the Riemannian norm (noted LR , ER , VR ) is:
LR (z) − LF (z)
1
ω
0≤
≤ max
+
v∈Tz ZLR (z)
D + ω (D + ω)2()
ER (z) − EF (z)2
1 + 2ω
2ω
ω2
0≤
≤ max
+
+
+
v∈Tz ZER (z)
D + ω (D + ω)2(D + ω)3(D + ω)4
!q
VR (z) − VF (z)1ω0≤≤ 1 − 1 − max+v∈Tz ZVR (z)
D + ω (D + ω)2ω1
Proof. Let’s call M = maxv∈Tx M { (D+ω)
2 + D+ω }.

From Proposition 3.3, we have:0 ≤ kvkR − kvkF ≤ M kvkR ,with

M = maxv∈Tx M1
ω
+D + ω (D + ω)21) By the monocity of the Lesbesgue integral, we can directly integrate the previous norms along a curve γ,
which immediately leads to: 0 ≤ LR (x) − LF (x) ≤ M LR (x).
2) Since all the functions are positive: 0 ≤ kvkF ≤ kvkR ≤ M kvkR + kvkF leads to:22
2
2
kvkF ≤ kvkR ≤ M 2 kvkR + 2M kvkF kvkR + kvkF , and replacing kvkF ≤ kvkR in the right hand
2
2
2
2
2
2
2
term: kvkF ≤ kvkR ≤ (M 2 + 2M )kvkR + kvkF , and finally: 0 ≤ kvkR − kvkF ≤ (M 2 + 2M )kvkR . Again,2
by continuity of the Lebesgue integral, we directly obtain: 0 ≤ ER (x) − EF (x) ≤ (M + 2M )ER (x).

29

Published in Transactions on Machine Learning Research (10/2023)

3) In order to compare the volume between the Finsler and the Riemannian metric, we need to compare
the volume of their indicatrices, noted: vol(Ig ) and vol(IF ) respectively. We write the vectors v ∈ Tx M
in hypershperical coordinates, with v = re, r = kvk the radial distance and e the angular coordinates.
The volume of the indicatrices obtained in dimension q (dimension of the latent space) can be written as:dq V = rq−1 drdΦ, with Φ defining the different angles. We will note rF and rg the radial distances of theFinsler and Riemann metrics such that: kvkF = rF kekF = 1 and kvkR = rg kekR = 1 obtained for a specificangle e.


!q 
!q 

Z Z rf
Z rg
Z q
Z q
r
r
r
r
g
g
f
f
1 −
 dΦ ≤
,
vol(IF )−vol(Ig ) =rq−1 dr −rq−1 dr dΦ =dΦ·1 −
rf
rf
Φ
0
0
Φ q
Φ q
R
and by definition: vol(IF ) = Φ (rfq /q)dΦ. Furthermore, for a specific angle e, we have: rg /rF =pg(e)/F (e) ≥ 1 − M , from Proposition 3.3. We have:vol(IF ) − vol(Ig )0≤≤ ·1 −
vol(IF )rgrf

and by the definition of the Busemann Hausdorff volume:
proof.

B.2.3

!q
q

≤ 1 − (1 − M ) ,VF (x)−VG (x)VF (x)=

vol(IF )−vol(Ig ), we conclude thevol(IF )Implications in High Dimensions

In this section, we want to show that the difference between the Finsler norm and the Riemannian induced
1
norm, as well as their respective functionals, tend to zero at a rate of O( D). We need to be sure that ωdoesn’t grow faster than D, in other terms: ω = O(D). This can be obtained if we assume that every
element of the expectation of Jacobian is upper bounded (∃m ∈ R∗+ , ∀i, j E[Jij ] ≤ m). This happensin at least two cases: (1) E[f ] is somehow Lipschitz continuous; or (2) if f is a Gaussian Process andits covariance is upper bounded. The latter case happens when the process is defined over a bounded domain.
√
Lemma B.2.2. Our Finsler metric v → E[ v > Gv] is defined with v > Gv ∼ W1 (D, v > Σv, ω), and ω =(v > Σv)−1 (v > E[J]> E[J]v).
If the Finsler manifold is bounded, then: ω ≤ DM , with M ∈ R+ .
Proof. By definition, Σ does not depend on D. We assume the manifold is bounded, which means that every
element of the expected Jacobian is upper bounded: E[J]ij ≤ m, with m ∈ R∗+ . We call σ = v > Σv ∈ R∗+ .
We have:ω = σ −1

q X
q
D X
X

vi E[J]ki E[J]kj vj ≤ σ −1

k=1 i=1 j=1

DX2m2 kvk ≤ DM,

k=1

2

with M = σ −1 m2 kvk ∈ R∗+ , and M does not depend on D.
Corollary 3.3. Let f be a Gaussian Process. In high dimensions, we have:LR (z) − LF (z)=O
LR (z)



1
D


,

ER (z) − EF (z)=O
ER (z)



1
D


,

and

VR (z) − VF (z)=O
VR (z)

When D converges toward infinity: LR ∼ LF , ER ∼ EF and VR ∼ VF .
+∞

+∞

+∞

Proof. From Corollary 3.2, we directly obtained the results in high dimensions.
30



q
D


.

Published in Transactions on Machine Learning Research (10/2023)

We assume that our latent space is bounded, then by B.2.2, we have: 0 ≤ ω ≤ M D, with M ∈ R+.
For the length, we have:LG (x) − LF (x)
1
ω
≤ max
+
v∈Tx MLG (x)
D + ω (D + ω)2
1+M≤DFor the energy functional, we have:EG (x) − EF (x)≤ maxv∈Tx MEG (x)(

2
1 + 2ω
2ω
ω2
+
+
+
2
3
D + ω (D + ω)
(D + ω)(D + ω)4)

2 + 2M
1 + 2M + M 2+
DD2
EG (x) − EF (x)M 2 + 2M + 1
lim sup D ×≤ lim sup 2(1 + M ) +→ 2(1 + M )EG (x)
D2
D→∞
D→∞
≤

For the volume, we have:
VG (x) − VF (x)≤1−
VG (x)1
ω+
D + ω (D + ω)2

1 − max

1+M≤1− 1−
Dqv∈Tx M!qq

Using Taylor series expansion, when x ∼ 0, we have: 1 − (1 − x) = qx + o(x2 ). Let’s call ε  1, and rewrite
the Taylor series:
VG (x) − VF (x)
1+M
1+M
≤q
+ εq
VG (x)DD
D VG (x) − VF (x)lim sup ×≤ (1 + M )(1 + ε)
VG (x)
D→∞ q
The difference between the functionals can converge to zero if they are similar in high dimensions, or if they
all diverge to infinity. This latter case does not happen as we assume the latent manifold being bounded,and so the metrics are then finite, which concludes the proof.
Corollary 3.4. Let f be a Gaussian Process. In high dimensions, the relative ratio between the Finsler
norm k·kF and the Riemmanian norm k·kR is: 
kvkR − kvkF
1
=O
kvkR
D
And, when D converges toward infinity: ∀v ∈ Tz Z, kvkR ∼ kvkF .
+∞Proof. Similar to the 3.3, assuming that our latent space is bounded, from B.2.2, we have 0 ≤ ω ≤ M D.
From 3.3, we deduce:
0≤

kvkR − kvkF
1
ω
≤
+
kvkR
D + ω (D + ω)2
1+M
≤
D
31

Published in Transactions on Machine Learning Research (10/2023)

In a bounded manifold, the metric are finite. We can deduce that they converge to each other in highdimensions.

C

Experiments

C.1

Datasets

C.1.1

Font data

The dataset represents 46 different font for each letter (upper and lower case) whose contour is parametrisedby a spline (or two splines, depending on the letter used) obtained from at least 500 points Campbell &Kautz (2014).
In our case, we choose to learn the manifold of the letter f. The dataset is composed of 46 different fonts,
each letter being drawn by 1024 points. We reduce this number from 1024 to 256 by sampling one point
every 4. The dimension of the observational space is then 256.
C.1.2

qPCR

The qPCR data, gathered from Guo et al. (2010), was used to illustrate the training of a GPLVM in Pyro
Pyro (2022) and is available at the Open Data Science repository Ahmed et al. (2019). It consists of 437
single-cell qPCR data for which the expression of 48 genes has been measured during 10 different cell stages.
We then have 437 data points, 48 observations, and 10 classes. Before training the GP-LVM, the data isgrouped by the capture time, as illustrated in the Pyro documentation.
C.1.3

Pinwheel on a sphere

A pinwheel in 2-dimension is created and then projected onto a sphere using a stereographic projection
method. The final dataset is composed of 1000 points with their coordinates in 3-dimensions.
C.2

GP-LVM training

We learn our two-dimensional latent space by training a GP-LVM Lawrence (2003) with Pyro Bingham et al.
(2019). The Gaussian Process used is a Sparse GP, defined with a kernel (RBF, or Matern) composed of aone-dimensional lengthscale and variance. The parameters are learnt with the Adam optimiser Kingma &Ba (2014). The number of steps and the initialisation of the latent space vary with the dataset.
datasets
Number of data points
Number of observations
initialisation
kernelsteps
learning rate
lengthscalevariancenoise

pinwheel
500
3
PCA
RBF
17000
1e-3
0.24
0.95
1e-4

font data
46
256
PCA
Matern52
5000
1e-4
0.88
0.30
1e-3

qPCR
437
48
customMatern52
5000
1e-4
0.15
0.75
1e-3

Table 2: Description of the datasets trained with a GP-LVM.

C.3

Computing indicatrices

An indicatrix of a function g at a point x is defined such that: v ∈ Tx M |gx (v) < 1. In other terms,
the indicatrix is the representation of a unit ball in our latent space. If we use an euclidean metric, ourindicatrix in our 2-dimensional latent space would be a unit ball, as we need to solve: v ∈ Tx M , kvk < 1.
32

Published in Transactions on Machine Learning Research (10/2023)

For a Riemannian metric, our indicatrix is necessarily an ellipse, whose semi axis are the square-roots of theeigenvalues of the metric tensor G:: v ∈ Tx M , v > Gv < 1. For our Finsler metric, we don’t have an analyticalsolution, and so it’s difficult to predict the shape of the convex polygon.
In this paper, the indicatrices are drawn the following way: for a single point in our latent space, we computethe value of v > Gv and F (x, v) for v varying over the space. We then extract the contour when v > Gv andF (x, v) are equal to 1. Computing the area of the indicatrices will be used in the section C.4 to computethe volume measures.
C.4

Computing the volume forms

For the figures used in this paper, by default, the backgroundof the latent space represents the volumepmeasure of the expected Riemannian metric (VG = E[G]) on a logarithm scale. In figure 3, the volume
measure of the Finsler metric is also computed.
C.4.1

Finsler metric

To compute the volume measure of our Finsler metric, we choose the Busemann-Hausdorff definition, whichis the ratio of a unit ball over the volume of its indicatrix: V = vol(Bn (1))/vol({v ∈ Tx M |F (x, v) < 1}).
While our Finsler function has an analytical form, its expression doesn’t allow to directly solve the equation:v ∈ Tx M , F (x, v) < 1. Instead we approximate its indicatrix as describe in C.3, using a contour plot andextracting the paths vertices. We can then compute the area of the obtained polygon, and divide with thevolume of a unit ball: vol(B2 (1)) = π.
The volume measure can then be computed for each point over a grid (32 x 32, in figure 3), and we interpolate
all the other points. Note that this method can only be used when our latent space is of dimension 2.
C.4.2

Expected Riemannian metric

There is two ways to computep
the volume measure of the expected Riemannian metric. One way is to directly
use the metric tensor: VG = E[G]. Another one is to remember that any Riemannian metric is a Finsler
metric, and thus, the Busemann-Hausdorff definition also applied for our metric: VG = vol(Bn (1))/vol({v ∈Tx M |v > E[G]v < 1}). Solving v > E[G]v < 1 for v ∈ Tx M is equivalent to solving the area of an ellipse.
For the first method, we can either sample multiple times the metric, which is computationally expensive, or
use the fact that our metric tensor is a non-central Wishart matrix: G = J > J ∼ Wq (D, Σ, Σ−1 E[J]> E[J]),with Σ the covariance of the Jacobian J and D the dimension of the observational space. In this case, its
expectation is: E[G] = E[J]> E[J] + DΣ. We can access the derivatives of the function f (detailed in sectionD.2), and compute both quantities E[J] and Σ needed to estimate the expected metric and its determinant.
For the second method, we can compute the area of the ellipse in the same way we compute the Finsler
volume measure.
C.5

Experiments when increasing the number of dimensions

In Figure 4, we computed the volume ratio and draw indicatrices while varying the number of dimensions,to illustrate that both our Finsler metric and the expected Riemannian metric seem to converge when D
increases.
The main issue with this experiment is to vary only one factor, the number of dimensions D, while keeping theother factors unchanged. This is difficult for two reasons: 1) Even with a very low dimensional observational,
both metrics are already very similar to each other. It would be difficult to illustrate a convergence while
increasing the number of dimensions. 2) the function f needs to be learnt again each time we increase thenumber of dimensions of the observational space, and the parameters of the Gaussian Process will change
too.
Instead, we try to illustrate our results by computing empirically the stochastic metric tensor G = J > J,
using its Jacobian J ∼ N (E[J], Σ), a D × q matrix. The number of dimensions is modified by simply
33

Published in Transactions on Machine Learning Research (10/2023)

truncatenating the Jacobian J. In Figure 4, the volume ratio is computed for 12 Jacobians obtained withdifferent random parameters E[J] and Σ. The Finsler and Riemannian indicatrices (lower right) are drawn
for only one Jacobian selected randomly.

D

Computations

D.1

Computing geodesics with Stochman and minimising the curve energy functionals

An essential task is to compute shortest paths, or geodesics, between data points in the latent space. Thoseshortest paths can be obtained in two ways: either by solving a corresponding system of ODEs, or byminimising the curve energy in the latent space. The former being computationally expensive, we favour
the second approach which consists in optimising a parameterised spline on the manifold. This method isalready implemented in Stochman Detlefsen et al. (2021), where we can easily optimise splines by redefining
the curve energy function of a manifold class.
We need two curvge energy functional: one for the expected Riemannian metric and one for the Finsler
metric.
D.1.1

Curve energy for the Riemannian metricWe know that the stochastic metric tensor Gt defined on a point t follows a non-central Whishart distribution.
Thus, we can compute its expectation E[Gt ] knowing the Jacobian covariance and expectation: E[Jt ] and Σ.
The next Section D.2 explains how to compute those quantities.
Assuming the spline is discretized into N points, we can compute the curve energy with:
Z 1
N


X
>
EG (γ(t)) =γ̇(t) E[Gt ]γ̇(t) dt ≈γ̇i> E[Ji ]T E[Ji ] + DΣi γ̇i .
0

D.1.2

i=1

Curve energy for the Finsler metric

In order to compute of the curve energy E(γ), we must first derive the expectation E[Jt ]and covariance Σof the Jacobian of f , which should follow a normal distribution: Ji = ∂fi ∼ N E[J], Σ . We assume thepoints ∂fi are independent samples with the same variance drawn from a normal distribution. We can thencompute the Finsler metric which follows a non-central Nakagami distribution (See Proposition 2.2):E(γ(t)) =Z 1

F (t, γ̇(t)) dt ≈
2

0

N
X

2γ̇i> Σi γ̇i

i=1

1
Γ( D
2 + 2)
Γ( D
2)

!2


2
1 D ωi,−,1 F1 − ,2 22with 1 F1 the confluent hypergeometric function of the first kind and ωi = (γ̇i> Σi γ̇i )−1 (γ̇i> Ωi γ̇i ) and Ωi =>Σ−1
i E[Ji ] E[Ji ].
This function has been implemented in Pytorch using the known gradients for the hypergeometric function:∂a∂x 1 F1 (a, b, x) = b 1 F1 (a + 1, b + 1, x).
D.2

Accessing the posterior derivatives

We assume that the probabilitic mapping f from the latent variables X to the observational variables Y
follows a normal distribution. We would like to obtain the posterior kernel Σ∗ and expectation µ∗ such thatp(∂t f |Y, X) ∼ N (µ∗ , Σ∗ ).
We make the hypothesis the observed variables are modelled with a gaussian noise  whose variance is thesame in every dimension. In particular, for the nth latent (x) and observed (y) variable in the j th dimension:yn,j = fj (xn,: ) + n . Thus, the output variables have the same variance, and the posterior kernel Σ∗ is thenisotropic with respect to the output dimensions: Σ∗ = σ∗2 · ID .
There are two ways of obtaining the posterior variance and expectation:34

Published in Transactions on Machine Learning Research (10/2023)• We use the gaussian processes to predict the derivative (∂c f ) of the mapping function f , andwe multiply the obtained posterior kernel by the curve derivative (∂t c), following the chain rule:df (c(t))
df dc
= dc
· dt (Section: D.2.1)
dt• We discretize the derivative of the mapping function as the difference of this function evaluatedat two close points. We use a linear operation to obtain the posterior variance and expectation:∂t f (c(t)) ∼ f (c(ti+1 )) − f (c(ti )). (Section: D.2.2)
D.2.1

Closed-form expressions

We assume that f is a Gaussian process. Hence, because the differentiation is a linear operation, thederivative of a Gaussian Process is also a Gaussian Process Rasmussen & Williams (2005).
The data Y ∈ RN ×D follows a normal distribution, so we can infer the partial derivative of one data point∂yi(J T )ji = ∂x
, with i = 1 . . . D and j = 1 . . . d. We have:
j
"

Y
(J)>

#
=

D
Y


N

i=1

µy
µ∂y

 
,

K(x, x)∂K(x, x∗ )∂K(x∗ , x) ∂ 2 K(x∗ , x∗ )

!
.

and J > can be predicted:p(J > |Y, X) =DYN (µ∗ , Σ∗ ) ,

i=1

with:
µ∗ = ∂K(x∗ , x) · K(x, x)−1 · (y − µy ) + µ∂yΣ∗ = ∂ 2 K(x∗ , x∗ ) − ∂K(x∗ , x) · K(x, x)−1 · ∂K(x, x∗ ).
Finally, ∂t f is obtained:p(∂t f (c(t))|f (x), x) =

D
Y



N ċµ∗ , ċ> Σ∗ ċ · ID .

i=1

D.2.2

Discretization

One can notice that: ∂t f (c(t)) ∼ f (c(ti+1 )) − f (c(ti )). We know that f (c(ti+1 )) and f (c(ti )) both follows anormal distribution.
f (c(ti ))f (c(ti+1 ))



=

D
Y
j=1



N

µi
µi+1

 "
,

2
σii
2
σi+1,i2
σi,i+12
σi+1,i+1#.

If Y = AX affine transformation of a multivariate Gaussian X ∼ N (µ, σ 2 ), then Y is also a multivariate
Gaussian with: Y ∼ N (Aµ, AT σ 2 A). In our case, we choose AT = [−1, 1]. We have:f (c(ti+1 )) − f (c(ti )) ∼ N (µ∗ , σ∗2 · ID ),with:µ∗ = µi+1 − µi222σ∗2 = σii
+ σi+1,i+1− 2σi,i+135

Published in Transactions on Machine Learning Research (10/2023)

Figure 12: Illustration of the derivatives obtained with a trained GP on a simple parametrised function: both methods
give the correct derivatives if enough points are sampled.

36