# VolterraNet: A higher order convolutional network with group equivariance
for homogeneous manifolds

Monami Banerjee†, Rudrasis Chakraborty†, Jose Bouza, and Baba C. Vemuri  M.
Banerjee is with FaceBook, CA, R. Chakraborty is with University of
California, Berkeley, USA. † denotes equal contributions. J. Bouza and B. C.
Vemuri are with University of Florida, Gainesville, FL, USA.
E-mail: {monamie.b, rudrasischa<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Convolutional neural networks have been highly successful in image-based
learning tasks due to their translation equivariance property. Recent work has
generalized the traditional convolutional layer of a convolutional neural
network to non-Euclidean spaces and shown group equivariance of the
generalized convolution operation. In this paper, we present a novel higher
order Volterra convolutional neural network (VolterraNet) for data defined as
samples of functions on Riemannian homogeneous spaces. Analagous to the result
for traditional convolutions, we prove that the Volterra functional
convolutions are equivariant to the action of the isometry group admitted by
the Riemannian homogeneous spaces, and under some restrictions, any non-linear
equivariant function can be expressed as our homogeneous space Volterra
convolution, generalizing the non-linear shift equivariant characterization of
Volterra expansions in Euclidean space. We also prove that second order
functional convolution operations can be represented as cascaded convolutions
which leads to an efficient implementation. Beyond this, we also propose a
dilated VolterraNet model. These advances lead to large parameter reductions
relative to baseline non-Euclidean CNNs.

To demonstrate the efficacy of the VolterraNet performance, we present several
real data experiments involving classification tasks on spherical-MNIST,
atomic energy, Shrec17 data sets, and group testing on diffusion MRI data.
Performance comparisons to the state-of-the-art are also presented.

###### Index Terms:

Homogeneous spaces, Volterra Series, Convolutions, Geometric Deep Learning,
Equivariance

##  1 Introduction

CNNs were introduced in the 1990s by Lecun [1] and gained enormous popularity
in the past decade especially after the demonstration of the significant
success on Imagenet data by Krizhevsky et al. [2]. At the heart of CNNs
success is its ability to learn a rich class of features from data using a
combination of convolutions and nonlinear operations such as ReLU or softmax
functions. The success of CNNs however is achieved at the expense of a large
number of parameters that need to be learned and a computational burden in the
training time. It is well known now that a multi-layer perceptron can
approximate any function to the desired level of accuracy with a finite number
of neurons in the hidden layer. It is therefore natural to consider parameter
efficiency as one of the network design goals to strive for in a deep netowrk.
The higher order Volterra series can capture a richer class of features and
hence significantly reduce the total number of parameters while maintaining
comparable or better classification accuracy relative to the baseline models.

In computer vision and medical imaging, many applications deal with data
domains that are non-Euclidean. For instance, the n-sphere ($n\geq 2$), the
manifold of symmetric positive definite matrices, the Grassmannian, Stiefel
manifold, flag manifolds etc. Most of these manifolds belong to the class of
(Riemannian) homogeneous spaces (manifolds). Thus, our goals here are to  1.
Introduce a principled framework for defining CNNs on general homogeneous
Riemannian manifolds.  2. Introduce a novel higher order convolution layer
using Volterra theory [3] on homogeneous Riemannian manifolds which provides
significant parameter-efficiency improvements for non-Euclidean CNNs.  3.
Establish empirical evidence demonstrating the applicability of our
homogeneous Riemannian manifold CNNs and the performance boost provided by the
Volterra convolutions.

Much of the recent work in this problem domain has focused on generalizing
CNNs to homogeneous spaces by exploiting the weight sharing that the
symmetries of the underlying manifold allow. The 2-sphere is a particularly
important example. In the recent past, CNNs have been reported in literature
[4, 5, 6] which are designed to handle data that are samples of functions
defined on a 2-sphere and hence are equivariant to 3D rotations which are
members of the $\textsf{SO}(3)$ group. The spherical convolution111As has been
pointed out several times in the literature, the convolution operation in CNNs
is actually a correlation and not a convolution. Hence, in this paper, we will
use the term convolution and correlation interchangeably but always imply
correlation. network presented in [5, 7] is named Spherical CNN. Recently,
Kondor et al. in [8] proposed the Clebsch-Gordan net by replacing the repeated
forward and backward Fourier transform operations used in [5]. They showed
that by using the Clebsch-Gordan transform as the source of nonlinearity,
better performance can be achieved by avoiding the repetitive forward and
inverse Fourier transform operations. In [9], authors present polar
transformer networks, which are equivariant to rotations and scaling
transformations. By combining them with the spatial transformer [10], they
achieved the required equivariance to translations as well. Recently, the
equivariance of convolutions to more general classes of group actions has been
reported in literature [11], and later in [12]. In [7], Esteves et al. used
the correlation defined in [13] to propose an $\textsf{SO}(3)$ equivariant
operator and in turn define a spherical convolution. In this paper, we will
define correlations and the Volterra series on a homogeneous manifold and show
that the equivariance property holds for both.

Volterra kernels were first proposed in image classification literature in
[14, 15]. In [15], authors learn the kernels in a data driven fashion and
formulate the learning problem as a generalized eigenvalue problem. Volterra
theory of nonlinear systems was applied more than two decades ago to a single
hidden layer feed-forward neural network with a linear output layer and a
fully dynamic recurrent network in [16]. Most recent use of Volterra kernels
in deep networks was reported in [17], where, authors presented a single layer
of Volterra kernel based convolutions followed by conventional CNN layers.
They however did not explore equivariance properties of the network or
consider non-Euclidean input domains.

In this paper, we define a Volterra kernel to replace traditional convolution
kernels. We present a novel generalization of the convolution group-
equivariance property to higher order convolutions expressed using Volterra
theory of functional convolution on non-Euclidean domains, specifically, the
Riemannian homogenous spaces [18] referred to earlier. Most of these manifolds
are commonly encountered in mathematical formulations of various computer
vision tasks such as action recognition, covariance tracking etc., and in
medical imaging for example, in diffusion magnetic resonance imaging (dMRI),
elastography etc. By generalizing traditional CNNs in two possible ways,  1.
to cope with data domains that are non-Euclidean and  2. to higher order
convolutions expressed using Volterra series, we expect to extend the success
of CNNs in yet unexplored ways.

We begin with a significant extension of prior work by the authors of [19],
where the authors defined a correlation operation for homogeneous manifolds.
Specifically, our extension consists of a proof that not only is the
correlation operation group equivariant, but additionally any linear group
equivariant function can be written as a correlation on the manifold (Banerjee
et al. [19] only showed the first fact). We present experiments to demonstrate
better performance of the proposed VolterraNet on spherical-MINST and the
Shrec17 data with less number of parameters than previously shown in
literature for the Spherical-CNN and the Clebsch-Gordan net. We then present a
dilated convolution model based on the VolterraNet and demonstrate its
efficacy in group testing on diffusion magnetic resonance data acquired from
patients with movement disorders. The domain of this data is another example
of a Riemannian homogeneous space.

In summary, our key contributions in this paper are:  1. A principled method
for choice of basis in designing a deep network architecture on a Riemannian
homogeneous manifold $\mathcal{M}$.  2. A proof of a generalization of the
classical linear shift invariance (in our terminology, equivariance)
characterization theorem for correlation operations on Riemannian homogeneous
manifolds.  3. A novel generalization of convolution operations to higher
order Volterra series on non-Euclidean domains specifically, Riemannian
homogeneous manifolds which are often encountered both in computer vision and
medical imaging applications.  4. A generalization of the classical non-linear
shift invariance (in our terminology, equivariance) characterization theorem
for Volterra convolution operations on Riemannian homogeneous manifolds.  5.
Experiments on real data sets that are publicly available such as the
spherical-MNIST, atomic energy and Shrec17. For these real data, we present
comparisons to the state-of-the-art methods.  6. An extension of the
VolterraNet to Dilated VolterraNet and demonstrate its efficiency via group
testing on diffusion MRI brain scans from controls (normal subjects) and
movement disorder patients. Further, ablation studies on VolterraNet to
demonstrate the usefulness of the higher order convolution operations.

The rest of the paper is organized as follows: In section 3.3 we define the
correlation operation on homogenous manifolds and prove a generalization of
the Euclidean linear shift invariance (LSI) theorem for this correlation
operation. Then, in section 3.4 we present a framework for principled choice
of basis in representing functions on a Riemmanian homogeneous manifold. In
section 4.1, we define the Volterra higher-order convolution operation and
prove a generalization of the non-linear shift invariance theorem for this
Volterra operation. Following this, we present a detailed description of the
proposed VolterraNet architecture in 5 and a description of the proposed
dilated VolterraNet in 6. Finally, section 7 contains the experimental results
and section 8 the conclusions.

##  2 List of Notations

We now summarize the list of notations that will be used throughout this
paper.

$\mathcal{M}$ | Riemannian homogeneous space (manifold)
---|---
$G$ | a group
$\textsf{SO}(n)$ | n-dimensional special orthogonal group
| of matrices
$I(\mathcal{M})$ | Isometry group admitted by $\mathcal{M}$
$L^{2}(\mathcal{M},\mathbf{R})$ | Space of real-valued square integrable
| functions on $\mathcal{M}$
$L^{2}(G,\mathbf{R})$ | Space of real-valued square integrable
| functions on $G$
$\omega^{\mathcal{M}}$ | Volume form of $\mathcal{M}$
$\mu_{G}$ | Haar measure of $G$
$g\cdot x/L_{g}(x)$ | Action of $g\in G$ on $x\in\mathcal{M}$
$gh/L_{g}(h)$ | Action of $g\in G$ on $h\in G$
$g\cdot f/L^{*}_{g^{-1}}(f)$ | Action of $g\in G$ on $f:\mathcal{M}\rightarrow\mathbf{R}$
| and is given by $x\mapsto f(g^{-1}\cdot x)$
$\mathbf{S}^{n}$ |  $n$-sphere
$\mathbf{R}^{+}$ | Space of positive reals
$P_{3}$ | Space of $3\times 3$ symmetric
| positive-definite matrices
$\textsf{GL}(n)$ | General linear group of $n\times n$ matrices
$\textsf{O}(n)$ | Space of $n\times n$ orthogonal matrices
$\mathbf{R}\setminus\left\\{0\right\\}$ | Space of reals without the origin
$\text{Stab}(x)$ | Stabilizer of an element $x\in\mathcal{M}$
$g^{\mathcal{M}}$ | Riemannian metric on the manifold $\mathcal{M}$
$logm$ | Matrix log operation

##  3 Correlation on Riemannian homogeneous spaces

In this section, we define a correlation operation which generalizes the
Euclidean convolution layer to arbitrary Riemannian homogeneous spaces.
Further, we prove a generalization to Riemannian homogeneous spaces of the
linear shift-invariant system (LSI) characterization of Euclidean
convolutions. Similar theorems were first proved in [11] and later in [12].
This result is not meant to be novel but to motivate the analogous result for
higher order convolutions on Riemannian homogeneous spaces that we prove
subsequently.

###  3.1 Background

We will briefly review the differential geometry of Riemannian homogeneous
spaces from an informal perspective. Formal definitions will be deferred to
the appendix for conceptual clarity.

As mentioned earlier, Riemannian homogeneous spaces are Riemannian manifolds
which ‘look’ the same locally at each point with respect to some symmetry
group $G$, meaning that the action of $G$ on $\mathcal{M}$ is transitive. We
will specifically consider Riemannian manifolds with a transitive action of
the isometry group $I(\mathcal{M})$. For the rest of the paper, we use
$G=I(\mathcal{M})$ unless mentioned otherwise. For example, the $2$-sphere is
a homogeneous space with $G=\textsf{SO}(3)$. An important fact about
homogeneous spaces is that they can be identified as a quotient space. In
general, if $\mathcal{M}$ is a homogeneous space with group $G$ acting on it,
and $H_{x}$ is some stabilizer (see definition in Appendix A) of of a point
$x\in\mathcal{M}$ then $\mathcal{M}\simeq G/H_{x}$. Returning to the
$2$-sphere example, if we take $H$ to be the stabilizer of the north pole, a
subgroup of $\textsf{SO}(3)$ isomorphic to $\textsf{SO}(2)$, then
$G/H\simeq\textsf{SO}(3)/\textsf{SO}(2)\simeq\mathbf{S}^{2}$. For a detailed
exposition on these concepts, we refer the reader to [18].

####  3.1.1 Assumptions

For the remainder of this paper we assume $\mathcal{M}$ to be a Riemannian
homogeneous space admitting a transitive action of the group $G$, which we
call the symmetries of $\mathcal{M}$. We also assume that $G$ is a locally
compact topological group. Further we assume that any function
$f:\mathcal{M}\to\mathbf{R}$ is square integrable, i.e.
$\int_{\mathcal{M}}|f(x)|^{2}\omega^{\mathcal{M}}(x)<\infty$, where
$\omega^{\mathcal{M}}$ is a suitable volume form on $\mathcal{M}$. As
mentioned before, we denote the space of square integrable functions on
$\mathcal{M}$ by $L^{2}(\mathcal{M},\mathbf{R})$.

###  3.2 Euclidean LSI Theorem

We begin this section by recalling the Euclidean Linear Shift Invariant (LSI)
theorem.

######  Definition 1.

Let $F:U\to V$ be a bounded linear operator between spaces $U$ and $V$
consisting of functions $\mathbf{R}^{n}\to\mathbf{R}$. For $f\in U\cup V$ and
$x\in\mathbf{R}^{n}$ we define $\tau_{x}(f)(z)=f(z-x)$. The set
$\\{\tau_{x}\\}_{x\in\mathbf{R}^{n}}$ forms a group under composition. We say
$F$ is translation equivariant (i.e. shift invariant in the traditional
literature) if

$\tau_{x}(F(g))=F(\tau_{x}(g))$

for all $g\in U$, $x\in\mathbf{R}^{n}$.

######  Theorem 1.

Let $w:\mathbf{R}^{n}\to\mathbf{R}$ be a weight kernel, then the operator
given as $G_{w}:U\to V$ is defined by $G_{w}(f)=f\star w$, where $\star$ is
the Euclidean convolution operation which is a bounded, linear, and
translation equivariant operator. Further, if $F$ is any bounded linear
translation equivariant operator, then there exists
$w:\mathbf{R}^{n}\to\mathbf{R}$ such that $F=G_{w}$, i.e. $F(f)=f\star w$, for
all $f\in U$.

Thus, Euclidean convolutions with a weight kernel have an interesting and
powerful characterization as linear shift invariant operators. Next we show
that the correlation operation on any Riemannian homogeneous manifolds satisfy
a generalization of the aforementioned LSI theorem.

###  3.3 Generalizing Convolutions and the LSI Theorem to Riemannian
homogeneous spaces

We begin by defining the correlation operation for arbitrary homogeneous
Riemannian manifolds. Some equivalent definitions have been made several times
in the literature, first for specific manifolds as in [5], [7], then in more
generality such as in [11] and later in [12]. We then state a generalization
of the LSI theorem for this correlation operation, which we call the Linear
Group Equivariant (LGE) theorem. Note that similar theorems were first proved
in [11] and later in [12]. We present this theorem not as a novel result, but
as motivation for a non-linear version of the theorem which we will prove in
section 4. Regardless, we present a much simpler proof (compared to [11],
[12]) of the result in the appendix.

######  Definition 2 (Correlation).

The correlation between $f:\mathcal{M}\to\mathbf{R}$ and
$w:\mathcal{M}\to\mathbf{R}$ is given by, $\left(f\star
w\right):G\rightarrow\mathbf{R}$ defined as follows:

$\displaystyle\left(f\star
w\right)\left(g\right):=\int_{\mathcal{M}}f(x)\left(g\cdot
w\right)(x)\omega^{\mathcal{M}}(x)$ (1)

The correlation between $f:G\to\mathbf{R}$ and $w:G\to\mathbf{R}$ is given by,
$\left(f\star w\right):G\rightarrow\mathbf{R}$ defined as follows:

$\displaystyle\left(f\star w\right)\left(g\right):=\int_{G}f(h)\left(g\cdot
w\right)(h)\mu_{G}(h)$ (2)

where $\mu_{G}$ is the Haar measure on $G$ (which is guaranteed to exist based
on our assumption that $G$ is a locally compact topological group). Please see
the discussion at the end of this subsection for details.

Equation 1 is described in words as follows: the weight kernel $w$ is
“shifted” using the action of the symmetry group, and the point-wise product
of the shifted weight kernel and the function $f$ is integrated over the
manifold. A similar interpretation can be given to the correlation on groups
in Eq. 2. This generalizes the work on the $2$-sphere presented in [5], [7]
for an arbitrary Riemannian homogeneous space $\mathcal{M}$.

We show that this correlation operation is equivariant to the isometry group
$G$ of the underlying homogeneous space $\mathcal{M}$. In order to state this
theorem, we first formally define equivariance.

######  Definition 3 (Equivariance).

Let $X$ and $Y$ be sets and $G$ be a group acting on $X$ and $Y$ (in
literature these sets are termed as $G$ sets [20]). Then, $F:X\rightarrow Y$
is said to be equivariant to the action of $G$ if

$\displaystyle F(g\cdot x)=g\cdot F(x)$ (3)

for all $g\in G$ and all $x\in X$.

We are now ready to state the following theorems:

######  Theorem 2.

Let $S$ and $U$ be $L^{2}(\mathcal{M},\mathbf{R})$ or $L^{2}(G,\mathbf{R})$
and $F:S\rightarrow U$ be a function given by $f\mapsto\left(f\star w\right)$.
Then $F$ is equivariant with respect to the pullback action of $G$, i.e.

$(\phi\cdot f)\star w=\phi\cdot(f\star w)$

for $\phi\in G$ a symmetry of $\mathcal{M}$ where

$\phi\cdot h\coloneqq h\circ\phi^{-1}$

for any $h:\mathcal{M}\to\mathbf{R}$ square-integrable.

###### Proof.

See appendix B. ∎

This constitutes the forward direction of the LGE theorem. Now we show the
converse statement, namely, every linear group equivariant function is a
correlation.

######  Theorem 3.

Let $S$ and $U$ be $L^{2}(\mathcal{M},\mathbf{R})$ or $L^{2}(G,\mathbf{R})$
and $F:S\rightarrow U$ be a linear equivariant function with respect to the
pullback action of $I(\mathcal{M})$. Then, $\exists w\in S$ such that,
$\left(F(f)\right)(g)=\left(f\star w\right)(g)$, for all $f\in S$ and $g\in
G$.

###### Proof.

See appendix B. ∎

Together with these two theorems, we can generalize the LSI theorem to
homogeneous spaces using the correlation defined in Def. 2 .

A note on volume forms / measures

In Def. 2, we specify the Haar measure for integration of a function on $G$.
If $f$ and $w$ are functions on $G$, then the Haar measure $\mu_{G}$ has
several desirable properties. For example, the Haar measure is invariant to
“translations”, i.e. if $S\subset G$ is measurable then
$\mu_{G}(S)=\mu_{G}(gS)$ for any $g\in G$. Further, using the Haar measure
provides a convolution theorem which makes correlation a simple multiplication
under the generalized Fourier transform for groups. This particular property
is vital for efficient implementations. Note that on the other hand, we do not
specify a specific volume form $\omega^{\mathcal{M}}$ for integration of
function on $\mathcal{M}$ in definition 2. In many cases, the Haar measure on
$G$ will induce a $G$-invariant volume form on $\mathcal{M}\simeq G/H$, but
stating the exact conditions for this to be possible requires some work.
Instead, we define the correlation using an arbitrary volume form. In the next
section we will give a construction which induces such a $G$-invariant volume
form on $\mathcal{M}$.

###  3.4 Basis functions for $L^{2}$-functions on homogeneous spaces

Our goal in this section is to induce a natural basis on
$L^{2}\left(\mathcal{M},\mathbf{R}\right)$ from the canonical basis on
$L^{2}\left(G,\mathbf{R}\right)$ where $G$ is the group acting on the
homogeneous manifold $\mathcal{M}$. The basis on $G$ consists of matrix
elements of irreducible unitary representations, which provides a Fourier
transform on $G$ (for more details reader is referred to [21]). We show that
this construction matches the commonly used basis for specific manifolds, e.g.
the spherical harmonics and Wigner-D functions in [5]. This construction can
be used to induce basis on arbitrary Riemannian homogeneous spaces.

####  3.4.1 Basis on $L^{2}\left(\mathcal{M},\mathbf{R}\right)$ induced from
$L^{2}\left(G,\mathbf{R}\right)$

To induce a basis on $L^{2}\left(\mathcal{M},\mathbf{R}\right)$, we use the
principal fiber bundle structure of the homogeneous manifold $\mathcal{M}$. A
fiber bundle is a space that locally looks like a product space. It is
expressed as a base space $B$ with the fibers making up a fiber space $F$, and
their union being the total space denoted by $E$. There is a projection map
$\pi:E\to B$ mapping fibers to their "base point" on $B$. A principal fiber
$G$-bundle is a fiber bundle with a continuous (right) action of a group $G$,
such that the action of $G$ is free, transitive and preserves the fibers. For
more details on fiber bundle theory see [22].

As mentioned in 3.1, $\mathcal{M}$ can be identified with $G/H$ for $G$ the
group action on $\mathcal{M}$ and $H$ the stabilizer of a point
$x\in\mathcal{M}$, usually called the "origin". It is well known that this
identification induces a principal fiber $G$-bundle structure on $\mathcal{M}$
via the projection map.

Figure 1: Fiber bundle $(B,E,\pi)$.

######  Proposition 1.

[18] The homogeneous space, $\mathcal{M}$ identified as $G/H$ together with
the projection map $\pi:G\rightarrow G/H$ is a principal bundle with $H$ as
the fiber. Furthermore there exists a diffeomorphism
$\psi:G/H\rightarrow\mathcal{M}$ given by $gH\mapsto g\cdot o$, where $o$ is
the “origin” of $\mathcal{M}$.

Moreover, a section is a continuous right inverse of $\pi$, which is denoted
by $\sigma:B\rightarrow E$. In literature [18], a zero section (denoted by
$S\subset G$) is the section containing the identity element of $H$. Let
$\sigma_{0}:S\rightarrow\mathcal{M}$ be a diffeomorphism. Given
$\left\\{v_{\alpha}:G\rightarrow\mathbf{R}\right\\}$ be the set of basis of
$L^{2}\left(G,\mathbf{R}\right)$. Then, we can get the induced basis on
$L^{2}\left(\mathcal{M},\mathbf{R}\right)$ as
$\left\\{\widetilde{v}_{\alpha}=v_{\alpha}\circ\sigma_{0}^{-1}\right\\}$. A
schematic of an example fiber bundle is shown in Fig. 1.

Example: Consider the example of $\mathcal{M}$ $=\mathbf{S}^{2}$, where $G=$
$\textsf{SO}(3)$, $H=\textsf{SO}(2)$. A choice of basis on
$L^{2}\left(G,\mathbf{R}\right)$ is Wigner D-functions denoted by
$\left\\{D_{l,m}^{j}|j\in\\{0,1,\cdots,\infty\\},-j\leq l,m\leq j\right\\}$.
Let $(\alpha,\beta,\gamma)$ be the parametrization of $\textsf{SO}(3)$ and the
zero section ($S$) be denoted by $\\{\alpha,\beta,0\\}$. Then,
$\left\\{D_{l,0}^{j}\right\\}$ are the choice of basis on $S\subset G$, which
gives the induced basis on $\mathbf{S}^{2}$ as
$\widetilde{D}_{l}^{j}(\theta,\phi)=\sqrt{\frac{2l+1}{4\pi}}D_{l,0}^{j}(\phi,\theta,0)$.
Further, observe that $\left\\{\widetilde{D}_{l}^{j}\right\\}$ are the
spherical harmonics basis.

##  4 Higher order correlation on Riemannian homogeneous spaces

In this section, we define a higher order correlation operator on Riemannian
homogeneous spaces using the Volterra series, state a theorem demonstrating
it’s symmetry equivariance and show how to compute it efficiently using first
order correlation operations. Further, we prove that the set of functions
which can be written as sums of products of linear operators and are
$G$-equivariant can be expressed as a Volterra series. This partially
generalizes the non-linear shift equivariance characterization of Volterra
expansions in Euclidean space.

###  4.1 Volterra Series on Homogenous Spaces

We now generalize the Volterra Series to Riemannian homogeneous spaces.

######  Definition 4 (Volterra series expansion).

We define the Volterra expansion of a function
$f:\mathcal{M}\rightarrow\mathbf{R}$ or $f:G\rightarrow\mathbf{R}$ by
$F(f)=\sum_{n=1}^{\infty}\left(f\star_{n}w_{n}\right)$. If
$f:\mathcal{M}\to\mathbf{R}$ and $w_{n}:\left(\mathcal{M}\right)^{\oplus
n}\rightarrow\mathbf{R}$ then
$\left(f\star_{n}w_{n}\right):G\rightarrow\mathbf{R}$ is defined as,

$\displaystyle\left(f\star_{n}w_{n}\right)(g)$
$\displaystyle:=\int_{\mathcal{M}}\cdots\int_{\mathcal{M}}f(x_{1})\cdots
f(x_{n})\left(g\cdot w_{n}\right)$
$\displaystyle(x_{1},\cdots,x_{n})\omega^{\mathcal{M}}(x_{1})\cdots\omega^{\mathcal{M}}(x_{n})$

If instead $f:G\to\mathbf{R}$ and $w_{n}:\left(G\right)^{\oplus
n}\rightarrow\mathbf{R}$ then
$\left(f\star_{n}w_{n}\right):G\rightarrow\mathbf{R}$ is defined as,

$\displaystyle\left(f\star_{n}w_{n}\right)(g)$
$\displaystyle:=\int_{G}\cdots\int_{G}f(h_{1})\cdots f(h_{n})\left(g\cdot
w_{n}\right)$
$\displaystyle(h_{1},\cdots,h_{n})\mu_{G}(h_{1})\cdots\mu_{G}(h_{n})$

where $\mu_{G}$ is the Haar measure on $G$ (which again, is guaranteed to
exist based on our assumption that $G$ is a locally compact topological
group).

One can easily see that, Definition 2 is a special case of definition 4 when
$n=1$. When $n>1$, we will call it the $n^{th}$ order Volterra expansion.
Higher order terms of the Volterra expansion express polynomial relationships
between function values. An illustration of the second order Volterra kernel
is provided in Figure 2. As we can see, the second order Volterra kernel has a
regular correlation weight kernel at each location on the manifold
$\mathcal{M}$. The results of applying these weight kernels get multiplied
together to get the output of $f\star_{2}w_{2}$. A biological motivation is
provided in [17] for the (Euclidean) Volterra series. Now, we prove that $F$
as defined in Definition 4 is equivariant to the symmetry group actions
admitted by a homogeneous space.

######  Theorem 4.

Let $S$ and $U$ be $L^{2}(\mathcal{M},\mathbf{R})$ or $L^{2}(G,\mathbf{R})$
and $F:S\rightarrow U$ be a function given by
$f\mapsto\sum_{n=1}^{\infty}\left(f\star_{n}w_{n}\right)$. Then, $F$ is
equivariant.

###### Proof.

Observe that the sum of equivariant operators is equivariant. Hence, we only
need to check that $f\star_{n}w_{n}$ is equivariant for all $n$. Let $g,h\in
G$, let $n\in\mathbf{N}$. Then,

$\displaystyle\left(g.f\star_{n}w_{n}\right)\left(h\right)$
$\displaystyle=\left(L^{*}_{g^{-1}}f\star_{n}w_{n}\right)\left(h\right)$
$\displaystyle=\int_{\mathcal{M}}\cdots\int_{\mathcal{M}}L^{*}_{g^{-1}}f(x_{1})\cdots
L^{*}_{g^{-1}}f(x_{n})$
$\displaystyle\left(L^{*}_{h^{-1}}w_{n}\right)(x_{1},\cdots,x_{n})\omega^{\mathcal{M}}(x_{1})\cdots\omega^{\mathcal{M}}(x_{n})$
$\displaystyle=\int_{\mathcal{M}}\cdots\int_{\mathcal{M}}f(y_{1})\cdots
f(y_{n})w_{n}\left((h^{-1}g)\cdot y_{1}\right.$
$\displaystyle\left.,\cdots,(h^{-1}g)\cdot
y_{n}\right)\omega^{\mathcal{M}}(g\cdot
y_{1})\cdots\omega^{\mathcal{M}}(g\cdot y_{n})$
$\displaystyle=\int_{\mathcal{M}}\cdots\int_{\mathcal{M}}f(y_{1})\cdots
f(y_{n})w_{n}\left((h^{-1}g)\cdot y_{1}\right.$
$\displaystyle\left.,\cdots,(h^{-1}g)\cdot
y_{n}\right)\omega^{\mathcal{M}}(y_{1})\cdots\omega^{\mathcal{M}}(y_{n})$
$\displaystyle=\left(f\star_{n}w_{n}\right)(g^{-1}h)$
$\displaystyle=L^{*}_{g^{-1}}\left(f\star_{n}w_{n}\right)\left(h\right)$
$\displaystyle=\left(g\cdot\left(f\star_{n}w_{n}\right)\right)\left(h\right)$

Here, $(L_{g}^{*}f)(h)=f(g^{-1}h)$ (see appendix for details), since, $g,h\in
G$ and $n$ are arbitrary $F$ is equivariant. ∎

In the other direction, we also show that for the aforementioned set of
functions, every $G$-equivariant function can be written as a Volterra series.

######  Theorem 5.

Let $S$ and $U$ be $L^{2}(\mathcal{M},\mathbf{R})$ or $L^{2}(G,\mathbf{R})$
and $F:S\rightarrow U$ be a non-linear $G$-equivariant function which can be
written as $F=\sum_{i\in I}F_{i}$, where each $F_{i}$ is a product of two
linear functions, i.e., $F_{i}=F_{i,1}F_{i,2}$. Then,
$\exists\left\\{w_{i}\right\\}_{i\in I}\subset S$ such that,
$\left(F(f)\right)(g)=\sum_{i\in I}\left(f\star_{2}w_{i}\right)(g)$, for all
$f\in S$ and $g\in G$.

###### Proof.

It suffices to show that for each term $F_{i}=F_{i,1}F_{i,2}$ (for $F_{i,k}$ a
linear $G$-equivariant function) there exists $w_{i}$ such that
$F_{i}=f\star_{2}w_{i}$. If $w_{i}(x,y)=w_{i,1}(x)w_{i,2}(y)$ (i.e. $w_{i}$ is
separable), then $f\star_{2}w_{i}=(f\star w_{i,1})(f\star w_{i,2})$. But by
the previous theorem, there exists $w_{i,k}$ such that $F_{i,k}=f\star
w_{i,k}$, completing the proof. ∎

These results partially generalize the well know non-linear shift equivariance
characterization of Volterra expansions in Euclidean space and justifies the
use of the Volterra series as a higher-order generalization of the correlation
operation Definition 2.

$(0,0)$$(0,1)$$(0,2)$$(1,0)$$(1,1)$$(1,2)$$(2,0)$$(2,1)$$(2,2)$$\theta$$\phi$$0$$0.2$$0.4$$0.6$$0.8$$1$
Figure 2:  Visualization (in the spirit of [17]) of second-order term
$w_{2}:\mathcal{M}^{2}\to\mathbf{R}$ of a Volterra kernel (here on a
$2$-manifold parametrized by $(\theta,\phi)$). The coordinates above each grid
represent the first entry $w_{2}(\mathbf{x},\cdot)$, and within each grid the
gray-scale value represents the weight of the associated kernel
$w_{2}(\mathbf{x},\mathbf{y})$.

###  4.2 Efficient Computation of the Second-Order Volterra Kernel

The Volterra series presented in the previous definition is significantly more
expressive than the correlation operation defined in Definition 2 since it
captures higher order relationships between inputs, but it requires the
computation of iterated integrals and does not have an efficient GPU
implementation. Note that for separable second order kernel $w_{2}$,
$\left(f\star_{2}w_{2}\right)(g)$ can be factored as
$\left(\left(f\star\tilde{w}_{2}\right)\left(g\right)\right)\left(\left(f\star\bar{w}_{2}\right)\left(g\right)\right)$.
Thus, we can compute the second order Volterra series with separable kernel as
a product of traditional correlation operations. In general, we can use a
convex combination of first order and second order terms of the Volterra
series to define second order Volterra network.

Figure 3: Second order Volterra correlation operator with first order kernel
$w_{1}$ and separable second order kernel $w_{2}$.

A schematic diagram for the second order Volterra correlation operator is
shown in Fig. 3.  This representation of a second order kernel using product
of two separable kernels is analogous to tensor product approximation of a
function and can be shown to achieve approximation error of an arbitrary
precision [23]. The separability assumption on the kernels leads to efficient
computation which is especially valuable in the network setting where these
operations are performed numerous times.

##  5 Architecture

We now present the basic modules for implementing our correlation and higher-
order Volterra operations as layers in a deep network.

###  5.1 Correlation on homogeneous spaces

Using definition 2 we can define:

Correlation on $\mathcal{M}$ \- $\textsf{Corr}^{\mathcal{M}}(f,w)$: Let $f\in
L^{2}\left(\mathcal{M},\mathbf{R}\right)$ be the input function and $w\in
L^{2}\left(\mathcal{M},\mathbf{R}\right)$ be the mask. Then, using definition
2, $\textsf{Corr}^{\mathcal{M}}(f,w)$ is defined as $\left(f\star
w\right):G\rightarrow\mathbf{R}$. We have shown in Theorem 2, that
$\textsf{Corr}^{\mathcal{M}}(f,w)$ is equivariant to the action of $G$. Hence,
we can use $\textsf{Corr}^{G}$ layer as the next layer.

Correlation on $G$ \- $\textsf{Corr}^{G}(f,w)$: Let $\widetilde{f}\in
L^{2}\left(G,\mathbf{R}\right)$ be the input function and $w\in
L^{2}\left(G,\mathbf{R}\right)$ be the mask. Then analogous to
$\textsf{Corr}^{\mathcal{M}}$, we can define $\textsf{Corr}^{G}(f,w)$ as
$\left(\widetilde{f}\star w\right):G\rightarrow\mathbf{R}$ using definition 2.
We have used Theorem 2 to show that $\textsf{Corr}^{G}(f,w)$ is equivariant to
the action of $G$. Since this is an operation equivariant to $G$, we can
cascade $\textsf{Corr}^{G}$.

###  5.2 Volterra on homogeneous spaces

We can see that because the basic architecture of second order Volterra series
consists of the following modules:

Second order Volterra on $\mathcal{M}$ \-
$\textsf{Corr}_{2}^{\mathcal{M}}(f,w_{1},w_{2})$: Let $f\in
L^{2}\left(\mathcal{M},\mathbf{R}\right)$ be the input function and
$w_{1}:\mathcal{M}\rightarrow\mathbf{R}$ and
$w_{2}:\left(\mathcal{M}\right)^{\oplus 2}\rightarrow\mathbf{R}$ be the
kernels. Then,
$\textsf{Corr}_{2}^{\mathcal{M}}(f,w_{1},w_{2}):=\sum_{j=1}^{2}\left(f\star_{j}w_{j}\right):G\rightarrow\mathbf{R}$.
We have shown in Theorem 4, that
$\textsf{Corr}_{2}^{\mathcal{M}}(f,w_{1},w_{2})$ is equivariant to the action
of $G$. Hence, we can use $\textsf{Corr}_{2}^{G}(f,w_{1},w_{2})$ layer as the
next layer.

Second order Volterra on $G$ \- $\textsf{Corr}_{2}^{G}(f,w_{1},w_{2})$: Let
$f\in L^{2}\left(G,\mathbf{R}\right)$ be the input function and
$w_{1}:G\rightarrow\mathbf{R}$ and $w_{2}:\left(G\right)^{\oplus
2}\rightarrow\mathbf{R}$ be the kernels. Then,
$\textsf{Corr}_{2}^{G}(f,w_{1},w_{2}):=\sum_{j=1}^{2}\left(f\star_{j}w_{j}\right):G\rightarrow\mathbf{R}$.
We have used Theorem 4 to show that $\textsf{Corr}_{2}^{G}(f,w_{1},w_{2})$ is
equivariant to the action of $G$. Since this is an operation equivariant to
$G$, we can cascade $\textsf{Corr}_{2}^{G}(f,w_{1},w_{2})$.

###  5.3 Other Layers

Activation function: Since the outputs of all the above layers are functions
from $G$ to $\mathbf{R}$, we will use the standard activation operation on
$\mathbf{R}$.

Invariant last layer: As both layers, $\textsf{Corr}_{2}^{\mathcal{M}}$ and
$\textsf{Corr}_{2}^{G}$ are equivariant to the action of $G$, so are the
cascaded layers. Since, if the input signal is transformed by a group element
$g\in G$, so is the output of $\textsf{Corr}_{2}^{\mathcal{M}}$ as this layer
is equivariant. Thus the output of $\textsf{Corr}_{2}^{\mathcal{M}}$ is
transformed by the same group element $g$. Hence, the input of
$\textsf{Corr}_{2}^{G}$ is transformed by $g$ and due to the equivariance so
is the output of $\textsf{Corr}_{2}^{G}$. This justifies that the cascaded
layers are equivariant to the action of $G$. Hence, after the cascaded
correlation layers, the output $\widetilde{f}\in
L^{2}\left(G,\mathbf{R}\right)$ lies on a $G$ set. Similar to the Euclidean
CNN, we want the last layer to be $G$ invariant. Hence, we will integrate
$\widetilde{f}$ on the domain $G$ and return a scalar. Note that in the
experiment, we learn multiple channels analogous to the Euclidean CNN, where
in each channel, we learn a $G$ equivariant $\widetilde{f}\in
L^{2}\left(G,\mathbf{R}\right)$. Thus, after the integration, we have $c$
scalars, where $c$ is the number of channels, which will be input to the
softmax fully connected layer similar to the Euclidean CNN. We abbreviate this
last layer as iL (invariant layer).

A schematic diagram of our proposed VolterraNet is shown in Fig. 4.

Figure 4: Schematic diagram of a second order Volterranet

##  6 Dilated VolterraNet

In this section, we propose a dilated VolterraNet framework which is suitable
for sequential data. Sequential data here refers to a sequence of data points
(signal measurements at voxels) along the neuronal fiber tracts that are
extracted from diffusion MRI data sets. Neuronal fiber tracts in certain
regions of the brain are disrupted by movement disorders such as Parkinsons
disease. The sensory motor area tract (pathway) in the brain is one such
neuronal pathway where the disease caused changes are expected to be observed.
By treating this pathway as a sequence of points where diffusion sensitized MR
signal is acquired, we propose to apply the dilated VolterraNet (described
below) to analyze this data.

It is known that the sequential data should involve recurrent structure [24],
but as pointed out in [25], convolutional architectures often outperform
recurrent models in sequential data analysis. Furthermore, recurrent models
are computationally more expensive than the convolutional models. But, note
that in order to mimic the infinite memory capabilities of a recurrent model,
one needs to increase the receptive field by using the dilated convolutions.
We will first recap the definition of Euclidean dilated convolution [25] and
then describe the proposed dilated VolterraNet.

###  6.1 Euclidean Dilated Convolution:

Given a one-dimensional input sequence
$\mathbf{x}:\mathbf{N}\rightarrow\mathbf{R}^{n}$ and a kernel
$w:\left\\{0,\cdots,k-1\right\\}\rightarrow\mathbf{R}$, the dilated
convolution function
$\left(\mathbf{x}\star_{d}w\right):\mathbf{N}\rightarrow\mathbf{R}^{n}$ is
defined as,
$\left(\mathbf{x}\star_{d}w\right)(s)=\sum_{i=0}^{k-1}w(i)\mathbf{x}(s-d\times
i),$ where $\mathbf{N}$ is the set of natural numbers and $k$ and $d$ are the
kernel size and the dilation factor respectively. Note that with $d=1$, we get
the normal convolution operator. In a dilated CNN, the receptive field size
will depend on the depth of the network as well as on the choice of $k$ and
$d$.

###  6.2 Dilated VolterraNet

Now we present a dilated VolterraNet model by combining the VolterraNet with
the dilated CNN model. Given a one-dimensional input sequence
$\left\\{f_{i}:\mathcal{M}\rightarrow\mathbf{R}\right\\}$, we will first apply
$\textsf{Corr}_{2}^{\mathcal{M}}$ and cascaded $\textsf{Corr}_{2}^{G}$ layers
to each point in the sequence independently. The output of a
$\textsf{Corr}_{2}^{G}$ layer is a function $G\rightarrow\mathbf{R}$. Let the
output of the last $\textsf{Corr}_{2}^{G}$ layer be
$\left\\{g_{i}:G\rightarrow\mathbf{R}\right\\}$. Then, we discretize the group
$G$, to represent each $g_{i}$ by a vector $\mathbf{x}_{i}$ (as shown in Fig.
5).  The steps of discretization, i.e., length of $\mathbf{x}_{i}$, are chosen
via grid search in the experimental section. This is analogous to the standard
practice in literature [5, 7]. Polar coordinates on $G$ are used to discretize
$G$ and then we use the dilated CNN by treating each sample as a vector. This
essentially amounts to choosing a uniform grid in the parameter space using
Rodrigues vectors [26], although more sophisticated techniques can be employed
in this context [27]. Now, we input $\left\\{\mathbf{g}_{i}\right\\}$ to the
Euclidean dilated CNN (since the components of $\mathbf{g}_{i}$ are real) to
construct a dilated VolterraNet framework. In Fig. 5, we present a schematic
of dilated VolterraNet with input $\left\\{f_{i}\right\\}$ followed by
$\textsf{Corr}_{2}^{\mathcal{M}}$ $\rightarrow$ReLU $\rightarrow$
$\textsf{Corr}_{2}^{G}$ $\rightarrow$ ReLU $\rightarrow$
$\textsf{Corr}_{2}^{G}$.

A self explanatory schematic diagram of the dilated VolterraNet architecture
is shown in Fig. 5.

Figure 5: Schematic diagram of dilated VolterraNet

##  7 Experiments

In this section, we present experiments on spherical MNIST, atomic energy and
Shrec17 data sets respectively. We present comparisons of performance of our
VolterraNet to Spherical CNN by Cohen et al. [5] and Clebsch-Gordan net by
Kondor et al. [8]. Further, we also present a separate comparison with the
spherical CNN presented most recently in [7] on the shrec17 data set. The
separate comparison was necessary due to the fact that the loss function used
in [7] was distinct from the one used in [5, 8]. Finally, we extend the
VolterraNet to a dilated version, a higher order analogue of the dilated CNN
and use it to demonstrate its efficacy in group testing on diffusion MRI
(dMRI) data acquired from movement disorder patients. The data in this example
reside in a product space, $\mathbf{S}^{2}\times\mathbf{R}^{+}$, which is a
Riemannian homogeneous space distinct from $\mathbf{S}^{2}$. This experiment
serves as an example demonstrating the ability of VolterraNet to cope with
manifolds other than the sphere.

Choice of basis: In our experiments, we have three examples of manifolds,
$\mathbf{S}^{2}$, $\mathbf{S}^{2}\times\mathbf{R}^{+}$ and $P_{3}$. For
$\textsf{SO}(3)$, we use the Weigner basis and for
$\mathbf{S}^{2}\simeq\textsf{SO}(3)/\textsf{SO}(2)$, we use the induced basis,
i.e., the Spherical Harmonics basis. For $\mathbf{S}^{2}\times\mathbf{R}^{+}$,
we use the product basis of each of the spaces, i.e., Spherical Harmonics for
$\mathbf{S}^{2}$ and the canonical basis for $\mathbf{R}^{+}$. Since $P_{3}$
can be written as $\textsf{GL}(3)/\textsf{O}(3)$ , we use the induced basis on
$P_{3}$ which are induced from the canonical basis on $\textsf{GL}(3)$.

In all the experiments we compare the VolterraNet architecture to the state of
the art models, and additionally compare it to an architecture which we call
Homogeneous CNN (HCNN) which replaces the Volterra non-linear
convolutions/correlations with the correlation operations from definition 2.
We have released an implementation of the VolterraNet architecture with the
Spherical MNIST experiment which can be found at,
https://github.com/cvgmi/volterra-net.

###  7.1 Synthetic Data Experiment: Classification of data on $P_{3}$

In this section, we first describe the process of synthesizing functions
$f:P_{3}\rightarrow\left[0,1\right]$. In this experiment, we generated data
samples drawn from distinct Gaussian distributions defined on $P_{3}$ [28].
Let $\mathcal{X}$ be a $P_{3}$ valued random variable that follows
$\mathcal{N}\left(M,\sigma\right)$, then, the p.d.f. of $X$ is given by [28]:

$\displaystyle
f_{\mathcal{X}}\left(X;M,\sigma\right)=\frac{1}{C(\sigma)}\exp\left(-\frac{d^{2}(M,X)}{2\sigma^{2}}\right),$
(4)

where, $d(.,.)$ is the affine invariant geodesic distance on $P_{3}$ as given
by
$d(M,X)=\sqrt{\text{trace}\left(\left(\text{logm}\left(M^{-1}X\right)\right)^{2}\right)}$.

We first chose two sufficiently spaced apart location parameters $M_{1}$ and
$M_{2}$ and then for the $i^{th}$ class we generate Gaussian distributions
with location parameters that are perturbations of $M_{i}$ and with variance
$1$. This gives us two clusters in the space of Gaussian densities on $P_{3}$,
which we will classify using HCNN and VolterraNet. In this case, the HCNN
network architecture is given by: $\textsf{Corr}^{P_{3}}$ $\rightarrow$ ReLU
$\rightarrow$ $\textsf{Corr}^{\textsf{GL}(3)}$ $\rightarrow$ ReLU
$\rightarrow$ $\textsf{Corr}^{\textsf{GL}(3)}$ $\rightarrow$ ReLU
$\rightarrow$ iL $\rightarrow$ FC. and for VolterraNet the correlation
operations are replaced with the corresponding Volterra convolutions.

Model | mean acc. | std. acc.
---|---|---
VolterraNet | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>
HCNN | $86.50$ | ${0.02}$
TABLE I: Comparative mean and stdev. on the synthetic data

The data consists of $500$ samples from each class, where each sample is drawn
from a Gaussian distribution on $P_{3}$. The classification accuracies in a
ten-fold partition of the data are shown in Table I. In most deep learning
applications, one is used to seeing a high classification accuracy, but we
believe that this can be achieved here as well by increasing the number of
layers and possibly overfitting the data. The purpose of this synthetic
experiment was not to seek an “optimal” classification accuracy but to provide
a flexible framework which if “optimally” tuned can yield a good testing
accuracy for data whose domain is a non-compact Riemannian homogeneous space.

###  7.2 Spherical MNIST Data Experiment

The spherical MNIST data are generated using the scheme described in [5].
There are two instances of this data, one in which we project MNIST digits on
the northern hemisphere (denoted by ‘NR’) and the other where we apply random
rotation afterwards (denoted by ‘R’). The spherical signal is discretized
using a bandwidth of 60.

We selected the same baseline model as was chosen in [5], which is a Euclidean
CNN with $5\times 5$ filters and $32,64,10$ channels with a stride of $3$ in
each layer. This CNN is trained by mapping the digits from the northern
hemisphere onto the plane. The Spherical CNN model [5] we used has the
following architecture (as was reported in [5]),
$\textsf{Corr}^{\mathbf{S}^{2}}$ $\,\to\,$ ReLU $\,\to\,$
$\textsf{Corr}^{\textsf{SO}(3)}$ $\,\to\,$ ReLU $\,\to\,$ FC with bandwidths
$20$, $12$ and the number of channels $20$, $40$ respectively. We used the
same architecture for Clebsch-Gordan net as was reported in [8].

For our method, we used a second order Volterra network with the following
architecture: $\textsf{Corr}_{2}^{\mathbf{S}^{2}}$ $\,\to\,$ ReLU $\,\to\,$ iL
$\,\to\,$ FC with bandwidth $30$, $20$ respectively and number of features
$25$, $10$ respectively. We chose a batchsize of $32$ and learning rate of
$5\times 10^{-3}$ with ADAM optimization [29].

We performed three sets of experiments: non rotated training and test sets
(denoted by ‘NR/NR’), non rotated training and randomly rotated test sets
(denoted by ‘NR/R’) and randomly rotated both training and test sets (denoted
by ‘R/R’). The comparative results in terms of classification accuracy are
shown in Table II.

Method | NR/NR | NR/R | R/R | # params.
---|---|---|---|---
Baseline CNN | 97.67 | 22.18 | 12.00 | 68000
Spherical CNN [5] | 95.59 | 94.62 | 93.40 | 58550
Clebsch-Gordan net[8] | 96.00 | 95.86 | 95.80 | 342086
VolterraNet | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{46010}}$
TABLE II: Comparison of classification accuracy on Spherical MNIST data

We can see that the VolterraNet performed better than all the three competing
networks for both the ‘R/R’ and ’NR/R’ cases. Note that in terms of number of
parameters, VolterraNet used 46010, while Spherical CNN used 58550 and
Clebsch-Gordan net used 342086. The baseline CNN used 68000 parameters. Thus
in comparison, we have approximately an $86\%$ reduction in parameters over
the Clebsch-Gordan net with almost equal or better classification accuracy. In
comparison to the Spherical CNN, we have approximately a $21\%$ reduction in
the parameters over the Spherical CNN while achieving significantly better
performance. This clearly depicts the usefulness of our proposed VolterraNet
in comparison to existing networks used in processing this type of data in a
non-Euclidean domain.

###  7.3 3D Shape Recognition Experiment

We now report results for shape classification using the Shrec17 dataset [30]
which consists of $51300$ 3D models spread over $55$ classes. This dataset is
divided into a $70/10/20$ split for train/validation/test. Following the
method in [5], we perturbed the dataset using random rotations. We processed
the dataset as in [5]. Basically, we represented each 3D model by a spherical
signal using a ray casting scheme. For each point on the sphere, a ray towards
the origin is sent which collects the ray length, cosine and sine of the
surface angle. Additionally, the convex hull of the 3D shape gives 3 more
channels, which results in 6 input channels. The spherical signal is
discretized using Discoll-Healy grid [13] grid with a bandwidth of 128.

The Spherical CNN model [5] we used has the following architecture (as was
reported in [5]): $\textsf{Corr}^{\mathbf{S}^{2}}$ $\,\to\,$ BN $\,\to\,$ ReLU
$\,\to\,$ $\textsf{Corr}^{\textsf{SO}(3)}$ $\,\to\,$ BN $\,\to\,$ ReLU
$\,\to\,$ $\textsf{Corr}^{\textsf{SO}(3)}$ $\,\to\,$ BN $\,\to\,$ ReLU
$\,\to\,$ FC with bandwidths $32$, $22$ and $7$ and the number of channels
$50$, $70$ and $350$ respectively. We used the same architecture for Clebsch-
Gordan net as was reported in [8].

In our method, we used a second order Volterra network with the following
architecture: $\textsf{Corr}^{\mathbf{S}^{2}}$ $\,\to\,$ BN $\,\to\,$ ReLU
$\,\to\,$ $\textsf{Corr}_{2}^{\textsf{SO}(3)}$ $\,\to\,$ BN $\,\to\,$ ReLU
$\,\to\,$ iL $\,\to\,$ FC with bandwidths $10$, $8$, $8$ respectively and
number of features $60$, $80$, $100$ respectively. We chose a batch size of
$100$ and a learning rate of $5\times 10^{-3}$ with ADAM optimization [29].
Table III summarizes comparison of VolterraNet with other existing deep
network architectures that reported results on this data in literature. From
this table, it is evident that VolterraNet almost always yields classification
accuracy results within the top three methods, while having the best parameter
efficiency.

Method | P@N | R@N | F1@N | mAP | NDCG | # params.
---|---|---|---|---|---|---
Tasuma_ReVGG [31] | 0.70 | 0.76 | 0.72 | 0.69 | 0.78 | 3M
Furuya_DLAN [32] | 0.81 | 0.68 | 0.71 | 0.65 | 0.75 | 8.4M
SHREC16-bai_GIFT [33] | 0.68 | 0.66 | 0.66 | 0.60 | 0.73 | 36M
Deng_CM-VGG5-6DB | 0.41 | 0.70 | 0.47 | 0.52 | 0.62 | -
Spherical CNN [5] | 0.70 | 0.71 | 0.69 | 0.67 | 0.76 | 1.4Mil
Clebsch-Gordan net[8] | 0.70 | 0.72 | 0.70 | 0.68 | 0.76 | -
Ours (VolterraNet) |  ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(2nd) |  ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(3rd) |  ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(3rd) |  ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(3rd) |  ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(4th) | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{396297}}$
[7] (w triplet loss) | 0.72 | 0.74 | 0.69 | - | - | 0.5M
Ours (w triplet loss) | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{396297}}$
TABLE III: Comparison results in terms of classification accuracy on the
shrec17 data

Comparison with Esteves et al. [7]: We also compared our VolterraNet with
recent work of Esteves et al. [7] using an extra in-batch triplet loss [34]
(as used in Esteves et al. [7]). We show the comparison results in Table III
(last two rows), which clearly shows that,  (a) The VolterraNet outperforms
the network in [7] (which is the state-of-the-art algorithm in terms of
parameter efficiency).  (b) The triplet loss boosts the performance of
VolterraNet relative to the baseline loss of cross entropy.

###  7.4 Regression Experiment: Prediction of Atomic Energy

Here, we report the application of our VolterraNet to the QM7 dataset [35,
36], where the goal is to regress over atomization energies of molecules given
atomic positions $(\mathbf{p}_{i})$ and charges $(\mathbf{z}_{i})$. Each
molecule consists of at most $23$ atoms and the molecules are of $5$ types (C,
N, O, S, H). We use the Coulomb Matrix (CM) representation proposed by[36],
which is rotation and translation invariant but not permutation invariant. We
used a similar experimental setup to that described in [5] for this regression
problem. We define a sphere $\mathbf{S}_{i}$ around $\mathbf{p}_{i}$ for each
$i^{th}$ atom. We define the potential functions

$\displaystyle U_{\mathbf{z}}(\mathbf{x})=\sum_{i\neq
j,\mathbf{z}_{j}=\mathbf{z}}\frac{\mathbf{z}_{i}^{t}\mathbf{z}}{\|\mathbf{x}-\mathbf{p}_{i}\|},$
(5)

for every $\mathbf{z}$ and for every $\mathbf{x}$ on the sphere
$\mathbf{S}_{i}$. This yields a spherical signal consisting of $5$ features
which were discretized using the Discoll-Healy grid [13] with a bandwidth of
20. For the VolterraNet, we used one $\mathbf{S}^{2}$ and $\textsf{SO}(3)$
second order Volterra block with bandwidths $12,8,8,4$ and number of features
$8,10,20,50$ respectively.

We compute the loss and report it in Table IV. We can see that VolterraNet
performs better than the competing methods. For Spherical CNN [5] and Clebsch-
Gordan net[8], we used similar architectures as described in the respective
papers. Spherical CNN [5] and Clebsch-Gordan net[8] use $1.4M$ and $1.1M$
parameters respectively, while the VolterraNet used $128460$, nearly an order
of magnitude reduction of parameters, while achieving the best classification
accuracy. This illustrates the parameter efficiency gains that we get from
using a higher order correlation, a richer feature, in the VolterraNet.

Method | MSE
---|---
MLP/ Random CM [37] | 5.96
LGIKA (RF) [38] | 10.82
RBF Kernels/ Random CM [37] | 11.42
RBF Kernels/ Sorted CM [37] | 12\. 59
MLP/ Sorted CM [37] | 16.06
Spherical CNN [5] | 8.47
Clebsch-Gordan net[8] | 7.97
Ours (VolterraNet) |  ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>(1st)
TABLE IV: Comparison results on atomic energy prediction

###  7.5 Network architecture for dMRI data Using Dilated VolterraNet

Diffusion MRI (dMRI) is an imaging modality that non-invasively measures the
diffusion of water molecules in tissue samples being imaged. It serves as an
interesting example of our framework since dMRI data can naturally be
described by functions on a Riemannian homogeneous space. In this section we
describe the dMRI data and its processing using the framework presented in
this paper, which will help the reader understand the results of the following
subsections.

In each voxel of a dMRI data set, the signal magnitude is represented by a
real number along each gradient magnetic field over a hemi-sphere of
directions in 3D. Hence, in each voxel, we have a function
$f:\mathbf{S}^{2}\times\mathbf{R}^{+}\rightarrow\mathbf{R}$. The proposed
network architecture has two components: intra-voxel layers and inter-voxel
layers. The intra-voxel layers extract features from each voxels, while the
inter-voxel layers use dilated convolution to capture the interaction between
extracted features. In our application in the next section we extract a
sequence of voxels lying along a nerve fiber bundle in the brain known to be
affected in Parkinson disease. Hence we have a sequence of functions along the
fiber bundle
$\\{f_{i}:\mathbf{S}^{2}\times\mathbf{R}^{+}\rightarrow\mathbf{R}\\}$, making
the application of the dilated VolterraNet in section 6.2 appropriate.

####  7.5.1 Extracting intra-voxel features

We extract intra-voxel features (independently) from each voxel. As mentioned
before, in each voxel we have a function
$f:\mathbf{S}^{2}\times\mathbf{R}^{+}\rightarrow\mathbf{R}$. Since
$\mathbf{S}^{2}\times\mathbf{R}^{+}$ is a Riemannian homogeneous space
(endowed with the product metric), we will use a cascade of the Volterra
correlation layers defined earlier (with standard non-linearity between
layers) to extract features which are equivariant to the action of
$\mathsf{SO}(3)\times\left(\mathbf{R}\setminus\left\\{0\right\\}\right)$.
These features are extracted independently within each voxel. Observe that
this equivariance property is natural in the context of dMRI data. Since in
each voxel of the dMRI data, the signal is acquired in different directions
(in 3D), we want the features to be equivariant to the 3D rotations and
scaling.

####  7.5.2 Extracting inter-voxel features

After the extraction of the intra-voxel features (which are equivariant to the
action of $G$), we seek to derive features based on the interactions between
the voxels. Here we use the standard dilated convolution (as described in 6.1)
layers to capture the interaction between features extracted from voxels.

Now, we are ready to give the details of the data used for the experiment of
our proposed Dilated-VolterraNet. For this experiment, we used a second order
Dilated-VolterraNet with $3$ dilated layers of kernel size $(5\times 5)$ and
dilation factors of $1$, $2$ and $4$ respectively.

###  7.6 Dilated VolterraNet Experiment: Group testing on movement disorder
patients

This dMRI data was collected from $50$ PD patients and $44$ controls at the
University of Florida and are accessible via request from the NIH-NINDS
Parkinson’s Disease Biomarker Program portal https://pdbp.ninds.nih.gov/. All
images were collected using a 3.0 T MR scanner (Philips Achieva) and
32-channel quadrature volume head coil. The parameters of the diffusion
imaging acquisition sequence were as follows: gradient directions = 64,
b-values = 0/1000 s/mm2, repetition time =7748 ms, echo time = 86 ms, flip
angle = $90^{\circ}$, field of view = $224\times 224$ mm, matrix size =
$112\times 112$, number of contiguous axial slices = 60, slice thickness = 2
mm. Eddy current correction was applied to each data set by using standard
motion correction techniques.

Figure 6: M1 Template [39]

We first extracted the sensory motor area tracts called M1 fiber tracts (as
shown in Fig.6) using the FSL software [40] from both the left (‘LM1’) and
right hemispheres (‘RM1’). We applied the Dilated-Volterra to the raw signal
measurements along the fiber tracts. Our model was trained on both the control
(normal subjects) group and the PD group data sets, i.e., we learned two
Dilated VolterraNet models, one each for the control and the PD groups
respectively. Using the method from [41] we compute the distance between these
two models, denoted by $d$. Now we permute the class labels between the
classes, retrain two models and compute the network distance $d_{j}$. If there
are significant differences between the classes we should expect that
$d>d_{j}$. We repeat this experiment for $j=1,\ldots,1000$ and let $p$ be the
proportion of experiments for which $d\leq d_{j}$. This is a permutation test
of the null hypothesis: there is no significant difference between the tract
models learned from the two different classes. We also performed ablation
studies with regards to the order of the model in the Dilated-VolterraNet to
study the effect of higher order convolutions. We used the following
architecture $\textsf{Corr}_{2}^{\mathbf{S}^{2}}$ $\,\to\,$ BN $\,\to\,$ ReLU
$\,\to\,$ $\textsf{Corr}_{2}^{\textsf{SO}(3)}$ $\,\to\,$ BN $\,\to\,$ ReLU as
our baseline model and then replaced $\textsf{Corr}_{2}^{\textsf{SO}(3)}$ and
$\textsf{Corr}_{2}^{\mathbf{S}^{2}}$ to $\textsf{Corr}^{\textsf{SO}(3)}$ and
$\textsf{Corr}^{\mathbf{S}^{2}}$ respectively in an alternating fashion. The
ablation study result is presented in Table V. The ‘N’ in
$\textsf{Corr}^{\textsf{SO}(3)}$ ($\textsf{Corr}^{\mathbf{S}^{2}}$) indicates
that we used second order for the respective convolution operator. The table
shows that a second order representation in later layers is very useful and
hence a model with $\textsf{Corr}^{\textsf{SO}(3)}$ performs poorly but a
model with $\textsf{Corr}^{\mathbf{S}^{2}}$ and
$\textsf{Corr}_{2}^{\textsf{SO}(3)}$ performs as good as the model with both
second order kernels. Both models reject the null hypothesis with $95\%$
confidence.

We compared our dilated VolterraNet with the standard (no dilation)
VoletrraNet and as expected we needed $\approx 1.5\times$ parameters in case
of standard VolterraNet to achieve p-values of $0.03$ and $0.04$ for LM1 and
RM1 respectively, which is similar in performance to its dilated counterpart.
Additionally, we compared our network’s performance to the performance of a
similar dMRI architecture (recurrent model) namely, the SPD-SRU [39] and the
baseline model used for comparison in [39] (see section 5.2 of [39] for
details on the baseline model). We found that the baseline method yielded a
p-value of $0.17$ and $0.34$ respectively for ’LM1’ and ’RM1’. Whereas, the
SPD-SRU architecture yielded a p-values of $0.01$ and $0.032$ respectively. We
can conclude that both using standard and Dialted VolterraNet we can reject
the null hypothesis with $95\%$ confidence whereas Dilated VoletrraNet can
achieve the statistically significant result with $\approx 33\%$ reduction in
number of parameters compared to its standard counterpart.

$\textsf{Corr}^{\mathbf{S}^{2}}$ | $\textsf{Corr}^{\textsf{SO}(3)}$ | p-val.
---|---|---
|  | ‘LM1’ | ‘RM1’
N | N | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>
Y | N | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>| ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>
N | Y | 0.13 | 0.24
Y | Y | 0.15 | 0.26

TABLE V: Ablation studies for Dilated-VolterraNet

##  8 Conclusions

In this paper, we presented a novel generalization of CNNs to non-Euclidean
domains specifically, Riemannian homogeneous spaces. More precisely, we
introduced higher order convolutions – represented using a Volterra series –
on Riemannian homogeneous spaces. We call our network a Volterra homogeneous
CNN abbreviated as VolterraNet. The salient contributions of our work are: (i)
A proof of equivariance of higher order convolutions to group actions on
homogeneous Riemannian manifolds. Proofs of generalized Linear Shift Invariant
(equivariant) and Nonlinear Shift Invariant (eqivariant) theorems for
correlations and Volterra series defined on Riemannian homogeneous spaces.
(ii) We prove that second order Volterra convolutions can be expressed as a
cascade of convolutions. This allows for efficient implementation of second-
order Volterra representation used in the VolterraNet. (iii) In support of our
conjecture on the reduced number of parameters, real data experiments
empirically demonstrate that VolterraNet requires less number of parameters to
achieve the baseline accuracy of classification in comparison to both
Spherical-CNN and Clebsch-Gordan net. (iv) We also presented a dilated
VolterraNet that was shown to be effective on a group testing experiment on
movement disorder patients. Our future work will be focused on performing more
real data experiments to demonstrate the power of VolterraNet for a variety of
data domains that are Riemannian homogeneous spaces.

## Acknowledgements

This research was in part funded by the NSF grant IIS-1724174 to BCV. We thank
Professor David Vaillancourt of the University of Florida, Department of
Applied Physiology and Kinesiology for providing us with the diffusion MRI
scans used in this work.

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|  Monami Banerjee received her Ph.D. in computer science from the Univ. of
Florida in 2018. She is currently a research staff member at Facebook Oculus,
Menlo Park. Her research interests lie at the intersection of Geometry,
Computer Vision and Medical Image Anlaysis.
---|---
|  Rudrasis Chakraborty received his Ph.D. in computer science from the Univ.
of Florida in 2018. He is currently a post doctoral researcher at UC Berkeley.
His research interests lie in the intersection of Geometry, ML and Computer
Vision.
---|---
|  Jose Bouza is a fourth year Mathematics and Computer Science undergraduate
at the University of Florida. His primary interests encompass computer vision
and applied topology.
---|---
|  Baba C. Vemuri received his PhD in Electrical and Computer Engineering from
the University of Texas at Austin. Currently, he holds the Wilson and Marie
Collins professorship in Engineering at the University of Florida and is a
full professor in the Department of Computer and Information Sciences and
Engineering. His research interests include Statistical Analysis of Manifold-
valued Data, Medical Image Computing, Computer Vision and Machine Learning. He
is a recipient of the IEEE Technical Achievement Award (2017) and is a Fellow
of the IEEE (2001) and the ACM (2009).
---|---