TANGENT BUNDLE FILTERS AND NEURAL NETWORKS FROM MANIFOLDS TO CELLULAR SHEA VES AND BACK C. Battiloro12 Z. Wang1 H. Riess3 P . Di Lorenzo2 A. Ribeiro1

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TANGENT BUNDLE FILTERS AND NEURAL NETWORKS:

FROM MANIFOLDS TO CELLULAR SHEAVES AND BACK

C. Battiloro1,2, Z. Wang1, H. Riess3, P. Di Lorenzo2, A. Ribeiro1

1ESE Department, University of Pennsylvania, Philadelphia, USA

2DIET Department, Sapienza University of Rome, Rome, Italy

3ECE Department, Duke University, Durham, USA

E-mail: claudio.battiloro@uniroma1.it, zhiyangw@seas.upenn.edu

ABSTRACT

In this work we introduce a convolution operation over the tangent

bundle of Riemannian manifolds exploiting the Connection Lapla-

cian operator. We use the convolution to deﬁne tangent bundle ﬁl-

ters and tangent bundle neural networks (TNNs), novel continuous

architectures operating on tangent bundle signals, i.e. vector ﬁelds

over manifolds. We discretize TNNs both in space and time do-

mains, showing that their discrete counterpart is a principled vari-

ant of the recently introduced Sheaf Neural Networks. We formally

prove that this discrete architecture converges to the underlying con-

tinuous TNN. We numerically evaluate the effectiveness of the pro-

posed architecture on a denoising task of a tangent vector ﬁeld over

the unit 2-sphere.

Index Terms—Geometric Deep Learning, Tangent Bundle Sig-

nal Processing, Tangent Bundle Neural Networks, Cellular Sheaves

1. INTRODUCTION

The success of deep learning is mostly the success of Convolutional

Neural Networks (CNNs) [1]. CNNs have achieved impressive per-

formance in a wide range of applications showing good generaliza-

tion ability. Based on shift operators in the space domain, one (but

not the only one) key attribute is that the convolutional ﬁlters sat-

isfy the property of shift equivariance. Nowadays, data deﬁned on

irregular (non-Euclidean) domains are pervasive, with applications

ranging from detection and recommendation in social networks pro-

cessing [2], to resource allocations over wireless networks [3], or

point clouds for shape segmentation [4], just to name a few. For

this reason, the notions of shifts in CNNs have been adapted to

convolutional architectures on graphs (GNNs) [5, 6] as well as a

plethora of other structures, e.g. simplicial complexes [7–10], cell

complexes [11, 12], and manifolds [13]. In [14], a framework for al-

gebraic neural networks has been proposed exploiting commutative

algebras. In this work we focus on tangent bundles, a formal tool for

describing and processing vector ﬁelds on manifolds, which are key

elements in tasks such as robot navigation or ﬂocking modeling.

Related Works. The renowned manifold assumption states that high

dimensional data examples are sampled from a low-dimensional

Riemannian manifold. This assumption is the fundamental block

of manifold learning, a class of methods for non-linear dimension-

ality reduction. Some of these methods approximate manifolds

with k-NN or geometric graphs via sampling points, i.e., for a ﬁne

enough sampling resolution, the graph Laplacian of the approxi-

mating graph “converges” to the Laplace-Beltrami operator of the

manifold [15]. These techniques rely on the eigenvalues and eigen-

vectors of the graph Laplacian [16], and they give rise to a novel

perspective on manifold learning. In particular, the above approx-

imation leads to important transferability results of graph neural

networks (GNNs) [17,18], as well as to the introduction of Graphon

and Manifold Neural Networks, continuous architectures shown to

be limit objects of GNNs [19, 20]. However, most of the previ-

ous works focus on scalar signals, e.g. one or more scalar values

attached to each node of graphs or point of manifolds; recent devel-

opments [21] show that processing vector data deﬁned on tangent

bundles of manifolds or discrete vector bundles [22, 23] comes with

a series of beneﬁts. Moreover, the work in [24] proves that it is

possible to approximate both manifolds and their tangent bundles

with certain cellular sheaves obtained from a point cloud via k-NN

and Local PCA, such that, for a ﬁne enough sampling resolution,

the Sheaf Laplacian of the approximating sheaf “converges” to the

Connection Laplacian operator. Finally, the work in [25] generalizes

the result of [24] by proving the spectral convergence of a large class

of Laplacian operators via the Principal Bundle set up.

Contributions. In this work we deﬁne a convolution operation over

the tangent bundles of Riemannian manifolds with the Connection

Laplacian operator. Our deﬁnition is consistent, i.e. it reduces to

manifold convolution [19] in the one-dimensional bundle case, and

to the standard convolution if the manifold is the real line. We intro-

duce tangent bundle convolutional ﬁlters to process tangent bundle

signals (i.e. vector ﬁelds over manifolds), we deﬁne a frequency

representation for them and, by cascading layers consisting of tan-

gent bundle ﬁlters banks and nonlinearities, we introduce Tangent

Bundle Neural Networks (TNNs). We then discretize the TNNs in

the space domain by sampling points on the manifold and build-

ing a cellular sheaf [26] representing a legit approximation of both

the manifold and its tangent bundle [24]. We formally prove that

the discretized architecture over the cellular sheaf converges to the

underlying TNN as the number of sampled points increases. More-

over, we further discretize the architecture in the time domain by

sampling the ﬁlter impulse function in discrete and ﬁnite time steps,

showing that space-time discretized TNNs are a principled variant

of the very recently introduced Sheaf Neural Networks [23, 27, 28],

discrete architectures operating on cellular sheaves and generalizing

graph neural networks. Finally, we numerically evaluate the perfor-

mance of TNNs on a denoising task of a tangent vector ﬁeld of the

unit 2-sphere.

Paper Outline. The paper is organized as follows. We start with

some preliminary concepts in Section 2. We deﬁne the tangent bun-

dle convolution and ﬁlters in Section 3, and Tangent Bundle Neural

Networks (TNNs) in Section 4. In Section 5, we discretize TNNs in

space and time domains, showing that discretized TNNs are Sheaf

Neural Networks and proving the convergence result. Numerical re-

sults are in Section 6 and conclusions are in Section 7.

arXiv:2210.15058v3 [eess.SP] 18 Nov 2022

2. PRELIMINARY DEFINITIONS

Manifolds and Tangent Bundles. We consider a compact and

smooth d−dimensional manifold Misometrically embedded in Rp.

Each point x∈ M is endowed with a d−dimensional tangent (vec-

tor) space TxM∼

=Rd,v∈ TxMis said to be a tangent vector at

xand can be seen as the velocity vector of a curve over Mpassing

through the point x(formal deﬁnitions can be found in [29]). The

disjoint union of the tangent spaces is called the tangent bundle

T M =Fx∈M TxM. The embedding induces a Riemann structure

on M; in particular, it equips each tangent space TxMwith an inner

product, called Riemann metric, given, for each v,w∈ TxM, by

hv,wiTxM=iv•iw,(1)

where iv∈ TxRpis the embedding of v∈ TxMin TxRp⊂Rp

(the d-dimensional subspace of Rpwhich is the embedding of TxM

in Rp), with i:T M → TxRpbeing an injective linear mapping

referred to as differential [29], and •is the dot product. The Riemann

metric induces also a probability measure µover the manifold.

Tangent Bundle Signals. A tangent bundle signal is a vector ﬁeld

over the manifold, thus a mapping F:M → T M that associates

to each point of the manifold a vector in the corresponding tangent

space. An inner product for tangent bundle signals Fand Gis

hF,GiT M =ZMhF(x),G(x)iTxMdµ(x),(2)

and the induced norm is ||F||2

T M =hF,FiT M . We denote with

L2(T M)the Hilbert Space of ﬁnite energy (w.r.t. ||·||T M ) tangent

bundle signals. In the following we denote h·,·iT M with h·,·i when

there is no risk of confusion.

Connection Laplacian. The Connection Laplacian is a (second-

order) operator ∆ : L2(T M)→ L2(T M), given by the trace of

the second covariant derivative deﬁned (for this work) via the Levi-

Cita connection [24]. The connection Laplacian ∆has some desir-

able properties: it is negative semideﬁnite, self-adjoint and elliptic.

The Connection Laplacian characterizes the heat diffusion equation

∂U(x, t)

∂t −∆U(x, t) = 0,(3)

where U:M × R+

0→ T M and U(·, t)∈ L2(T M)∀t∈R+

(see [21] for a simple interpretation of (3)). With initial condition set

as U(x, 0) = F(x), the solution of (3) is given by

U(x, t) = et∆F(x),(4)

which provides a way to construct tangent bundle convolution, as

explained in the following section. The Connection Laplacian ∆has

a negative spectrum {−λi,φi}∞

i=1 with eigenvalues λiand corre-

sponding eigenvector ﬁelds φisatisfying

∆φi=−λiφi,(5)

with 0< λ1≤λ2≤. . . . The λis and the φis can be interpreted as

the canonical frequencies and oscillation modes of T M.

3. TANGENT BUNDLE CONVOLUTIONAL FITLERS

In this section we deﬁne the tangent bundle convolution of a ﬁlter

impulse response e

hand a tangent bundle signal F.

Deﬁnition 1. (Tangent Bundle Filter) Let e

h:R+→Rand let

F∈ L2(T M)be a tangent bundle signal. The manifold ﬁlter with

impulse response e

h, denoted with h, is given by

G(x) = hF(x) := e

h ?T M F=Z∞

h(t)U(x, t)dt, (6)

where U(x, t)is the solution of the heat equation in (3) with

U(x, 0) = F(x). Injecting (4) in (6), we obtain

G(x) = hF(x) = Z∞

h(t)et∆F(x)dt=h(∆)F(x).(7)

The convolution in Deﬁnition 1 is consistent, i.e. it generalizes the

manifold convolution [19] and the standard convolution in Euclidean

domains (see Appendix A.4). The frequency representation ˆ

Fof F

can be obtained by projecting Fonto the φis basis

ˆ

Fi=hF,φii=ZMhF(x),φi(x)iTxMdµ(x)(8)

Deﬁnition 2. (Bandlimited Tangent Bundle Signals) A tangent bun-

dle signal is said to be λM-bandlimited with λM>0if ˆ

Fi= 0

for all i such that λi> λM.

Proposition 1. Given a tangent bundle signal Fand a tangent bundle

ﬁlter h(∆) as in Deﬁnition 1, the frequency representation of the

ﬁltered signal G=h(∆)Fis given by

ˆ

Gi=Z∞

h(t)e−tλidtˆ

Fi.(9)

Proof. See Appendix A.1.

Deﬁnition 3. (Frequency Response)The frequency response ˆ

h(λ)

of the ﬁlter h(∆) is deﬁned as

h(λ) = Z∞

h(t)e−tλdt. (10)

This leads to ˆ

Gi=ˆ

h(λi)ˆ

Fi, meaning that the tangent bundle

ﬁlter is point-wise in the frequency domain. Therefore, we can write

the frequency representation of the tangent bundle ﬁlter as

G=h(∆)F=

∞

i=1

h(λi)hF,φiiφi.(11)

We note that the frequency response of the tangent bundle ﬁlter gen-

eralizes the frequency response of a standard time ﬁlter as well as a

graph ﬁlter [30].

4. TANGENT BUNDLE NEURAL NETWORKS

We deﬁne a layer of a Tangent Bundle Neural Network (TNN) as a

bank of tangent bundle ﬁlters followed by a pointwise non-linearity.

In this setting, pointwise informally means “pointwise in the ambi-

ent space”. We introduce the notion of differential-preserving non-

linearity to formalize this concept.

Deﬁnition 4. (Differential-preserving Non-Linearity) Denote with

Ux⊂ TxRpthe image of the injective differential iin x. A mapping

σ:L2(T M)→ L2(T M)is a differential-preserving non-linearity

if it can be written as σ(F(x)) = i−1◦eσx◦iF(x), where eσx:Ux→

Uxis a point-wise non-linearity in the usual (euclidean) sense.

Furthermore, we assume that eσx=eσfor all x∈ M. Thus, the

l−th layer of a TNN with Flinput signals {Fq

l}Fl

q=1,Fl+1 output

signals {Fu

l+1}Fl+1

u=1 , and point-wise non linearity σ(·)is written as

l+1(x) = σ Fl

q=1

h(∆)u,q

lFq

l(x)!, u = 1, ..., Fl+1.(12)

A TNN of depth Lwith input signals {Fq}F0

q=1 is built as the stack

of Llayers deﬁned in (12), where Fq

0=Fq. To globally represent

the TNN, we collect all the ﬁlter impulse responses in a function set

H=ˆ

hu,q

ll,u,q and we describe the TNN u−th output as a map-

ping Fu

L=ΨuH,∆,{Fq}F0

q=1to enhance that it is parameterized

by ﬁlters Hand Connection Laplacian ∆.

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TANGENTBUNDLEFILTERSANDNEURALNETWORKS:FROMMANIFOLDSTOCELLULARSHEAVESANDBACKC.Battiloro1;2,Z.Wang1,H.Riess3,P.DiLorenzo2,A.Ribeiro11ESEDepartment,UniversityofPennsylvania,Philadelphia,USA2DIETDepartment,SapienzaUniversityofRome,Rome,Italy3ECEDepartment,DukeUniversity,Durham,USAE-mail:claudio.battilor...

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TANGENT BUNDLE FILTERS AND NEURAL NETWORKS FROM MANIFOLDS TO CELLULAR SHEA VES AND BACK C. Battiloro12 Z. Wang1 H. Riess3 P . Di Lorenzo2 A. Ribeiro1

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