Convolutional Neural Networks on Manifolds From Graphs and Back Zhiyang Wang Luana Ruiz Alejandro Ribeiro

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Convolutional Neural Networks on Manifolds:

From Graphs and Back

Zhiyang Wang Luana Ruiz Alejandro Ribeiro

Abstract—Geometric deep learning has gained much attention

in recent years due to more available data acquired from non-

Euclidean domains. Some examples include point clouds for

3D models and wireless sensor networks in communications.

Graphs are common models to connect these discrete data points

and capture the underlying geometric structure. With the large

amount of these geometric data, graphs with arbitrarily large size

tend to converge to a limit model – the manifold. Deep neural

network architectures have been proved as a powerful technique

to solve problems based on these data residing on the manifold.

In this paper, we propose a manifold neural network (MNN)

composed of a bank of manifold convolutional ﬁlters and point-

wise nonlinearities. We deﬁne a manifold convolution operation

which is consistent with the discrete graph convolution by

discretizing in both space and time domains. To sum up, we focus

on the manifold model as the limit of large graphs and construct

MNNs, while we can still bring back graph neural networks by

the discretization of MNNs. We carry out experiments based on

point-cloud dataset to showcase the performance of our proposed

MNNs.

Index Terms—Manifold convolution, manifold neural networks,

geometric deep learning

I. INTRODUCTION

Convolutional neural networks (CNNs) have achieved impres-

sive success in a wide range of applications, including but not

limited to natural language processing [1], image denoising [2]

and video analysis [3]. Convolution operations are implemented

to capture the local information and features based on the

characteristics of the dataset. The remarkable success provides

the support that CNNs are recognized as powerful techniques

when processing traditional signals such as sound, image or

video, which all lie in the Euclidean domains. As we have

more access to larger scale data and stronger computing power,

increasing attention is being paid to processing data lying in

the non-Euclidean domains.

Many practical problems rely on non-Euclidean data. There is

the case, for example, detection and recommendation in social

networks [4], resource allocations over wireless networks [5],

point clouds for shape segmentation [6]. There have been works

that extend the CNN architecture to non-Euclidean domains

[7]–[9], which reproduce the success of CNNs in Euclidean

domains. Among these models, graphs are commonly used

to construct the underlying data structure, while the graph

size scales with the amount of data. In this work, we aim to

construct CNNs on this more general model – the manifold.

Supported by NSF CCF 1717120, Theorinet Simons and ARL DCIST

CRA under Grant W911NF-17-2-0181. Zhiyang and Alejandro are with

Department of Electrical and Systems Engineering, University of Pennsylvania,

Philadelphia, Pennsylvania, USA. Luana is with Simons-Berkeley Institute,

California, USA.

Graphs with well-deﬁned limits are shown to converge to a

manifold model [10], [11], which makes the manifold capable

of capturing properties for a series of graphs. The convolution

operation is not taken for granted in non-Euclidean domains

due to the lack of global parametrization and shift invariance.

We deﬁne a manifold convolution operation based on the heat

diffusion process controlled by the Laplace-Beltrami operator.

We construct a manifold convolutional ﬁlter to process manifold

signals. By cascading the layers consisting of manifold ﬁlter

banks and nonlinearities, we can deﬁne the manifold neural

networks (MNNs) as a deep learning framework on the

manifold. To motivate the practical implementations of our

proposed MNNs, we ﬁrst discretize the MNN in the space

domain by sampling points on the manifold. The proposed

MNN can be transferred to this discretized manifold as a

discretized MNN which converges to the underlying MNN

when the manifold signal is bandlimited. We further carry

out discretization in the time domain by sampling the ﬁlter

impluse function in discrete and ﬁnite time steps. In this way,

we can not only execute our proposed MNNs, but also recover

the graph convolutions and graph neural networks [7]. This

concludes our thought starting from a graph sequence to the

limit as a a manifold and back to the graphs. We ﬁnally verify

the performance of our proposed MNN with a point cloud

based model classiﬁcation problem.

Related works include neural networks built on graphons

[12], [13], which are limits of a sequence of dense graphs.

Different from manifolds, graphons only represent the limits

for graphs with unbounded degrees [14]. Stability of MNNs

have been studied considering the perturbations to the Laplace-

Beltrami operator [9], [15]. A general framework for algebraic

neural networks has been proposed for architectures uniﬁed

with commutative algebras [16].

The rest of the paper is organized as follows. We start

with some preliminary concepts and deﬁne the manifold

convolutions in Section II. We construct the MNNs based

on manifold ﬁlters in Section III. In Section IV, we implement

the discretization in space and time domains to make the MNNs

realizable which also bring back to graph convolutions. Our

proposed MNN is veriﬁed in a model classiﬁcation problem

in Section V. The conclusions are presented in Section VI.

II. MANIFOLD CONVOLUTION

A. Preliminary Deﬁnitions

In this paper, we consider a compact, smooth, and differen-

tiable

-dimensional submanifold

embedded in

. The

embedding induces a Riemannian structure [17] on

which

arXiv:2210.00376v1 [eess.SP] 1 Oct 2022

endows a measure

over the manifold. Manifold signals

supported on

are smooth scalar functions

f:M → R

We consider manifold signals in a Hilbert space in which we

deﬁne the inner product as

hf, giL2(M)=ZM

f(x)g(x)dµ(x)(1)

with the norm deﬁned as kfk2

L2(M)=hf, fiL2(M).

The manifold is locally Euclidean, which elicits intrinsic

gradient for differentiation as a local operator [11]. The

local Euclidean space around

x∈ M

containing all of

the vectors tangent to

is denoted as tangent space

TxM

. We use

to represent the disjoint union of all

tangent spaces on

. The intrinsic gradient can thus be

written as an operator

∇:L2(M)→L2(TM)

mapping

scalar functions to tangent vector functions on

. The

adjoint operator of intrinsic gradient is the intrinsic divergence

deﬁned as

div :L2(TM)→L2(M)

. Based on these two

differentiation operators, the Laplace-Beltrami (LB) operator

L:L2(M)→L2(M)

can be deﬁned as the intrinsic

divergence of the intrinsic gradient [18], formally as

Lf=−div ◦ ∇f=−∇ · ∇f. (2)

Similar to the Laplacian operator in Euclidean domains or the

Laplace matrix in graphs [19], the LB operator evaluates how

much the function value at point

differs from the average

function value of its neighborhood [11].

The LB operator provides a basis for expressing and solving

physical tasks by Partial Differential Equations (PDEs). One of

the remarkable applications is characterizing the heat diffusion

over manifolds by the heat equation

∂u(x, t)

∂t +Lu(x, t)=0, (3)

where

u(x, t)∈L2(M)

measures the temperature at

x∈ M

at time

t∈R+

. With initial condition given by

u(x, 0) = f(x)

the solution can be expressed as

u(x, t) = e−tLf(x), (4)

which provides an essential element to construct manifold

convolution in the following.

Due to the compactness of

, the LB operator

is self-

adjoint and positive-semideﬁnite. This means that

possesses

a real positive spectrum

{λi}∞

i=1

with the eigenvalues

λi

and

the corresponding eigenfunctions φisatisfying

Lφi=λiφi(5)

The eigenvalues are ordered as

0< λ1≤λ2≤. . .

According to Weyl’s law [20], we have

λi∝i2/d

for a

dimensional manifold. The orthonormal eigenfunctions

φi

form

a general eigenbasis of L2(M)in the intrinsic sense.

Since

is a total variation operator, the eigenvalues

λi

can be

interpreted as the canonical frequencies and the eigenfunctions

φias the canonical oscillation modes of M.

B. Manifold Convolutional Filters

Convolution operation is a powerful technique to give zero

state response to any input signal with the ﬁlter impulse re-

sponse of the system [21]. Similar to time signals are processed

with time convolutions and graph signals are processed by graph

convolutions [22], we deﬁne manifold convolution with a ﬁlter

impulse response ˜

hand manifold signal f.

Deﬁnition 1 (Manifold ﬁlter)

Let

h:R+→R

and let

f∈

L2(M)

be a manifold signal. The manifold ﬁlter with impulse

response ˜

h, denoted h, is given by

g(x)=(hf)(x) := Z∞

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

where

u(x, t)

is the solution of the heat equation

(3)

with

u(x, 0) = f(x)

. Substitute the solution

u(x, t)

with

(4)

, and

we can derive a parametric form of has

g(x) = (hf)(x) = Z∞

h(t)e−tLf(x)dt=h(L)f(x).(7)

Manifold ﬁlters are local spatial operators operating directly

on points on the manifold based on the LB operator. The

exponential term

e−tL

can be interpreted as a shift operator

like the time delay in a Linear-Time Invariant (LTI) ﬁlter [21]

and the graph shift in a Linear-Shift Invariant (LSI) graph

ﬁlter [22]. In fact, manifold ﬁlters can recover graph ﬁlters by

discretization, which we discuss thoroughly in Section IV.

The LB operator

possesses the eigendecomposition

{λi, φi}∞

i=1

. Eigenvalue

λi

can be interpreted as the canonical

frequency and the eigenfunction

φi

as the canonical oscillation

mode. By projecting a manifold signal

onto the eigenfunction,

we can write the frequency representation ˆ

fas

[ˆ

f]i=hf, φiiL2(M)=ZM

f(x)φi(x)dµ(x). (8)

Deﬁnition 2 (Bandlimited manifold signals)

A manifold

signal is deﬁned as

λM

-bandlimited with

λM>0

[ˆ

f]i= 0

for all isuch that λi> λM.

The spectrum and eigenbasis of the LB opertor help to

understand the frequency behavior of the manifold ﬁlter

h(L)

The frequency representation of manifold ﬁlter output

can

be similarly written as

[ˆg]i=ZMZ∞

h(t)e−tLf(x)dtφi(x)dµ(x). (9)

By substituting e−tLφi=e−tλiφi, we can get

[ˆg]i=Z∞

h(t)e−tλidt[ˆ

f]i. (10)

The function solely dependent on

λi

is deﬁned as the frequency

response of the ﬁlter h(L).

Deﬁnition 3 (Frequency response)

The frequency response

of the ﬁlter h(L)is given by

h(λ) = Z∞

h(t)e−tλdt,(11)

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ConvolutionalNeuralNetworksonManifolds:FromGraphsandBackZhiyangWangLuanaRuizAlejandroRibeiroAbstractGeometricdeeplearninghasgainedmuchattentioninrecentyearsduetomoreavailabledataacquiredfromnon-Euclideandomains.Someexamplesincludepointcloudsfor3Dmodelsandwirelesssensornetworksincommunications.Graph...

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Convolutional Neural Networks on Manifolds From Graphs and Back Zhiyang Wang Luana Ruiz Alejandro Ribeiro

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