# The minimum orientable genus of the repeated Cartesian product of families
graphs

M. Galea,111Department of Mathematics, University of Malta, Msida, Malta
<EMAIL_ADDRESS>J. B. Gauci222Department of Mathematics, University
of Malta, Msida, Malta<EMAIL_ADDRESS>
(January 15, 2024)

###### Abstract

Determining the minimum genus of a graph is a fundamental optimisation problem
in the study of network design and implementation as it gives a measure of
non-planarity of graphs. In this paper, we are concerned with determining the
smallest value of $g$ such that a given graph $G$ has an embedding on the
orientable surface of genus $g$. In particular, we consider the Cartesian
product of graphs since this is a well studied graph operation which is often
used for modeling interconnection networks. The $s$-cube $Q_{i}^{(s)}$ is
obtained by taking the repeated Cartesian product of $i$ complete bipartite
graphs $K_{s,s}$. We determine the genus of the Cartesian product of the
$2r$-cube with the repeated Cartesian product of cycles and of the Cartesian
product of the $2r$-cube with the repeated Cartesian product of paths.

##  1 Introduction

Ever since their introduction by Sabidussi in 1959 [7], Cartesian product
graphs have been applied in many areas, including in the study of
interconnection networks and coding theory. Many important classes of graphs
such as hypercubes, Hamming graphs, and prisms, are Cartesian products. This
graph product provides a quick, effective and efficient way of constructing
bigger graphs from smaller ones in such a way that the structure of the
smaller graphs is preserved. Hence, Cartesian product graphs play a key role
in the design and analysis of interconnection networks.

Various disciplines which deal with network design are interested in
determining the optimal surface on which a network can be drawn without any of
the links between its nodes intersecting each other. Mathematically, this
problem revolves around determining the surface of minimum genus that permits
a given graph to be drawn on it without edge-crossings. In computer science,
the minimum genus of a graph is also a useful parameter in algorithm design
since faster algorithms can be designed when the topological structure of a
graph is known. These algorithms usually assume that the input graph is
embedded in some surface, as for example in [1]. The problem of determining
the minimum genus of a graph, albeit very relevant and applicable, has been
proved to be substantially difficult from the theoretical, practical and
structural perspectives. Garey and Johnson listed its complexity as one of the
most important open problems in [2].

The problem of embedding graphs on surfaces other than the sphere dates back
to the Map Colour Conjecture due to Heawood in 1890 [3]. Throughout this
paper, a surface is a 2-manifold (that is, each point has a neighbourhood
homeomorphic to a sphere) and compact (that is, closed and bounded). Surfaces
can be classified into two categories: orientable and nonorientable. The
former are obtained by adding handles to the sphere, whereas the latter are
obtained from the sphere by adding crosscaps. In this work we restrict our
attention to orientable surfaces $S_{g}$, where $g$ is the genus of the
surface, that is, the number of handles added to the sphere $S_{0}$. A handle
is added to the sphere by removing two disjoint open discs and identifying
their boundaries with the ends of a truncated cylinder.

An embedding of a graph $G$ is a drawing of a graph on a surface such that no
two edges cross. If every face in a drawing of $G$ is homeomorphic to an open
disc, the embedding is referred to as a 2-cell embedding. Such an embedding in
which each face has four sides is called a quadrilateral embedding. Given a
graph $G$, if $G$ has a 2-cell embedding on the surface $S_{g}$ but not on the
surface $S_{g-1}$, then $G$ is said to be of genus $g$ and we write $g(G)=g$.

Thomassen [8] showed that determining the genus of a graph is NP-complete.
Nevertheless, results on the genus can be obtained by considering families of
graphs with specific properties. One of the first results in this area is due
to Ringel [6], who determined the genus of the complete bipartite graph
$K_{m,n}$. White [9] considered Cartesian product graphs, where the Cartesian
product $G_{1}\square G_{2}$ of two graphs $G_{1}$ and $G_{2}$ has vertex set
$V(G_{1}\square G_{2})=\\{(v_{1},v_{2}):v_{1}\in V(G_{1})\text{ and }v_{2}\in
V(G_{2})\\}$ and edge set $E(G_{1}\square
G_{2})=\\{(v_{1},v_{2})(w_{1},w_{2}):(v_{1}=w_{1}\text{ and }v_{2}w_{2}\in
E(G_{2}))\text{ or }(v_{1}w_{1}\in E(G_{1})\text{ and }v_{2}=w_{2})\\}$. He
determined the genus of the Cartesian product of the complete bipartite graph
$K_{2m,2m}$ with $K_{2n,2n}$, for all natural numbers $m$ and $n$. White also
considered three families of graphs obtained by taking repeated Cartesian
products, namely the repeated Cartesian product of paths, the repeated
Cartesian product of cycles, and the repeated Cartesian product of complete
bipartite graphs, respectively defined more precisely as follows:

$\displaystyle H_{j}=H_{j-1}\square P_{m_{j}}\text{where }j\geq 2\text{ and
}H_{1}=P_{m_{1}},$ $\displaystyle G_{j}=G_{j-1}\square C_{2m_{j}}\text{where
}j\geq 2\text{ and }G_{1}=C_{2m_{1}},$ $\displaystyle
Q_{i}^{(s)}=Q_{i-1}^{(s)}\square K_{s,s}\text{where }i\geq 2\text{ and
}Q_{1}^{(s)}=K_{s,s}.$

We remark that

  1. (i)

$P_{n}$ and $C_{n}$ denote the path on $n$ vertices and the cycle on $n$
vertices, respectively;

  2. (ii)

the well-known hypercube $Q_{n}$ is $H_{n}$ with $m_{j}=2$ for all
$j\in\\{1,2,\ldots,n\\}$;

  3. (iii)

$Q_{i}^{(s)}$, also referred to at the $s$-cube, is a generalisation of the
hypercube $Q_{n}$ since $Q_{n}^{(1)}=Q_{n}$.

Pisanski [4] generalised White’s result by considering the repeated Cartesian
product of connected regular bipartite graphs on an even number of vertices.
In particular, letting $G(n,d)$, for $n\geq d\geq 1$, denote a connected
regular bipartite graph of degree $d$ on $2n$ vertices, Pisanski determined
the genus of the repeated Cartesian product of $G(n,d)\square
G_{1}(n_{1},d_{1})\square G_{2}(n_{2},d_{2})\square\ldots\square$
$G_{m}(n_{m},d_{m})$ provided that $\max\\{d_{1},d_{2},\ldots,d_{m}\\}\leq
d\leq d_{1}+d_{2}+\ldots+d_{m}$.

The aim of this paper is to generalise further the work done on the genus of
the repeated Cartesian product of graphs. In our main results, respectively
stated in Theorem 7 and Theorem 10 hereunder, we determine the minimum genus
of the Cartesian products $Q_{i}^{(2r)}\square G_{j}$ and $Q_{i}^{(2r)}\square
H_{j}$, where $i$ and $j$ are positive integers. We remark that most of these
cases are excluded from the result by Pisanski due to the condition imposed
therein on the degrees, as highlighted later on in this work.

##  2 Basic results

An important and well known result about the Cartesian product of two graphs
is given in the following theorem.

######  Theorem 1.

Let $G$ and $H$ be two non-empty graphs. The Cartesian product $G\square H$ is
bipartite if and only if $G$ and $H$ are bipartite.

It follows immediately that the repeated Cartesian product of bipartite graphs
is also bipartite.

A direct consequence of Euler’s generalized formula for 2-cell embeddings of a
bipartite graph in any surface is stated below.

######  Theorem 2.

If the bipartite graph $G$ with $n$ vertices and $m$ edges has a quadrilateral
embedding in a surface, then $g(G)=1+\frac{m}{4}-\frac{n}{2}$.

In [9], White proved the following results.

######  Proposition 3.

[9]

  1. 1.

$g(H_{j})=1+\frac{1}{4}\left(\prod_{k=1}^{j}m_{k}\right)\left(j-2-\sum_{k=1}^{j}\frac{1}{m_{k}}\right)$
for $j\geq 3$ and even $m_{1},m_{2},m_{3}$;

  2. 2.

$g(G_{j})=1+2^{(j-2)}(j-2)\prod_{k=1}^{j}m_{k}$ for $j\geq 2$;

  3. 3.

$g\left(Q_{i}^{(r)}\right)=1+2^{i-3}r^{i}(i-4)$ for $r$ even and $i\geq 1$ or
for $r\in\\{1,3\\}$ and $i\geq 2$.

Ringel [6] proved the following result, which we will need in our proofs.

######  Theorem 4.

[6] The complete bipartite graph $K_{2r,2r}$ has an embedding on a surface of
genus $(r-1)^{2}$.

Another useful result by White in [9] establishes a partition of a set of
faces found in a quadrilateral embedding for $K_{2r,2r}$, given below.

######  Lemma 5.

[9] For a quadrilateral embedding of $K_{2r,2r}$, the set of $2r^{2}$
quadrilateral faces may be partitioned into $2r$ subsets of $r$ faces each so
that each subset of $r$ faces contains all $4r$ vertices of the graph.

##  3 The Cartesian product of the $2r$-cube with cycles

In this section we use a technique developed by Pisanski [5] and White [9],
referred to as the White-Pisanski method, that constructs a quadrilateral
embedding for the Cartesian product of two regular bipartite graphs.

Note that the number of vertices of $Q_{i}^{(2r)}$ is $2^{2i}r^{i}$ and the
number of edges is $i2^{2i}r^{i+1}$, which can be easily derived using
induction or the Handshaking Lemma. We first construct a quadrilateral
embedding of $Q_{i}^{(2r)}\Box\ C_{2s}$ and then derive the genus of this
graph.

######  Lemma 6.

The genus of $Q_{i}^{(2r)}\Box\ C_{2s}$ is given by

$g(Q_{i}^{(2r)}\Box\ C_{2s})=1+2^{2i-1}sr^{i}(ir-1).$

###### Proof.

First, we minimally embed a copy of $Q_{i}^{(2r)}$ as follows. By Theorem 1,
$K_{2r,2r}\Box\ K_{2r,2r}$ is a bipartite graph. Using Theorem 4, we embed
$4r$ disjoint copies of $K_{2r,2r}$ in $4r$ surfaces of genus $(r-1)^{2}$ such
that each embedding is quadrilateral. Assign an orientation to $2r$ of these
copies and give the reverse orientation to the other half. This partition
corresponds to the vertex-set partition of $K_{2r,2r}$. All that is left is to
add edges between the $4r$ disjoint copies to acquire $K_{2r,2r}\Box\
K_{2r,2r}$. Note that $4r$ edges are required to be added between each pair of
oppositely oriented copies, which can be done by adding $r$ handles, each
carrying four edges. Such an operation will be referred to as a link. For each
surface, there are $2r$ links to be made and by Lemma 5, each link is
designated a subset of faces from the partition. We just need to check that
after partitioning the faces, the corresponding subsets can be matched
correctly. For $1\leq\overleftarrow{j}\leq 2r$, each copy $\overleftarrow{j}$
of $K_{2r,2r}$ is embedded with common orientation, whilst for
$1\leq\overrightarrow{j}\leq 2r$ each copy $\overrightarrow{j}$ of $K_{2r,2r}$
is embedded with the reverse orientation. For some copy $\overleftarrow{j}$,
its face partition consists of $2r$ subsets of $r$ faces each. Match subset
$\overleftarrow{k}$ of the face partition to subset $\overrightarrow{k}$ in
copy
$\overrightarrow{j}+\overrightarrow{k},1\leq\overrightarrow{j}+\overrightarrow{k}\leq
2r$ (mod $2r$). If
$\overrightarrow{j}+\overrightarrow{k}=\overrightarrow{j}^{\prime}+\overrightarrow{k}^{\prime}$
with $\overrightarrow{k}=\overrightarrow{k}^{\prime}$, then,
$\overleftarrow{j}=\overleftarrow{j}^{\prime}$, therefore, each subset of the
partition has exactly one handle attached at each face in that subset. Each
handle carries a maximum of four edges, so that each new face formed is
quadrilateral. This process is repeated by successively introducing the
complete bipartite graph $K_{2r,2r}$ until a quadrilateral embedding of
$Q_{i}^{(2r)}$ is obtained.

Now, to embed $Q_{i}^{(2r)}\Box\ C_{2s}$, embed $2s$ copies of $Q_{i}^{(2r)}$
as described above. We assign $s$ many copies one orientation and the other
$s$ copies the reverse orientation, corresponding to the vertex-set partition
of $C_{2s}$. From each copy of $Q_{i}^{(2r)}$, two links are to be made, both
to copies of reverse orientation. For each link, $2^{2i}r^{i}$ edges are to be
added and these are passed over $2^{2i-2}r^{i}$ handles with each handle
carrying four edges. By the construction given above and Lemma 5, each copy of
$Q_{i}^{(2r)}$ has $2r$ disjoint sets of $2^{2i-2}r^{i}$ mutually vertex-
disjoint quadrilateral faces that cover all $2^{2i}r^{i}$ vertices of
$Q_{i}^{(2r)}$. Since only two links are to be made from each copy of
$Q_{i}^{(2r)}$, we choose any two of the available sets. The number of
quadrilateral faces in each set is precisely the amount required to perform a
link, as previously described. Since the embedding of each copy of
$Q_{i}^{(2r)}$ is quadrilateral and each insertion of a handle introduces more
quadrilateral faces, the obtained embedding of $Q_{i}^{(2r)}\Box\ C_{2s}$ is
quadrilateral. Applying Theorem 2, we get

$\begin{split}g(Q_{i}^{(2r)}\Box\
C_{2s})&=1+\frac{2sir^{i+1}2^{2i}+2sr^{i}2^{2i}}{4}-\frac{2sr^{i}2^{2i}}{2}\\\
&=1+2^{2i-1}sr^{i}(ir-1).\end{split}$

∎

Let $M^{(j)}=\prod\limits_{k=1}^{j}m_{k}$. Note that the number of vertices of
$G_{j}$ is $2^{j}M^{(j)}$ and the number of edges is $j2^{j}M^{(j)}$, which
follows immediately by induction. We are now in a position to construct a
quadrilateral embedding of $Q_{i}^{(2r)}\Box\ G_{j}$ and, consequently,
determine the genus of this graph.

######  Theorem 7.

The genus of $Q_{i}^{(2r)}\Box\ G_{j}$ is given by

$g(Q_{i}^{(2r)}\Box\ G_{j})=1+M^{(j)}2^{2i+j-2}r^{i}(j+ir-2).$

###### Proof.

We let $G$ denote $Q_{i}^{(2r)}\Box\ G_{j}$. The proof proceeds by induction
on $j$, where the property $P(j)$ is as follows: There is a quadrilateral
embedding of $G$, with the number of faces
$f_{j}=r^{i}2^{2i+j-1}M^{(j)}(j+ir)$, including two disjoint sets each of
$r^{i}2^{2i+j-2}M^{(j)}$ mutually vertex-disjoint quadrilateral faces each
such that each set contains all the $r^{i}2^{2i+j}M^{(j)}$ vertices of $G$.

For $j=1$, the quadrilateral embedding of $Q_{i}^{(2r)}\Box\ C_{2m_{1}}$ is
given by Lemma 6. The number of faces $f_{1}$ is given by
$f_{1}=\frac{1}{2}\left|E(Q_{i}^{(2r)}\Box\ C_{2m_{1}})\right|$, and thus

$\displaystyle f_{1}$
$\displaystyle=\frac{(2m_{1})(2^{2i}r^{i})+(2m_{1})(i2^{2i}r^{i+1})}{2}$
$\displaystyle=m_{1}2^{2i}r^{i}+im_{1}2^{2i}r^{i+1}$
$\displaystyle=2^{2i}m_{1}r^{i}(1+ir).$

which conforms to the number of expected faces as stated by $P(1)$. The first
disjoint set of faces may be chosen by selecting opposite faces from alternate
links, and the second set of faces may be formed by choosing the remaining
faces from the same handles in each chosen link. Therefore, each disjoint set
would contain $2m_{1}(2^{2i-2}r^{i})=2^{2i-1}m_{1}r^{i}$ quadrilateral faces.
This follows since half of the $2m_{1}$ links are chosen, where each link has
$2^{2i-2}r^{i}$ handles and only two faces are chosen from each handle. The
faces in each set are mutually vertex-disjoint and each set contains all the
vertices of $Q_{i}^{(2r)}\Box\ C_{2m_{1}}$. Therefore, $P(1)$ holds.

Assuming $P(j)$ is true, we will show that $P(j+1)$ also holds. The graph we
are considering is $G^{\prime}=Q_{i}^{(2r)}\Box\ G_{j+1}$. First, minimally
embed $2m_{j+1}$ copies of $G$ as described by $P(j)$. Assign $m_{j+1}$ many
copies one orientation and the other $m_{j+1}$ copies the reverse orientation.
This partition corresponds to the vertex-set partition of $C_{2m_{j+1}}$. From
each copy of $G$, two links are to be made, both to copies of reverse
orientation. The number of vertices of $G$ is $r^{i}2^{2i+j}M^{(j)}$, and thus
we need to pass that many edges over a link joining two copies of $G$ with
different orientations. Therefore, the number of handles required for each
link is $\frac{1}{4}\left(r^{i}2^{2i+j}M^{(j)}\right)$ since each handle
carries four edges. Using $P(j)$, we have two disjoint sets each of
$r^{i}2^{2i+j-2}M^{(j)}$ mutually vertex-disjoint quadrilateral faces and thus
we have a sufficient number of faces such that we can construct these links
using $r^{i}2^{2i+j-2}M^{(j)}$ handles, where each handle carries four edges
so that each new face formed is a quadrilateral. In this manner, the graph
$G^{\prime}$ is embedded after constructing the $2m_{j+1}$ links.

The number of faces $f_{j+1}$ in $G^{\prime}$ is
$f_{j+1}=2m_{j+1}f_{j}+\tilde{f}$, where $\tilde{f}$ denotes the number of new
faces introduced by the handles. Note that
$\tilde{f}=4m_{j+1}(r^{i}2^{2i+j-2}$ $M^{(j)})$ since we have $2m_{j+1}$
links, with each link requiring $r^{i}2^{2i+j-2}M^{(j)}$ handles and the
difference in the number of faces is two per handle. Therefore,

$\begin{split}f_{j+1}&=2m_{j+1}(r^{i}2^{2i+j-1}M^{(j)}(j+ir))+4m_{j+1}(r^{i}2^{2i+j-2}M^{(j)})\\\
&=r^{i}2^{2i+j}M^{(j+1)}(j+ir)+r^{i}2^{2i+j}M^{(j+1)}\\\
&=r^{i}2^{2i+j}M^{(j+1)}(j+ir+1).\end{split}$

This satisfies the expected number of faces as stated by $P(j+1)$. Next, we
have to construct the two disjoint sets of faces. The first disjoint set of
faces may be chosen by selecting opposite faces from alternate links, and the
second set of faces may be formed by choosing the remaining faces from the
same handles in each chosen link. Therefore, each disjoint set would contain
$2m_{j+1}(r^{i}2^{2i+j-2}M^{(j)})=2^{2i+j-1}r^{i}M^{(j+1)}$ quadrilateral
faces. This follows since half of the $2s$ links are chosen, where each link
has $r^{i}2^{2i+j-2}M^{(j)}$ handles and only two faces are chosen from each
handle. The faces in each set are mutually vertex-disjoint and each set
contains all the vertices of $G^{\prime}$. Therefore, $P(j+1)$ follows from
$P(j)$ and since $G$ is bipartite by Theorem 1, we calculate the genus using
Theorem 2 to get

$\begin{split}g(Q_{i}^{(2r)}\Box\
G_{j})&=1+\frac{r^{i}2^{2i+j}M^{(j)}(j+ir)}{4}-\frac{2^{2i+j}r^{i}M^{(j)}}{2}\\\
&=1+M^{(j)}2^{2i+j-2}r^{i}(j+ir-2).\end{split}$

∎

As a direct corollary of the above result, we establish the genus of
$K_{2r,2r}\Box G_{j}$, addressing the case when $r>j$ which is excluded from
Pisanski’s result.

######  Corollary 8.

The genus of $K_{2r,2r}\Box G_{j}$ is given by

$g(K_{2r,2r}\Box\ G_{j})=1+r2^{j}M^{(j)}(j+r-2).$

##  4 The Cartesian product of the $2r$-cube with paths

Using Lemma 6, we can establish the genus of $Q_{i}^{(2r)}\Box\ P_{2s}$ as
follows.

######  Lemma 9.

The genus of $Q_{i}^{(2r)}\Box\ P_{2s}$ is given by

$g(Q_{i}^{(2r)}\Box\ P_{2s})=1+2^{2i-2}r^{i}(2s(ir-1)-1).$

###### Proof.

We let $G$ denote $Q_{i}^{(2r)}\Box\ P_{2s}$ and note that $G$ is bipartite by
Theorem 1. We start by first embedding $G^{\prime}=Q_{i}^{(2r)}\Box\ C_{2s}$
as given in Lemma 6. To obtain $G$, we only need to remove $4^{i}r^{i}$ edges
from $G^{\prime}$ between a copy of $Q_{i}^{(2r)}$ with one orientation and
another copy of reverse orientation. The only way these two copies are
connected is through $2^{2i-2}r^{i}$ handles, therefore, we remove these
$2^{2i-2}r^{i}$ handles and the edges on them. The faces we have reintroduced
are quadrilateral, since the embedding of $G^{\prime}$ is quadrilateral,
producing a quadrilateral embedding for $G$. The genus of $G$ is given by,

$\displaystyle g(Q_{i}^{(2r)}\Box\ P_{2s})$
$\displaystyle=1+2^{2i-1}sr^{i}(ir-1)-2^{2i-2}r^{i}$
$\displaystyle=1+2^{2i-2}r^{i}(2s(ir-1)-1).$

∎

Similarly to the repeated product of even cycles, we denote by $M^{(j)}$ the
product $\prod_{k=1}^{j}m_{k}$ and $m^{(j)}$ denotes
$\sum_{k=1}^{j}\frac{1}{2m_{k}}$. Using Lemma 9, we establish the genus of
$Q_{i}^{(2r)}\Box\ H_{j}$.

######  Theorem 10.

The genus of $Q_{i}^{(2r)}\Box\ H_{j}$ is given by

$g(Q_{i}^{(2r)}\Box\ H_{j})=1+2^{2i+j-2}r^{i}M^{(j)}(ir+j-m^{(j)}-2).$

###### Proof.

We let $G$ denote $Q_{i}^{(2r)}\Box\ H_{j}$. The proof proceeds by induction
on $j$, where the property $P(j)$ is as follows: There is a quadrilateral
embedding of $G$, with $f_{j}=2^{2i+j-1}r^{i}M^{(j)}(j+ir-m^{(j)})$, including
two disjoint sets of $r^{i}2^{2i+j-2}M^{(j)}$ mutually vertex-disjoint
quadrilateral faces each, such that each set contains all the
$r^{i}2^{2i+j}M^{(j)}$ vertices of $G$.

For $j=1$, the quadrilateral embedding of $Q_{i}^{(2r)}\Box\ P_{2m_{1}}$ is
given by Lemma 9. The number of faces $f_{1}$ is given by
$f_{1}=\frac{1}{2}\left|E(Q_{i}^{(2r)}\Box\ P_{2m_{1}})\right|$, and thus

$\displaystyle f_{1}$
$\displaystyle=\frac{(2m_{1}-1)(2^{2i}r^{i})+(2m_{1})(i2^{2i}r^{i+1})}{2}$
$\displaystyle=2^{2i-1}r^{i}(2m_{1}-1+2m_{1}ir)$
$\displaystyle=2^{2i}m_{1}r^{i}\left(1+ir-\tfrac{1}{2m_{1}}\right).$

which conforms to the number of expected faces as stated by $P(1)$. The first
disjoint set of faces may be chosen by selecting opposite faces from alternate
links, and the second set of faces may be formed by choosing the remaining
faces from the same handles in each chosen link. Therefore, each disjoint set
would contain $2m_{1}(2^{2i-2}r^{i})=2^{2i-1}r^{i}m_{1}$ quadrilateral faces.
This follows since from $2m_{1}-1$ links, $\lceil\frac{2m_{1}-1}{2}\rceil$ are
chosen, where each link has $2^{2i-2}r^{i}$ handles and only two faces are
chosen from each handle. The faces in each set are mutually vertex-disjoint
and each set contains all the vertices of $Q_{i}^{(2r)}\Box\ P_{2m_{1}}$,
therefore, $P(1)$ holds.

Assuming $P(j)$ is true, we will show that $P(j+1)$ also holds. The graph we
are considering is $G^{\prime}=Q_{i}^{(2r)}\Box\ H_{j+1}$. First, minimally
embed $2m_{j+1}$ copies of $G^{\prime}$ as described by $P(j)$. Assign
$m_{j+1}$ many copies one orientation and the other $m_{j+1}$ copies the
reverse orientation. This partition corresponds to the vertex-set partition of
$P_{2m_{j+1}}$. From each copy of common orientation, except the end copy, of
$G$, two links are to be made, all to copies of reverse orientation. We can
construct these links using $r^{i}2^{2i+j-2}M^{(j)}$ handles, where each
handle carries four edges so that each new face formed is a quadrilateral. In
this manner, the graph $G^{\prime}$ is embedded after constructing the
$2m_{j+1}-1$ links.

The number of faces $f_{j+1}$ in $G^{\prime}$ is
$f_{j+1}=2m_{j+1}f_{j}+\tilde{f}$, where $\tilde{f}$ denotes the number of new
faces introduced by the handles. Note that
$\tilde{f}=2(2m_{j+1}-1)(r^{i}2^{2i+j-2}M^{(j)})$ since we have $2m_{j+1}-1$
links, with each link requiring $r^{i}2^{2i+j-2}M^{(j)}$ handles and the
difference in the number of faces is two per handle. Therefore,

$\begin{split}f_{j+1}&=2m_{j+1}(2^{2i+j-1}r^{i}M^{(j)}(j+ir-m^{(j)}))+(2m_{j+1}-1)(r^{i}2^{2i+j-1}M^{(j)})\\\
&=r^{i}2^{2i+j}M^{(j+1)}\left(ir+j-m^{(j)}+\frac{2m_{j+1}-1}{2m_{j+1}}\right)\\\
&=r^{i}2^{2i+j}M^{(j+1)}(ir+j+1-m^{(j+1)}).\end{split}$

This satisfies the expected number of faces as stated by $P(j+1)$. Next, we
have to construct the two disjoint sets of faces. The first disjoint set of
faces may be chosen by selecting opposite faces from alternate links. The
second set of faces may be formed by choosing the remaining faces from the
same handles in each chosen link. Therefore, the disjoint set would contain
$2\left\lceil\frac{2m_{j+1}-1}{2}\right\rceil(r^{i}2^{2i+j-2}M^{(j)})=m_{j+1}2^{2i+j-1}r^{i}M^{(j)}=2^{2i+j-1}r^{i}M^{(j+1)}$
quadrilateral faces. This follows since half of the $2m_{j+1}-1$ links are
chosen, where each link has $r^{i}2^{2i+j-2}M^{(j)}$ handles and only two
faces are chosen from each handle. The faces in each set are mutually vertex-
disjoint and each set contains all the vertices of $G^{\prime}$. Therefore,
$P(j+1)$ follows from $P(j)$ and since $G$ is bipartite by Theorem 1, we
calculate the genus using Theorem 2 to get

$\begin{split}g(Q_{i}^{(2r)}\Box\
H_{j})&=1+\frac{r^{i}2^{2i+j}M^{(j)}(j+ir-m^{(j)})}{4}-\frac{2^{2i+j}r^{i}M^{(j)}}{2}\\\
&=1+r^{i}2^{2i+j-2}M^{(j)}(j+ir-m^{(j)})-2^{2i+j-1}r^{i}M^{(j)}.\\\
&=1+2^{2i+j-2}r^{i}M^{(j)}(ir+j-m^{(j)}-2).\end{split}$

∎

We remark that Pisanski’s result also excludes Theorem 10 since the path is
not a regular graph.

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# CoroNetGAN: Controlled Pruning of GANs via Hypernetworks

Aman Kumar  OPPO Mobiles R & D Center, Hyderabad, India ††thanks:
E-mail:{aman.kumar1, khushboo.anand, shubham.mandloi, ashutosh.mishra1
avinash.thakur<EMAIL_ADDRESS>Khushboo Anand  OPPO Mobiles R & D
Center, Hyderabad, India ††thanks: E-mail:{aman.kumar1, khushboo.anand,
shubham.mandloi, ashutosh.mishra1
avinash.thakur<EMAIL_ADDRESS>Shubham Mandloi  OPPO Mobiles R & D
Center, Hyderabad, India ††thanks: E-mail:{aman.kumar1, khushboo.anand,
shubham.mandloi, ashutosh.mishra1
avinash.thakur<EMAIL_ADDRESS>Ashutosh Mishra  OPPO Mobiles R & D
Center, Hyderabad, India ††thanks: E-mail:{aman.kumar1, khushboo.anand,
shubham.mandloi, ashutosh.mishra1
avinash.thakur<EMAIL_ADDRESS>Avinash Thakur  OPPO Mobiles R & D
Center, Hyderabad, India ††thanks: E-mail:{aman.kumar1, khushboo.anand,
shubham.mandloi, ashutosh.mishra1
avinash.thakur<EMAIL_ADDRESS>Neeraj Kasera  OPPO Mobiles R & D
Center, Hyderabad, India ††thanks: E-mail:{aman.kumar1, khushboo.anand,
shubham.mandloi, ashutosh.mishra1
avinash.thakur<EMAIL_ADDRESS>Prathosh A P  Indian Institute of
Science, Bangalore††thanks<EMAIL_ADDRESS>
###### Abstract

Generative Adversarial Networks (GANs) have proven to exhibit remarkable
performance and are widely used across many generative computer vision
applications. However, the unprecedented demand for the deployment of GANs on
resource-constrained edge devices still poses a challenge due to huge number
of parameters involved in the generation process. This has led to focused
attention on the area of compressing GANs. Most of the existing works use
knowledge distillation with the overhead of teacher dependency. Moreover,
there is no ability to control the degree of compression in these methods.
Hence, we propose CoroNetGAN for compressing GAN using the combined strength
of differentiable pruning method via hypernetworks. The proposed method
provides the advantage of performing controllable compression while training
along with reducing training time by a substantial factor. Experiments have
been done on various conditional GAN architectures (Pix2Pix and CycleGAN) to
signify the effectiveness of our approach on multiple benchmark datasets such
as $\text{Edges}\rightarrow\text{Shoes}$,
$\text{Horse}\leftrightarrow\text{Zebra}$ and
$\text{Summer}\rightarrow\text{Winter}$. The results obtained illustrate that
our approach succeeds to outperform the baselines on
$\text{Zebra}\rightarrow\text{Horse}$ and
$\text{Summer}\rightarrow\text{Winter}$ achieving the best FID score of 32.3
and 72.3 respectively, yielding high-fidelity images across all the datasets.
Additionally, our approach also outperforms the state-of-the-art methods in
achieving better inference time on various smart-phone chipsets and data-types
making it a feasible solution for deployment on edge devices.

##  1 Introduction

Computer vision applications such as image-to-image translation, image
synthesis, image generation, super resolution etc. have seen tremendous
progress yielding high-fidelity images with the advent of GANs[1]. The
development of image-based GAN applications have in-turn accelerated the
demand for deployment of such models on edge devices for the usage of the end
consumers. However, the complexity of training such parameter heavy models to
generate visually pleasing images result in high computational and memory
overhead which acts as a bottleneck in deployment of GANs on mobile devices.
For instance, the popular CycleGAN requires over 56.8G MACs (Multiply-
Accumulate Operations) for generating a single image of resolution $256\times
256$ pixels. On the other hand, Pix2Pix requires 18.6G MACs which is 4X
compared to traditional Res-Net50 [2] architecture. This huge number of
operations is not desirable for the deployment on edge devices. Hence, there
is a need for compressing these networks by removing the redundant parameters
and reducing the memory and computational consumption.

Discriminative approaches such as image classification, object detection and
semantic segmentation have been at the receiving end of undivided focus since
these networks have surpassed human imagination but still, for the learning to
saturate, these networks take huge amount of training time. For instance, the
popular image classification model, Alexnet [3] has 60 million parameters and
requires about 240 MB of memory while VGG16 [4] has 130 million parameters and
has takes around 500 MB of memory. The research community has given
unmitigated attention on the application of model compression techniques to
accelerate deployment of image classification and object detection networks
using techniques like weight quantization [5, 6], pruning [7, 8] and knowledge
distillation [9, 10]. However, these methods are not directly applicable to
generative models such as GANs. A lot of the recently proposed methods have
tried to compress generative adversarial networks using the combined
techniques of knowledge distillation [11, 12] and channel pruning[13, 14].
However, these approaches don’t allow controllable compression to happen
neither using a single technique nor through the combination of multiple
techniques.

To address the above-mentioned issues, we propose a novel method for
compressing the GAN using differentiable pruning method using the concept of
hypernetwork. The compression is performed during the training regime. The
proposed hypernetwork takes latent vector as an input and dynamically produces
weights of a given layer of the generator network. This input latent vector
decides the pruning rate for different layers in the network. Sparsification
of latent vector is achieved via proximal gradient. Post sparsification, the
latent vectors are passed through the hypernetwork that in turn generates the
weight of the generator network. Since the latent vector and the weights of
the generator network are covariant with each other, the sparsification of
latent vectors leads to the pruning of the weights of the concerned network.
The proposed method also helps in reducing the training time and inference
time as compared to that of conventional GAN training method. Through the
experiments on different conditional generative models on various datasets,
the potential of the proposed method is revealed. The main contributions of
the paper can be summarized as follows:

  1. 1.

We propose CoroNetGAN, an approach based on differentiable pruning via
hypernetworks for GAN compression. To the best of our knowledge, this is the
first work that achieves model compression using controllable pruning via
hypernetwork for conditional GANs. Our proposed approach compresses the GAN
network in a controlled way by providing the compression rate as an input to
the algorithm.

  2. 2.

Compression is achieved simultaneously alongside training unlike the
distillation based methods that involve teacher dependency [15]. CoroNetGAN
outperforms state-of-the-art compression technique [15] on training time on
all the datasets validating the effectiveness of our technique both on
training latency and visual appearance of the generated images. This will be
of great advantage in reducing the training time while maintaining the
accuracy when training GANs on bigger datasets containing billion of images.

  3. 3.

Our proposed approach, CoroNetGAN outperforms state-of-the-art conditional GAN
compression methods on widely used $\text{Zebra}\rightarrow\text{Horse}$ and
$\text{Summer}\rightarrow\text{Winter}$ datasets. CoroNetGAN obtains
reasonable qualitative and quantitative results on other datasets. CoroNetGAN
also outperforms state-of-the-art compression techniques [15] on inference
time.

##  2 Related Work

###  2.1 Generative Adversarial Networks

GANs [1] have proven to generate realistic results on a variety of tasks. For
instance, Isola et al. [16] propose Pix2Pix for paired image-to-image
translation trained via the combination of adversarial loss and pixel-wise
regression loss in order to ensure the visual quality of generated images.
Later, [17] is proposed that helps to increase the resolution of translated
images with multi-scale neural networks and edge maps. GANs have also been
proposed to perform image deblurring [18], style transfer [19, 20], image
super resolution [21] along with text-to-image generation [22]. Zhu et al.
[23] propose CycleGAN for unpaired image-to-image translation. The algorithm
trains generators on different domains of data through a weakly supervised
setting using cycle consistency loss. The final objective is to convert the
data from one domain to other without using any label information.

###  2.2 GAN Compression

The tremendous resource consumption by GANs has garnered recent attention
towards GAN compression. Wang et al.[24] proposes a novel quantization method
and multi-precision quantization algorithm considering different sensitivities
of discriminator and generator. Aguinaldo et al. [11] introduces the idea of
knowledge distillation in GANs between large over-parameterized network and
small few parameter networks optimized using joint and mean squared error loss
functions. However, the only focus here is to compress the generator keeping
the discriminator intact. Most usage of GANs in mobile devices is based on the
application of image-to-image translation task. [12] distills the student
discriminator to assist training of the student generator and also focused on
image translation problem using Pix2Pix framework. Chang et al. in [25]
focuses to mimic the functionality of BigGAN with a smaller compressed network
and fewer parameters. Different devices with varied computing power require
generators of different sizes. In order to accommodate this trade-off,
Slimmable-GAN[26] proposes flexible switching between the multi-width
configurations. Further, Ren et al.[15] overcomes the complex multi-stage
compression process and proposes a single-stage GAN online distillation
strategy to obtain the compressed model. However, these approaches use images
from the teacher directly to distill knowledge. Zhang et al.[27] proposes the
idea of investigating GAN compression from frequency perspective and
introduces the idea of wavelet analysis. They decompose the image into
frequency bands and perform distillation only on bands with higher frequency
unlike naive methods that do not prioritize the high frequency. [28] aims to
find crucial regions in the image using attention module. Considering the
attention value important to the region, features are distilled from teacher
to student. Recent works such as [29] introduce an Inception-based Residual
block replacing the original residual blocks in CycleGAN and search for
student generator from teacher generator via pruning followed by Similarity
based Knowledge Distillation. Further, approaches integrating various
compression techniques are also proposed. [13] combines model distillation,
channel pruning and quantization and generate a unified optimization form
which achieves superior trade-off compared with standalone compression
techniques. Liu at al.[14] combines the idea of channel pruning and knowledge
distillation and mainly expands the focus on accelerating unconditional GANs.

Figure 1: Illustration of the proposed algorithm designed for compressing
GAN’s using controllable differentiable pruning. A latent vector is attached
to each of the convolution layer of the generator. The latent vector generates
the weights for the generator via hypernetwork. Sparsification of the latent
vector leads to pruning of the corresponding weights of the generator network.
The proposed design allows the latent vector and its corresponding weight
matrix to be covariant with each other. The generator generates visual results
using the computed weight matrix through the hypernetwork (Best viewed when
zoomed).

###  2.3 HyperNetworks

Hypernetworks are a group of smaller networks that generates the weights for a
larger network. These smaller neural networks have been used historically for
vision[30], functional representation[31] and bayesian inference tasks[32].

Albeit the word hypernetwork has been coined recently, the concept of using
dynamic parameter generation has been used by researchers for a long time
[33]. Von der Malsberg et al.[34] indicates a possibility of dynamic modelling
between a slow classical weight and a dynamic decaying connection. The
technique to model short term memory by computing weight changes of another
network was initiated by Schmidhuber et al.[35]. Parameter prediction through
co-relation between different parameters of the neural network was extensively
studied in [36]. A weight matrix is produced using a learnable lower
dimensional matrix using a linear operation.[37] uses weight matrices as a
factored representation and feed forward one-shot learners reducing the
dimensionality of the hypernetwork.[38] proposes an approach to calculate the
parameters for image transformation using a weight generating network. [39]
proposes an approach for generating weights for visual question answering
task. The parameter prediction network takes input the questions post which
the network predicts weights of the main network. In addition, they also use
hashing of parameters to reduce the size of the final matrix of parameters.
The concept of dynamic filters has been used for image super-resolution [30].
These filters are computed based on input using a similar concept to
hypernetwork.

##  3 Methodology

GAN consist of a generator and a discriminator network employed in a min-max
game. The proposed method allows compression of the generator network while
training. Compression is achieved using differentiable meta pruning which is
based on the idea of hypernetwork. Hypernetwork is responsible for generating
the weights of the generator network for each of its layer. The input to the
hypernetwork is a latent vector and the output is a weight matrix of the
generator network.

During the forward pass, latent vector is given as an input to hypernetwork to
generate the weights of the generator. During back-propagation, the gradient
flows in the hypernetwork instead of the main network. It is designed in a way
such that its output is covariant with the input latent vector. Proximal
Gradient helps in pruning of output channels of the generator network by
eliminating the redundant parameters automatically.

###  3.1 HyperNetwork Design

The hypernetwork consists of three layers. The latent layer takes as input the
latent vectors and computes a latent matrix. The embedding layer projects the
elements of the latent matrix to an embedding space. The final layer converts
the embedded vectors to the final output. The design is taken from [40]. As an
example, consider the generator to be an L-layer convolutional neural network.
Each layer of the network has its own corresponding latent vector that is
responsible for generating the weights of the corresponding layer. The size of
the latent vector is equal to the number of output channels in that layer. For
instance, consider an _$l$_ -th convolutional layer having _$n*c*w*h$_ number
of parameters, where _n_ and _c_ are the output and input channels and _w*h_
is the size of kernel respectively. Suppose that the latent vector
corresponding to that particular _$l-th$_ layer is _$v^{l}$_ $\in$ _$R^{c}$_.
Therefore, the previous layer has latent vector _$v^{l-1}$_ $\in$ _$R^{n}$_.
The hypernetwork takes latent vector of current layer (_$v^{l}$_) and its
previous layer (_$v^{l-1}$_) as input and will output the weights matrix of
the _l-th_ layer of the generator network. Initially, the first layer of the
hypernetwork computes a latent matrix using the two latent vectors:

$\textbf{V}^{l}=\textbf{v}^{l}.\textbf{v}^{{l-1}^{T}}+\textbf{B}_{0}$ (1)

where,

$\textbf{V}^{l},\textbf{B}_{0}\in R^{n*c}$

Here, [_T_] denotes transpose of the matrix while [.] denotes matrix
multiplication.

Subsequently, the second layer of the hypernetwork projects every element of
the latent matrix to a m-dimensional embedding space as follows:

$\textbf{S}^{l}_{ij}=\textbf{V}^{l}_{ij}\textbf{w}^{l}_{1}+\textbf{b}^{1}_{l}\hskip
14.22636pti=1..n,j=1...c$ (2)

where,

$\textbf{S}^{l}_{ij},\textbf{w}^{l}_{1},\textbf{b}^{1}_{l}\in R^{m}$

Here, we are considering $\textbf{w}^{l}_{1}$ and $\textbf{b}^{1}_{l}$ as
different for different elements of the matrix. The subscript (_i, j_) has not
been used for easier interpretation of the above mentioned equations. The
vectors $\textbf{w}^{l}_{1}$, $\textbf{b}^{1}_{l}$ and $\textbf{S}^{l}_{ij}$
for all the elements of the matrix together forms a 3D tensor, i.e.,
$\textbf{W}^{l}_{1}$, $\textbf{B}^{l}_{1}$ and $\textbf{S}^{l}_{1}$ $\in$
$R^{n*c*m}$.

After the second step, the final layer of the hypernetwork is responsible for
converting the embedding vectors to the output($\textbf{F}^{l}_{ij}$) which
can be used as weight matrix of the Generator network. This is done by
multiplying the embedded vectors $\textbf{S}^{l}_{ij}$ by an explicit matrix
as follows:

$\textbf{F}^{l}_{ij}=\textbf{w}^{l}_{2}.\textbf{S}^{l}_{ij}+\textbf{b}^{2}_{l}\hskip
14.22636pti=1..n,j=1...c$ (3)

where,

$\textbf{F}^{l}_{ij},\textbf{b}^{l}_{2}\in R^{wh}$

and,

$\textbf{w}^{l}_{2}\in R^{wh*m}$

$\textbf{w}^{l}_{2}$ and $\textbf{b}^{2}_{l}$ are different and unique for
each of the element and subscript (_i,j_) has not been used for easier
interpretations. Vectors $\textbf{w}^{l}_{2}$, $\textbf{b}^{2}_{l}$ and
$\textbf{F}^{l}_{ij}$ for all the elements together will be high-dimensional
tensors i.e., $\textbf{W}^{l}_{2}$ $\in$ $R^{n*c*wh*m}$ and
$\textbf{B}^{l}_{2}$ and $\textbf{F}^{l}$ $\in$ $R^{n*c*wh}$.

Combining 1, 2 and 3, the functionality of the proposed approach can be
collectively written as:

$\textbf{F}^{l}=h(\textbf{v}^{l},\textbf{v}^{l-1};\textbf{W}^{l},\textbf{B}^{l}),$
(4)

where, h(.) denotes the functionality of the above architecture. The final
output $\textbf{F}^{l}$ will be used as the weight parameter of the _l_ -th
layer. The hypernetwork is designed in such a way that the weight matrix of
the generator is covariant with it’s corresponding input latent vector as
pruning an element in the latent vector automatically leads to removal of
corresponding slice in the final weight matrix( $\textbf{F}^{l}$). Figure 1
depicts the overall workflow of the proposed CoroNetGAN.

While designing the hypernetwork, we also execute residual connections in the
network. In case of residual or skip connections, we take the input latent
vector as the combination of the latent vector of the previous layer and the
corresponding layer from which the skip connection originates. We concatenate
both the input latent vectors to create one single input latent vector. The
resultant input latent vector along with latent vector of the current layer is
used to create the latent matrix. The creation of latent matrix is followed by
execution of steps((2),(3)) for generating the weights matrix of the
convolution layer.

###  3.2 Vector Sparsity using Proximal Gradient

The differentiable property of the algorithm comes through the use of proximal
gradient. Proximal gradient helps in sparsification of the latent vector by
searching the potential candidates. Since latent vector is covariant with the
weight matrix of the Generator network, it leads to compression of the
Generator network. During training time, the parameters of the hypernetwork is
updated using Stochastic Gradient Descent (SGD) optimization algorithm. During
back-propagation, gradients flow from the Generator network to the
hypernetwork. The latent vectors are updated using the proximal gradient [40]
which leads to sparsified input latent vectors as follows:

$\textbf{v}[k+1]=\textbf{prox}_{\lambda\mu
R}(v[k]-\lambda\mu\nabla\emph{L}(v[k]))$ (5)

The proximal gradient algorithm forces the potential elements of the latent
vectors to approach zero quicker than the others without any human effort and
interference in this process. Due to the fact that proximal operator has
closed-form solution and use of SGD, the whole solution is recognized as
approximately differentiable.

###  3.3 Network Pruning

Figure 2: Graphical representation of training time(in minutes) and FID for
Pix2Pix(left) on $\text{Edges}\rightarrow\text{Shoes}$ and
CycleGAN(middle,right) on $\text{Horse}\rightarrow\text{Zebra}$ and
$\text{Summer}\rightarrow\text{Winter}$ datasets respectively. From the
graphs, it is evident that total training time for our proposed approach is
significantly lesser compared to OMGD [15]. For CycleGAN on
$\text{Summer}\rightarrow\text{Winter}$ dataset, our algorithm outperforms
OMGD [15] on both training time and FID (Best viewed when zoomed).

Our proposed method allows the weight matrix to be covariant with the it’s
corresponding latent vector. Hence, sparsifictaion of latent vector leads to
the pruning of the corresponding weights of the CNN layer in the generator
network. Our training regime consists of two stages, namely searching stage
and converging stage. During the searching stage, proximal gradients helps in
identifying the potential candidates of the latent vector. Therefore, after
the searching stage, we get the sparsified latent vector (_$\hat{v^{l}}$_).
Proximal gradient help in elements of the latent vector either be zero or
approaching towards zero. We use a mask(_$m^{l}$_) on the sparsified latent
vector with a predefined threshold($\tau$). This is followed by masking
operation that compares every element of the latent vector with the threshold
value. If greater than threshold, the returned value is one else zero. The
sparsified latent vector, _$\hat{v^{l}}$_ is pruned with the help of the
computed mask (_$m^{l}$_).

Once the target compression ratio is achieved, the algorithm shifts from
searching to the converging stage. In the converging stage, hypernetwork is
discarded, and the training of the generator follows the conventional GAN
training procedure. Upon extensive experimentation, it is observed that the
number of epochs in searching stage is much smaller than the number of epochs
in the converging stage. The pseudo-code of the proposed algorithm is
mentioned in Algorithm 1.

Algorithm 1 CoroNetGAN Pseudo Code

$total\\_epochs\leftarrow total\\_number\\_of\\_epochs$

$targetflops\leftarrow target\\_compression\\_ratio$

$latent\\_vectors(v_{1},v_{2},..,v_{i})$

$converging\leftarrow False$

$epochs\leftarrow 0$

Compression via Differentiable Pruning

while $converging\neq True$ do

$\bullet$ Sample m{$z_{1},z_{2},..,z_{i}$} images from given dataset

$\bullet$ Sample m{$x_{1},x_{2},..,x_{i}$} ground-truths

$\bullet$ Update Hypernetwork using SGD

$\triangledown_{\theta_{h}}\dfrac{1}{m}\sum_{i=1}^{m}log(1-D(G(z_{i})))$

$\bullet$ Update Discriminator using SGD

$\triangledown_{\theta_{d}}\dfrac{1}{m}\sum_{i=1}^{m}\\{logD(x^{i})+log(1-D(G(z_{i})))\\}$

$\bullet$ Compress latent vector using proximal gradient

$\textbf{v}[k+1]=\textbf{prox}_{\lambda\mu
R}(v[k]-\lambda\mu\nabla\emph{L}(v[k]))$

$\bullet$ $epochs\leftarrow epochs+1$

if $flops-target\\_flops\leq threshold$ then

$converging\leftarrow True$

end if

if $epochs\leq total\\_epochs$ then

$break$

end if

end while

Finetuning

while $epochs\leq total\\_epochs$ do

$\bullet$ Sample m{$z_{1},z_{2},..,z_{i}$} images from given dataset

$\bullet$ Sample m{$x_{1},x_{2},..,x_{i}$} ground-truths

$\bullet$ Update Generator using SGD

$\triangledown_{\theta_{g}}\dfrac{1}{m}\sum_{i=1}^{m}log(1-D(G(z_{i})))$

$\bullet$ Update Discriminator using SGD

$\triangledown_{\theta_{d}}\dfrac{1}{m}\sum_{i=1}^{m}\\{logD(x^{i})+log(1-D(G(z_{i})))\\}$

$\bullet$ $epochs\leftarrow epochs+1$

end while

##  4 Experiments

###  4.1 Experiment Setting

####  4.1.1 Models and Datasets

We evaluate our approach incorporating the following models to demonstrate the
effectiveness of the proposed method:

  1. 1.

Pix2Pix [16] for paired image-to-image translation with original U-Net
generator architecture.

  2. 2.

CycleGAN [23] for unpaired image-to-image translation using Res-Net
architecture to perform transformation on an image belonging to source domain
to desired target domain.

  3. 3.

Deep Convolutional Generative Adversarial Network (DCGAN) [41] that uses
convolutional and convolutional-transpose layers in the discriminator and
generator, respectively.

For the purpose of quantitative and qualitative evaluation, four datasets are
utilised including $\text{Edges}\rightarrow\text{Shoes}$,
$\text{Horse}\leftrightarrow\text{Zebra}$,
$\text{Summer}\rightarrow\text{Winter}$ and CIFAR10.

  1. 1.

$\text{Edges}\rightarrow\text{Shoes}$ [16] is a paired image-to-image
translation dataset including images edges of shoes to be mapped to their
corresponding complete image of shoes. The dataset consists of 49825 images.

  2. 2.

$\text{Horse}\leftrightarrow\text{Zebra}$ [23] dataset contains images
originally from ImageNet [42]. It is an unpaired image-to-image translation
dataset used for translating horse images to zebra and vice versa. In our
experiments, the training set includes 1067 horse images and 1334 zebra
images.

  3. 3.

$\text{Summer}\rightarrow\text{Winter}$ [23] is also unpaired image-to-image
translation dataset which translates summer images to winter. We have used
1231 summer images for training purpose.

  4. 4.

CIFAR10 dataset [43] consists of 50000 training images and 10000 test images
across 10 different classes.

Our approach CoroNetGAN with Pix2Pix architecture is bench-marked on
$\text{Edges}\rightarrow\text{Shoes}$ dataset. On the other hand, CoroNetGAN
with CycleGAN has been bench-marked on
$\text{Horse}\leftrightarrow\text{Zebra}$ and
$\text{Summer}\rightarrow\text{Winter}$ datasets.

Although our proposed approach focuses on the compression for conditional GAN,
we also made initial attempts to perform compression using our proposed
algorithm for unconditional GAN (specifically DCGAN).

Figure 3: Samples generated from our approach. First row contains translated
images from $\text{Zebra}\rightarrow\text{Horse}$ dataset. The second row
contains translated images from $\text{Horse}\rightarrow\text{Zebra}$ dataset
(Best viewed when zoomed). Figure 4: Qualitative comparison of CoroNetGAN with
CycleGAN architecture on $\text{Summer}\rightarrow\text{Winter}$ dataset
compared with original CycleGAN [23], GAN Compression [44] and OMGD [15]
algorithms. Our approach generates visually realistic images and outperforms
all the other algorithms on the FID metric (Best viewed when zoomed).

####  4.1.2 Implementation Details

We train our proposed approach using single NVIDIA Tesla V100 GPU on PyTorch
deep learning framework. For the algorithm to compress the network, a target
compression ratio needs to be selected. When the difference between the actual
compression and the target compression ratio falls below 2$\%$, pruning stops
and model moves to fine-tuning state from the compression state. The number of
parameters in the hypernetwork is proportional to the size of the embedding
space. For the experimentation, embedding space is set to 8 across all the
experiments. Learning rate is set to 0.0002. Batch size has been set to 4 and
1 for Pix2Pix and CycleGAN respectively on all the experiments across
different datasets. The sparsity regularization factor for the proximal
gradient is set to 0.5 across all the experiments.

####  4.1.3 Evaluation Setting

For the quantitative performance comparison, we adopt Frechet Inception
Distance (FID) [45] as common evaluation metric. FID is specifically developed
for assessing the performance of GANs. It is used to evaluate the quality of
the images generated by generative models by comparing the distribution of
features corresponding to real and generated images using an InceptionV3 [46]
network. A lower FID score is an indicator of high similarity between both the
distributions and thus better quality of generated images. We have evaluated
FID for different architectures with our approach on multiple datasets and
compared it against existing methodologies.

Figure 5: Qualitative comparison of CoroNetGAN with Pix2Pix architecture on
$\text{Edges}\rightarrow\text{Shoes}$ dataset compared with original Pix2Pix
[16], GAN Compression [44] and OMGD [15] algorithms. Our approach generates
visually plausible images compared to state-of-the-art methods (Best viewed
when zoomed in).

Model | Dataset | Paper | Params(M) | FLOPs(G) | MACs(G) | FID
---|---|---|---|---|---|---
Pix2Pix | $\text{Edges}\rightarrow\text{Shoes}$ | Original [16] | 54.4 | – | 18.6 | 34.31
Region-Aware [28] | 13.61 (4.00×) | 1.56 | – | 77.69±3.14
Wavelet KD [27] | 13.61 (4.00×) | 1.56 | – | 80.13±2.18
DMAD [47] | 2.13 (25.5×) | – | 2.99 (6.2×) | 46.95
OMGD [15] | 3.404 (16.0×) | – | 1.219 (15.3×) | 25
CoroNetGAN(75%) | 13.225 | 4.8879 | – | 39.1
CoroNetGAN(95%) | 4.721 | 1.2551 | – | 54.3
CycleGAN | $\text{Horse}\rightarrow\text{Zebra}$ | Original [23] | 11.3 | – | 56.8 | 61.53
Region-Aware [28] | 1.61 (7.08×) | 7.29 | – | 60.01±5.22
Wavelet KD [27] | 1.61 (7.08×) | 7.29 | – | 61.65±4.73
DMAD [47] | 0.42 (26.9×) | – | 3.97 (14.3×) | 62.41
Teachers Do More Than Teach [29] | – | – | 2.56 | 53.48
GAN Compression [44] | 0.34 (33.3×) | – | 2.67 (21.2×) | 64.95
Revisiting Discriminator in GAN Compression [48] | – | – | 2.40 | 59.31
OMGD [15] | 0.137 (82.5×) | – | 1.408 (40.3×) | 51.92
CoroNetGAN(75%) | 2.685 | 0.217 | – | 57.7
| CoroNetGAN(85%) | 1.670 | 0.1347 | – | 60.9
$\text{Zebra}\rightarrow\text{Horse}$ | Original [23] | 11.3 | 49.64 | – | 138.07±4.01
Region-Aware [28] | 1.61 (7.08×) | 7.29 (6.80×) | – | 137.03±3.03
Wavelet KD [27] | 1.61 (7.08×) | 7.29 (6.80×) | – | 138.84±1.47
DMAD [47] | 0.30 (37.7×) | – | 3.50 | 139.3
CoroNetGAN (75%) | 2.685 | 0.217 | – | 32.3
$\text{Summer}\rightarrow\text{Winter}$ | Original [23] | 11.3 | – | 56.8 | 79.12
DMAD [47] | 0.24 (47.1×) | – | 3.18 (17.9×) | 78.24
OMGD [15] | 0.137 (82.5×) | – | 1.408 (40.3×) | 73.79
Auto-GAN [49] | – | 4.34 | – | 78.33
| CoroNetGAN (75%) | 2.685 | 0.217 | – | 72.3
|  | CoroNetGAN (85%) | 1.670 | 0.1347 | – | 74.7

Table 1: Performance comparison of CoroNetGAN with state-of-the-art algorithms
on Pix2Pix and CycleGAN architectures. It can be observed that our approach
achieves best FID with CycleGAN on $\text{Zebra}\rightarrow\text{Horse}$ and
$\text{Summer}\rightarrow\text{Winter}$ datasets. Our results also outperform
[28, 27, 47, 11, 48] and achieve competitive FID on
$\text{Horse}\rightarrow\text{Zebra}$ dataset. We also achieve second best FID
score on $\text{Edges}\rightarrow\text{Shoes}$ dataset beating the results of
[28, 27, 47].

Model | Dataset | Method | Params(M) | FLOPs(G) | FID
---|---|---|---|---|---
Pix2Pix | $\text{Edges}\rightarrow\text{Shoes}$ | Original [16] | 54.41 | – | 34.31
CoroNetGAN(75%) | 13.225 (24.31%) | 4.8879 (26.94%) | 39.1
CoroNetGAN(G + D)(75%) | Generator 13.321 (24.48%) | Generator 4.8993 (27%) | 38.6
Discriminator 0.725 (26.23%) | Discriminator 0.4767 (26.74%)

Table 2: Generator and Discriminator compression in CoroNetGAN in Pix2Pix
architecture on $\text{Edges}\rightarrow\text{Shoes}$ dataset. It is evident
that compressing both generator and discriminator helps in improving the FID
score.

Chipset | d-type | Model | GPU Inference Time(CL)(ms)
---|---|---|---
Qualcomm Snapdragon SM8450 | 32-bit | Ours | 12.5419
OMGD [15] | 15.3378
16-bit | Ours | 11.794
OMGD [15] | 15.283
8-bit | Ours | 12.244
OMGD [15] | 16.0191
Dimensity 1200-Max Octa | 32-bit | Ours | 20.7268
OMGD [15] | 21.1919
16-bit | Ours | 20.1961
OMGD [15] | 20.9635
8-bit | Ours | 20.9972
OMGD [15] | 21.541

Table 3: Shows inference time comparison between the model compressed by our
methodology to 95% and the model compressed from OMGD on different processors.
Both the models are trained on $\text{Edges}\rightarrow\text{Shoes}$ dataset.
The inference time is computed for the input resolution of 256$\times$256\.
The 8-bit quantization of the compressed model results in increased processing
time due to the presence of a higher number of quantization and de-
quantization blocks compared to other data types.

###  4.2 Experimental Results

####  4.2.1 Quantitative Results

We evaluate our approach on different models and datasets using evaluation
setting mentioned in the previous section and report quantitative results
compared with the corresponding state-of-the-art methods. The results can be
summarized as follows:

Pix2Pix: We incorporate Pix2Pix with its original U-Net architecture in our
proposed approach and report the experimental results in Table 1. We observe
that our approach is able to achieve second best FID score on
$\text{Edges}\rightarrow\text{Shoes}$ dataset. We are also able to outperform
the results of [28, 27, 47] by achieving a better FID. Although, our FID score
is higher than [15] but our approach outperforms it in terms of training time.
Figure 2 illustrates that our approach significantly improves training time
however elevates FID score compared to [15].

CycleGAN: Similar to previous works, we include Res-Net style CycleGAN in our
method and report the results in Table 1. We observe that the results on
$\text{Zebra}\rightarrow\text{Horse}$ dataset outperform all the state-of-the-
art approaches by achieving the best FID score of 32.3 corresponding to 75%
compression. Additionally, our approach is also able to improve over all the
existing baselines by achieving FID score of 72.3 on
$\text{Summer}\rightarrow\text{Winter}$ dataset. As illustrated in Figure 2 we
also outperform [15] on both training time and FID.

Furthermore, we also observe that our approach with CycleGAN is able to beat
the results of [28, 27, 47, 44, 48] on $\text{Horse}\rightarrow\text{Zebra}$
dataset. Even though, we achieve greater FID score than [15, 29], CoroNetGAN
outperforms [15] on training time as illustrated in Figure 2. One thing to
note is that we compare CoroNetGAN quantitative results with compression rates
of 75$\%$ and 85 $\%$ using CycleGAN architecture unlike $95\%$ in Pix2Pix
since it becomes difficult to compress CycleGAN architecture beyond $85\%$ due
to its huge model complexity.

DCGAN: We demonstrate the applicability of our approach on unconditional GAN.
For the experimentation and evaluating our approach, we adopt DCGAN [50] on
CIFAR10 dataset [43] with original FID score of 45.8. As per the results, we
were able to achieve FID score of 56.3 with 50% compression ratio which
outperforms the results obtained with random pruning which achieves FID score
of 68.8.
Inference Time Comparisons: Additionally, a comparative analysis of the
inference time is conducted between the 95% compressed model obtained from our
approach and OMGD [15]. Both the models are trained on
$\text{Edges}\rightarrow\text{Shoes}$ and evaluated using diverse smartphone
chipsets and data types. The inference results, as presented in Table 3,
demonstrate that our proposed model achieves superior inference time
performance compared to OMGD.

####  4.2.2 Qualitative Results

We further show visualization results of our proposed method in comparison
with state-of-the-art methodologies in Figure 3, 4 and 5 demonstrating the
effectiveness of our approach. As illustrated, our method can generate high-
fidelity images comparable to other state-of-the-art approaches across
multiple datasets. The reason we believe our approach generates realistic
images is that the compression state of our algorithm forces the generator to
generate visually plausible images while competing in the min-max game.

####  4.2.3 Ablation Studies

Our proposed method for GAN compression shows promising results and outperform
state-of-the-art methods on some conditional GANs. We perform extensive
ablation studies to further demonstrate the effectiveness of hypernetworks on
GAN compression on U-Net based architecture for Pix2Pix.

We tried exhaustive hyperparamter search and fine-tuning the learning rate.
All the modifications result in a very negligible change of overall FID score.
We also enabled a learning rate scheduler to check the improvement in model
performance but quantitatively, no major change was observed. We also tried
increasing the layer structure of the hypernetwork employed by increasing the
dimension of the embedding space. However, this led to an increase in the
training time with a small change in the overall FID score.

Compression of both generator and discriminator: To evaluate the significance
of our approach, we design a variant CoroNetGAN(G+D compression) which
compresses both the generator and discriminator till 75$\%$ during training.
As mentioned in Table 2, this ablation results in improving the FID score from
39.1 to 38.6 as the generator is able to generate better results while
compression of discriminator.
Generator finetuning through HyperNetwork: We also tried to finetune the
weights of the generator generated from the compression state through the
HyperNetwork itself, but we did not find any significant improvements in the
evaluation metric.

##  5 Conclusion

In this work, we propose a novel method CoroNetGAN for GAN compression based
on differentiable pruning via hypernetworks. Unlike other approaches having
teacher dependency overhead or post-hoc compression, the compression in our
approach is done during the training time itself giving us the benefit of
reducing overall time. Experiments conducted on conditional GANs (Pix2Pix,
CycleGAN) substantiate the effectiveness of our proposed method where we have
been able to outperform various state-of-the-art techniques on multiple
datasets without compromising the visual quality of generated images. Our
approach offers significant improvement in training time and inference time as
compared to the existing methods. Additionally, we also demonstrate the
ability of our approach to be applicable for unconditional GANs (specifically
DCGAN). The applicability on other unconditional GANs are open avenues for
future work.

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