$p$-Laplace equations in conformal geometry

Huajie Liu, Shiguang Ma, Jie Qing, and Shuhui Zhong

###### Abstract.

In this paper we introduce the $p$-Laplace equations for the intermediate
Schouten curvature in conformal geometry. These $p$-Laplace equations provide
more tools for the study of geometry and topology of manifolds. First, the
positivity of the intermediate Schouten curvature yields the vanishing of
Betti numbers on locally conformally flat manifolds as consequences of the
Böchner formula as in [Nay97, GLW05]. Secondly and more interestingly, when
the intermediate Schouten curvature is nonnegative, these $p$-Laplace
equations facilitate the geometric applications of $p$-superharmonic functions
and the nonlinear potential theory. This leads to the estimates on Hausdorff
dimension of singular sets and vanishing of homotopy groups that is inspired
by and extends the work in [SY88, MQ22]. In the forthcoming paper [MQ23] we
will present our results on the precise asymptotic behavior of
$p$-superharmonic functions at singularities.

###### Key words and phrases:

Intermediate Schouten curvature tensor, $p$-Laplace equations,
$p$-superharmonic functions, the Wolff potentials, $p$-thinness, Hausdorff
dimensions

###### 1991 Mathematics Subject Classification:

53C21; 31B35; 53A30; 31B05

##  1\. Introduction

The potential theory is a classic and powerful approach to the study of
partial differential equations. Recently $n$-Laplace equations as natural
extensions of the Gauss curvature equations in conformal geometry have been
introduced and used for establishing Huber-type theorems in higher dimensions
in [MQ21a, MQ21b]. More applications of linear potential theory in conformal
geometry have been investigated in [MQ22]. Those works renewed and advocated
more geometric applications of the potential theory which are represented by
geometric applications of subharmonic functions and the potential theory in 2
dimensions, for instance, in [Hu57, AH73]. In this paper we introduce the
$p$-Laplace equations and explore the geometric applications of the nonlinear
potential theory in conformal geometry for $p\in(2,n)$. We in general assume
the dimension of manifolds is greater than $2$ in this paper unless otherwise
specified.

###  1.1. $p$-Laplace equations in conformal geometry and applications

In conformal geometry one often encounters the Schouten curvature tensor on
$n$ dimensional manifolds

$A=\frac{1}{n-2}(Ric-\frac{1}{2(n-1)}R\,g)$

where $Ric$ stands for the Ricci curvature tensor and $R$ stands for the
scalar curvature of the metric $g$ on $n$-manifolds respectively. For
simplicity, we often let $J=\text{Tr}A=\frac{1}{2(n-1)}R$. In this paper we
want to turn the attention to the following intermediate Schouten curvature
tensor

(1.1.1) $A^{(p)}=(p-2)A+J\,g$

for $p\in(1,\infty)$. This curvature tensor plays an intermediate role between
the scalar curvature $Jg$ and the Schouten curvature $A$ for $p\in[2,\infty)$.
For the convenience of readers, let us recall (please see, for instance,
[Lind06])

$\Delta_{p}u=\text{div}(|\nabla u|^{p-2}\nabla u)\text{ and
}\Delta_{\infty}u=u_{ij}u_{i}u_{j}.$

It turns out that the transformation formula of $A^{(p)}$-curvature under a
conformal change gives rise to the $p$-Laplace equation

(1.1.2) $-\Delta_{p}u+\frac{n-p}{2(p-1)}|\nabla u|^{p-2}A^{(p)}(\nabla
u)u=\frac{n-p}{2(p-1)}(|\nabla u|^{p-2}A^{(p)}(\nabla u))[\bar{g}]u^{q}$

for $\bar{g}=u^{\frac{4(p-1)}{n-p}}g$ and $p\in(1,n)$, where $A^{(p)}(\nabla
u)$ is $A^{(p)}$-curvature in $\nabla u$ direction and

$q=\frac{2p(p-1)}{n-p}+1\in(1,\infty).$

For $p=2$, this is simply the well-known scalar curvature equation

(1.1.3) $-\Delta u+\frac{n-2}{2}Ju=\frac{n-2}{2}J[\bar{g}]u^{\frac{n+2}{n-2}}$

for $\bar{g}=u^{\frac{4}{n-2}}\,g$. For $p=n$, we recover the $n$-Laplace
equations introduced in [MQ21a, MQ21b]

(1.1.4)
$-\Delta_{n}\phi+|\nabla\phi|^{n-2}Ric(\nabla\phi)=(|\nabla\phi|^{n-2}Ric(\nabla\phi))[\bar{g}]e^{n\phi}$

for $\bar{g}=e^{2\phi}\,g$. For $p\in(n,\infty)$, the $p$-Laplace equation
(1.1.2) remains valid for both $\frac{4(p-1)}{n-p}$ and
$q=\frac{2p(p-1)}{n-p}+1$ being negative. In fact, taking $p\to\infty$, it
becomes the infinite Laplace equation

(1.1.5) $-\Delta_{\infty}u-\frac{1}{2}|\nabla u|^{2}A(\nabla
u)u=-\frac{1}{2}(|\nabla u|^{2}A(\nabla u))[\bar{g}]u^{-7}$

for $\bar{g}=u^{-4}\,g$.

We want to mention at this point that there have been interests in the study
of quasilinear elliptic equations with the nonlinearities similar to those in
(1.1.2), for example, in [BV21, CHZ22]. In Section 2 we discuss how the
$p$-Laplace equation (1.1.2) interpolates the scalar curvature equation
(1.1.3), $n$-Laplace equations (1.1.4) on Ricci curvature $Ric$, and the
infinite Laplace equations (1.1.5) on the Schouten curvature $A$. We also
compare the positivity of $A^{(p)}$ with other intermediate positivities of
the Schouten curvature tensor $A$.

As a consequence of Lemma 2.3.2 on the comparison of the intermediate
positivities of the Schouten curvature tensor, following the application of
the Böchner formula for harmonic forms in [Nay97, GLW05], we observe (see also
Theorem 5.2.1)

######  Theorem 1.1.1.

Let $(M^{n},g)$ be a compact locally conformally flat manifold with
$A^{(p)}\geq 0$ for $p\in[2,n)$. Suppose that the scalar curvature is positive
somewhere on $M^{n}$. Then, for $\frac{n-p}{2}+1\leq k\leq\frac{n+p}{2}-1$,
the cohomology spaces $H^{k}(M^{n},R)=\\{0\\}$, unless $(M^{n},g)$ is
isometric to a compact quotient of $\mathbb{H}^{r}\times\mathbb{S}^{n-r}$ for
$r=\frac{n-p}{2}+1\leq\frac{n}{2}$.

Notice that the situation is well-understood when $p=n$ (see for instance,
[MQ21b] and references therein). Please check (2.1.2) and see how effective
and sharp the intermediate Schouten curvature $A^{(p)}$ is in detecting the
vanishing of cohomology and recognizing
$\mathbb{H}^{k}\times\mathbb{S}^{n-k}$.

More interestingly, we are able to use the Wolff potential to estimate the
asymptotic behavior of $p$-superharmonic functions at singularities and
conclude

######  Theorem 1.1.2.

Suppose that $S$ is a compact subset of a bounded domain $\Omega$ in
$\mathbb{R}^{n}$. And suppose that there is a metric $\bar{g}$ on
$\Omega\setminus S$ that is conformal to the Euclidean metric and that
$\bar{g}$ is geodesically complete near $S$. Assume that $A^{(p)}[\bar{g}]\geq
0$ for some $p\in[2,n)$. Then

$dim_{\mathcal{H}}(S)\leq\frac{n-p}{2}.$

Again, when $p=n$, it is known that $S$ can only be at most finitely many
points. Please also check again (2.1.2) and see how effective and sharp the
intermediate Schouten curvature $A^{(p)}$ is in estimating the Hausdorff
dimensions and recognizing $\mathbb{H}^{k}\times\mathbb{S}^{n-k}$. There have
been extensive studies on the Hausdorff dimensions in [CQY00, CH02, CHY04,
Gon05, Ca12, GMS12, GM14, Zh18, DY21, HS21, MQ22] following [SY88]. On a
Kleinian manifold $\Omega(\Gamma)/\Gamma$, conversely, a metric is constructed
in [Nay97] such that the scalar curvature of this metric is nonnegative
(positive) if the Hausdorff dimension of the set of limit points $L(\Gamma)$
is $\leq\frac{n-2}{2}$ ($<\frac{n-2}{2}$). The intermediate Schouten curvature
tensor $A^{(p)}$ of the Nayatani’s metrics in [Nay97] does not seem to be
easily understood solely in terms of the Hausdorff dimension $\delta(\Gamma)$
in general.

However, as a consequence of Theorem 1.1.2, similar to that in [SY88], we have
(see also Corollary 5.1.1),

######  Theorem 1.1.3.

Let $(M^{n},g)$ be a compact locally conformally flat manifold with
$A^{(p)}\geq 0$ for $p\in[2,n]$. Then, for $1<k<\frac{n+p}{2}-1$, the homotopy
groups $\pi_{k}(M^{n})$ are trivial.

Theorem 1.1.2 and Theorem 1.1.3 extends some related results in [SY88] (and
for $p=n$ in [MQ21a, MQ21b]) as the intermediate Schouten curvature and the
$p$-Laplace equations substitute the scalar curvature and the scalar curvature
equations.

###  1.2. The Wolff potentials and $p$-thinness

In Section 3 we review definitions and collect some useful facts in potential
theory. And, most importantly, we define the concept of $p$-thinness (cf.
Definition 3.2.2) for the study of singular behaviors of $p$-superharmonic
functions, which turns out to be different from the thinness defined by the
Wiener integral in nonlinear potential theory in general dimensions (cf.
[KM94] for instance). The Wolff potential for this paper is defined as
follows:

$W^{\mu}_{1,p}(x,r)=\int_{0}^{r}(\frac{\mu(B(x,t))}{t^{n-p}})^{\frac{1}{p-1}}\frac{dt}{t}$

for a nonnegative Radon measure $\mu$. It is well known that the Wolff
potential is the same as the classic Newton potential when $p=2$. We obtain in
Section 3 that (see also Theorem 3.3.1)

######  Theorem 1.2.1.

Suppose $\mu$ is a nonnegative finite Radon measure on a bounded domain
$\Omega\subset\mathbb{R}^{n}$. Assume that, for a point $x_{0}\in\Omega$, some
number $m\in(0,n-p)$ and $p\in[2,n)$,

$\mu(B(x_{0},t))\leq Ct^{m}$

for all $t\in(0,3r_{0})$ with $B(x_{0},3r_{0})\subset\Omega$. Then, for any
$\varepsilon>0$ small, there are a subset $E\subset\Omega$, which is $p$-thin
at $x_{0}$, and a constant $C>0$ such that

(1.2.1) $W_{1,p}^{\mu}(x,\,r_{0})\leq
C|x-x_{0}|^{-\frac{n-p-m+\varepsilon}{p-1}}\text{ for all $x\in\Omega\setminus
E$.}$

One crucial property of subsets $E$ which are $p$-thin at $x_{0}$ for us is
Theorem 3.2.1, which states that there is always a ray from $x_{0}$ that
escapes from a small neighborhood of $x_{0}$ without intersecting $E$. It is
remarkable this helps make use of the geodesic completeness in analysis, which
seemed first observed in [AH73] for a more straightforward proof of the main
theorem on open surfaces in [Hu57] (see also [MQ22]).

###  1.3. Organization

The paper is organized as follows: In Section 2 we introduce the $p$-Laplace
equations in conformal geometry and discuss the comparisons of the positive
cones associated with $A^{(p)}$ to other intermediate positivity of the
Schouten curvature. In Section 3 we collect definitions and basic relevant
facts in the study of $p$-superharmonic functions and the nonlinear potential
theory. Most importantly we introduce the concept of $p$-thin in the nonlinear
potential theory and prove Theorem 1.2.1. In Section 4 we prove Theorem 1.1.2
using Theorem 1.2.1. In Section 5 we derive the topological consequences and
prove Theorem 1.1.1 and Theorem 1.1.3.

##  2\. Introduction of $p$-Laplace equations in conformal geometry

Let us recall the transformation formula of Ricci curvature tensor under a
conformal change $\bar{g}=e^{2\phi}g$

$\bar{R}_{ij}=R_{ij}-\Delta\phi
g_{ij}+(2-n)\phi_{i,j}+(n-2)\phi_{i}\phi_{j}+(2-n)|\nabla\phi|^{2}g_{ij}.$

For the scalar curvature, we then have

$e^{2\phi}\bar{R}=R-2(n-1)\Delta\phi-(n-1)(n-2)|\nabla\phi|^{2}.$

###  2.1. The intermediate Schouten curvature and its transformation formula

We write

(2.1.1) $A^{(p)}=\frac{1}{n-2}((p-2)Ric+(n-p)\frac{R}{2(n-1)}\,g),$

where

$A^{(2)}=\frac{R}{2(n-1)}\,g,\ A^{(n)}=Ric,\text{ and
}\text{Tr}A^{(p)}=\frac{n+p-2}{2(n-1)}\,R.$

Notice that $\frac{n-p}{n-2}=1-\frac{p-2}{n-2}$ and $A^{(p)}$ interpolates
between $\frac{R}{2(n-1)}\,g$ and the Ricci curvature tensor $Ric$ as $p$
varies from $2$ to $n$.

For example, one may calculate the intermediate Schouten curvature tensor on
$\mathbb{H}^{k}\times\mathbb{S}^{n-k}$ for $1\leq k\leq\frac{n}{2}$ and
$p=n-2k+2$,

(2.1.2) $\displaystyle A^{(p)}$
$\displaystyle=(p-2)A+Jg=(p-2)(\overset{k}{\overbrace{-\frac{1}{2},\cdots,-\frac{1}{2}}},\overset{n-k}{\overbrace{\frac{1}{2},\cdots,\frac{1}{2}}})+\frac{n-2k}{2}g$
$\displaystyle=(\overset{k}{\overbrace{0,\cdots,0}},\overset{n-k}{\overbrace{n-2k,\cdots,n-2k}})\geq
0.$

setminus Here $\frac{n-p}{2}=k-1$ and $\mathbb{H}^{k}\times\mathbb{S}^{n-k}$
topologically is $\mathbb{S}^{n}\setminus\mathbb{S}^{k-1}$. It is also
interesting to observe that $A^{(q)}>0$ when $q<n-2k+2$ and $A^{(q)}$ is
negative in the first $k$ directions when $q>n-2k+2$.

For $p\in(1,n)$ and $\bar{g}=e^{2\phi}\,g=u^{\frac{4(p-1)}{n-p}}\,g$, we
calculate

(2.1.3) $\displaystyle
A^{(p)}_{ij}[\bar{g}]=A^{(p)}_{ij}-\frac{2(p-1)}{n-p}\left[\frac{\Delta
u}{u}g_{ij}+(p-2)\frac{u_{i,j}}{u}\right]\,$
$\displaystyle+\frac{2(p-1)}{n-p}$
$\displaystyle\left[(1-(n+p-4)\frac{p-1}{n-p})\frac{|\nabla
u|^{2}}{u^{2}}g_{ij}+((p-2)+2(p-2)\frac{p-1}{n-p})\frac{u_{i}u_{j}}{u^{2}}\right].$

Multiplying $u|\nabla u|^{p-2}\frac{u_{i}}{|\nabla u|}\frac{u_{j}}{|\nabla
u|}$ to and summing up on both sides of (2.1.3), we arrive at the $p$-Laplace
equation

(2.1.4)
$-\Delta_{p}u+\frac{n-p}{2(p-1)}S^{(p)}(u)u=\frac{n-p}{2(p-1)}S^{(p)}(u)[\bar{g}]u^{q},$

where

$S^{(p)}(u)=|\nabla u|^{p-2}A^{(p)}(\nabla u),$

$A^{(p)}(\nabla u)$ is the $A^{(p)}$-curvature in the direction of $\nabla u$,
and $q=\frac{2p(p-1)}{n-p}+1\in(1,\infty)$ monotonically increasing when
$p\in(1,n)$.

###  2.2. The $p$-Laplace equations as $p\in(1,\infty)$

Clearly, when $p=2$, (2.1.4) becomes the scalar curvature equation

$-\Delta
u+\frac{n-2}{4(n-1)}Ru=\frac{n-2}{4(n-1)}R[u^{\frac{4}{n-2}}\,g]u^{\frac{n+2}{n-2}}.$

On the other hand, as $p\to n$, we calculate

$u-1=e^{\frac{n-p}{2(p-1)}\phi}-1=\frac{n-p}{2(n-1)}\phi+o(n-p),$
$u^{q}=e^{\frac{(n-p)q}{2(n-1)}\phi}\to e^{n\phi},$ $\nabla
u=\frac{n-p}{2(n-1)}\nabla\phi\,e^{\frac{n-p}{2(n-1)}\phi},$
$(\frac{2(p-1)}{n-p})^{p-1}\Delta_{p}u\to\Delta_{n}\phi,\text{ and
}(\frac{2(n-1)}{n-p})^{p-2}S^{(p)}(u)\to|\nabla\phi|^{p-2}Ric(\nabla\phi).$

Therefore, from (2.1.4), as $p\to n$, we recover the $n$-Laplace equation

$-\Delta_{n}\phi+|\nabla\phi|^{n-2}Ric(\nabla\phi)=(|\nabla\phi|^{n-2}Ric(\nabla\phi))[e^{2\phi}\,g]e^{n\phi},$

which was introduced and studied in [MQ21a, MQ21b]. Thus $p$-Laplace equation
(2.1.4) in some way interpolates between the scalar curvature equation and the
$n$-Laplace equation on Ricci curvature for $p\in[2,n]$ in conformal geometry.

The $p$-Laplace equations (2.1.4) still make sense for $p\in(n,\infty)$. As we
mentioned in the introduction section, we may write

$A^{(p)}=(p-2)A+J\,g$

and therefore $\lim_{p\to\infty}\frac{1}{p-2}A^{(p)}=A$. Also notice that

$\lim_{p\to\infty}\frac{1}{p-2}\frac{1}{|\nabla
u|^{p-4}}\Delta_{p}u=\Delta_{\infty}u=u_{ij}u_{i}u_{j}$

(cf. [Lind06, Chapter 8]). Therefore, from (2.1.4), after dividing both sides
by $(p-2)|\nabla u|^{p-4}$ and taking limit as $p\to\infty$, we get the
infinite Laplace equation on Schouten curvature

(2.2.1) $-\Delta_{\infty}u-\frac{1}{2}|\nabla u|^{2}A(\nabla
u)u=-\frac{1}{2}(|\nabla u|^{2}A(\nabla u))[\bar{g}]u^{-7},$

where $\bar{g}=u^{-4}\,g$. Therefore $p$-Laplace equation (2.1.4) also
interpolates between $n$-Laplace equation on Ricci curvature and the infinite
Laplace equation on Schouten curvature for $p\in(n,\infty)$ in conformal
geometry.

###  2.3. Intermediate positivity of the Schouten curvature tensor

By the discussion about the interpolation nature of the $p$-Laplace equations
(2.1.4) in the previous subsection, the positivity of $A^{(p)}$ describes
stronger and stronger positivity of the Schouten curvature $A$ as $p$
increases in $[2,\infty)$. A way to measure the positivity of $A^{(p)}$ is to
consider the cone

(2.3.1)
$\mathcal{A}^{(p)}=\\{(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})\in\mathbb{R}^{n}:(p-2)\min\\{\lambda_{i}\\}_{i=1}^{n}+\sum_{i=1}^{n}\lambda_{i}\geq
0\\}.$

Obviously, we have

######  Lemma 2.3.1.

  * •

$\mathcal{A}^{(p_{2})}\subset\mathcal{A}^{(p_{1})}\text{ for $p_{1}<p_{2}$}$;

  * •

$\mathcal{A}^{(2)}=\\{(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})\in\mathbb{R}^{n}:\sum_{i=1}^{n}\lambda_{i}\geq
0\\}$ is the baseline;

  * •

$\mathcal{A}^{(n)}$ stands for $Ric\geq 0$ when
$(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})$ represents the curvature tensor
$A$;

  * •

$\mathcal{A}^{(p)}$ approaches the nonnegative cone
$\Gamma^{n}=\\{\lambda\in\mathbb{R}^{n}:\lambda_{i}\geq 0\\}$ as $p\to\infty$.

It is interesting to compare the intermediate positivity cones
$\mathcal{A}^{(p)}$ with those, for example, in connection to the curvature
operator in the Böchner formula for harmonic forms (5.2.1) on locally
conformally flat manifolds. In [Nay97, GLW05], for the positivity of the
curvature operator $\mathcal{R}$ on $r$-forms in (5.2.1), the following
symmetric cones are considered:

(2.3.2)
$\mathcal{R}^{(r)}=\\{(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})\in\mathbb{R}^{n}:\min\\{(n-r)\sum_{k=1}^{r}\lambda_{i_{k}}+r\sum_{k=r+1}^{n}\lambda_{i_{k}}\\}\geq
0\\}$

for $r\in\\{1,2,\cdots,n\\}$, where $(i_{1},i_{2},\cdots,i_{n})$ is any
permutation of $(1,2,\cdots,n)$ and the minimum is taken over all
permutations. We observe the following

######  Lemma 2.3.2.

(2.3.3) $\mathcal{R}^{(s)}\subset\mathcal{R}^{(r)}\quad\text{ for $0<s\leq
r\leq\frac{n}{2}$}\\\ $

and

(2.3.4) $\mathcal{A}^{(p)}\subset\mathcal{R}^{(r)}\quad\text{ for
$\frac{n-p}{2}+1\leq r\leq\frac{n}{2}$}.$

###### Proof.

Assume without loss of generality that
$\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}$. And let

$G(n,s)=(n-s)\sum_{i=1}^{s}\lambda_{i}+s\sum_{i=s+1}^{n}\lambda_{i}.$

We calculate, for $0<s<r\leq\frac{n}{2}$,

(2.3.5) $\displaystyle G(n,r)$
$\displaystyle=(n-r)\sum_{i=1}^{r}\lambda_{i}+r\sum_{i=r+1}^{n}\lambda_{i}$
$\displaystyle=\frac{n-r}{n-s}G(n,s)+\frac{n(r-s)}{n-s}\sum_{i=s+1}^{n}\lambda_{i}+(n-2r)\sum_{i=s+1}^{r}\lambda_{i}$
$\displaystyle\geq\frac{n-r}{n-s}G(n,s)+\frac{n(r-s)}{n-s}\sum_{i=s+1}^{n}\lambda_{i}+\frac{(n-2r)(r-s)}{s}\sum_{i=1}^{s}\lambda_{i}$
$\displaystyle=\frac{n-r}{n-s}G(n,s)+\frac{2r(r-s)}{n-s}\sum_{i=s+1}^{n}\lambda_{i}+\frac{(n-2r)(r-s)}{s(n-s)}G(n,s)$
$\displaystyle\geq\frac{n-r}{n-s}G(n,s)+\frac{2r(r-s)}{n}\sum_{i=1}^{n}\lambda_{i}+\frac{(n-2r)(r-s)}{s(n-s)}G(n,s).$

Hence we may conclude that $\mathcal{R}^{(r)}\subset\mathcal{R}^{(s)}$ when
$0<s<r\leq\frac{n}{2}$. To verify (2.3.4), we may assume the smallest
eigenvalue $\lambda_{1}<0$ for the sake of the argument, because otherwise
$G(n,r)\geq 0$. We calculate

(2.3.6) $\displaystyle\frac{1}{r}G(n,r)$
$\displaystyle=(n-2r)\frac{1}{r}\sum_{i=1}^{r}\lambda_{i}+\sum_{i=1}^{n}\lambda_{i}$
$\displaystyle\geq(n-2r)\lambda_{1}$
$\displaystyle+\sum_{i=1}^{n}\lambda_{i}\geq(p-2)\lambda_{1}+\sum_{i=1}^{n}\lambda_{i}$

if let $n-2r\leq p-2$ or equivalently $\frac{n-p}{2}+1\leq r\leq\frac{n}{2}$.
This completes the proof of this lemma. ∎

We will see in Section 5 that (2.3.4) has some topological consequences from
the Böchner formula for harmonic forms as observed in [Nay97, GLW05], which
further confirms that, more positive the curvature is, more vanishing theorems
on the cohomology there are.

##  3\. $p$-superharmonic functions and the Wolff potentials

In this section, we start with definitions and then collect some relevant
facts in the potential theory for the study of $p$-superharmonic functions. We
also state and prove the key upper bound estimates of the Wolff potentials
(cf. Theorem 3.3.1) for our uses in the following sections.

###  3.1. $p$-superharmonic functions and the Wolff potentials

First we want to give definitions of $p$-harmonic and $p$-superharmonic
functions. In this section we always assume $1<p<n$ unless specified
otherwise. Let us recall again what is the $p$-Laplace operator

$\Delta_{p}u=\text{div}(|\nabla u|^{p-2}\nabla u).$

######  Definition 3.1.1.

([Lind06, Definition 2.5]) We say that $u\in W^{1,p}_{\rm{loc}}(\Omega)$ is a
weak solution of the $p$-harmonic equation in $\Omega$, if

$\int\langle|\nabla u|^{p-2}\nabla u,\nabla\eta\rangle dx=0$

for each $\eta\in C_{0}^{\infty}(\Omega)$. If, in addition, $u$ is continuous,
we then say $u$ is a $p$-harmonic function.

######  Definition 3.1.2.

([Lind06, Definiton 5.1]) A function $v:\Omega\to(-\infty,\infty]$ is called
$p$-superharmonic in $\Omega$, if

  * •

$v$ is lower semi-continuous in $\Omega$;

  * •

$v\not\equiv\infty$ in $\Omega$;

  * •

for each domain $D\subset\subset\Omega$ the comparison principle holds, that
is, if $h\in C(\bar{D})$ is $p$-harmonic in $D$ and $h|_{\partial D}\leq
v|_{\partial D}$, then $h\leq v$ in D.

As stated in [KM92, Theorem 2.1], for $u$ to be a $p$-superharmonic function
in $\Omega$, there is a nonnegative finite Radon measure $\mu$ in $\Omega$
such that

(3.1.1) $-\Delta_{p}u=\mu.$

For more about $p$-superharmonic functions and nonlinear potential theory, we
refer to [KM94, HKM93, HK88, PV08, AH99, Lind06] and references therein. The
most important tool is the following Wolff potential to substitute the role of
Riesz potential in linear potential theory

(3.1.2)
$W^{\mu}_{1,p}(x,\,r)=\int_{0}^{r}(\frac{\mu(B(x,\,t))}{t^{n-p}})^{\frac{1}{p-1}}\frac{dt}{t}$

for any nonnegative Radon measure $\mu$ and $p\in(1,n]$. The fundamental
estimate for the use of the Wolff potential in the study of $p$-superharmonic
functions is as follows:

######  Theorem 3.1.1.

([KM94, Theorem 1.6]) Suppose that $u$ is a nonnegative $p$-superharmonic
function satisfying (3.1.1) for a nonnegative finite Radon measure $\mu$ in
$B(x,3r)$. Then

(3.1.3) $c_{1}W^{\mu}_{1,p}(x,r)\leq u(x)\leq
c_{2}(\inf_{B(x,r)}u+W^{\mu}_{1,p}(x,2r))$

for some constants $c_{1}(n,p)$ and $c_{2}(n,p)$ for $p\in(1,n]$.

It is often useful to have the following existence result in order to fully
use the above estimates (cf. [KM94, Theorem 1.6]).

######  Lemma 3.1.1.

([KM92, Theorem 2.4]) Suppose $\Omega\subset\mathbb{R}^{n}$ is a bounded
domain and $\mu$ is a nonnegative finite Radon measure. There is a nonnegative
$p$-superharmonic function $u(x)$ satisfying (3.1.1) and
$\min\\{u,\lambda\\}\in W^{1,p}_{0}(\Omega)$ for any $\lambda>0$.

###  3.2. $p$-capacity and $p$-thinness

The $p$-Laplace equation (3.1.1) is naturally related to the energy functional
$\int_{\Omega}|\nabla u|^{p}dx$. Therefore the following $p$-capacity in terms
of the “energy" defined, for example, in [HKM93, Chapters 2 and 4] and [KM94,
Section 3.1], replaces the capacity in terms of the “charges" for studying
Riesz potentials.

######  Definition 3.2.1.

([KM94, Section 3.1]) For a compact subset $K$ of a domain $\Omega$ in
$\mathbb{R}^{n}$, we define

(3.2.1) $cap_{p}(K,\Omega)=\inf\\{\int_{\Omega}|\nabla u|^{p}dx:\text{ $u\in
C_{0}^{\infty}(\Omega)$ and $u\geq 1$ on $K$}\\}.$

Then $p$-capacity for arbitrary subset $E$ of $\Omega$ is

(3.2.2) $cap_{p}(E,\Omega)=\inf_{\text{open}\ G\supset E\text{ \&
}G\subset\Omega}\ \ \sup_{\text{compact}\ K\subset G}cap_{p}(K,\Omega).$

We would like to mention some basic facts about the capacity
$cap_{p}(E,\Omega)$ which are important for the arguments in this paper.
Readers are referred to [HKM93, KM94].

######  Lemma 3.2.1.

  1. (1)

Countable sub-additivity:

$cap_{p}(\bigcup_{k}E_{k},\Omega)\leq\sum_{k}cap_{p}(E_{k},\Omega)$

for a countable collection of subsets $\\{E_{k}\\}$ of $\Omega$.

  2. (2)

Scaling property:

$cap_{p}(E_{\lambda},\Omega_{\lambda})=\lambda^{n-p}cap_{p}(E,\Omega)$

where $A_{\lambda}=\\{\lambda x\in\mathbb{R}^{n}:x\in A\\}$ is the scaled set
of $A\subset\mathbb{R}^{n}$.

  3. (3)

Lipschitz property: Let

$\Phi:\mathbb{R}^{n}\to\mathbb{R}^{n}$

be a Lipschitz map, that is, there is a constant $C_{0}>0$ such that

$|\Phi(x)-\Phi(y)|\leq C_{0}|x-y|\text{ for all $x,y\in\mathbb{R}^{n}$}.$

Then, there is a constant $C>0$ such that

$cap_{p}(\Phi(E),\mathbb{R}^{n})\leq Ccap_{p}(E,\mathbb{R}^{n}).$

  4. (4)

Positivity:

$cap_{p}(\partial B(0,r),B(0,2r))=c(n,p)r^{n-p}>0.$

###### Proof.

The proof and references are here: (1) is from [HKM93, Theorem 2.2]; (2) is
easily verified; (3) is easily been verified based on Definition 3.2.1 (see
also [AH99, Theorem 5.2.1]); (4) has been explicitly computed, for instance,
[HKM93, Eample 2.12]. ∎

The following is also very important to us.

######  Lemma 3.2.2.

([KM94, Lemma 3.9]) Suppose that $u\in W_{0}^{1,p}(\Omega)$ is
$p$-superharmonic functions. Then for $\lambda>0$ it holds

(3.2.3)
$\lambda^{p-1}cap_{p}(\\{x\in\Omega:u(x)>\lambda\\},\Omega)\leq\mu(\Omega).$

The notions of thinness in the potential theory are vitally important. The
readers are referred to [Br40, Br44, AM72, AH73, HK88, KM94, MQ21a] for
detailed discussions and references therein. The discussion in [MQ21a, Section
2.3] for the two different notions of thinness, which turns out to be
equivalent in dimension 2 due to [Br44, Theorem 2], is relevant. To deal with
the singular behavior of $p$-superharmonic functions, like [MQ21a, Definition
3.1] (see also [Mi96, Section 2.5]), we propose a thinness that is less
restrictive than that by the Wiener type integral when $p\in(2,n)$. Let

$\displaystyle\omega_{i}(x_{0})$ $\displaystyle=$
$\displaystyle\\{x\in\mathbb{R}^{n}:2^{-i-1}\leq|x-x_{0}|\leq 2^{-i}\\};$
$\displaystyle\Omega_{i}(x_{0})$ $\displaystyle=$
$\displaystyle\\{x\in\mathbb{R}^{n}:2^{-i-2}\leq|x-x_{0}|\leq 2^{-i+1}\\}.$

######  Definition 3.2.2.

A set $E\subset\mathbb{R}^{n}$ is said to be $p$-thin for $p\in(1,n)$ at
$x_{0}\in\mathbb{R}^{n}$ if

(3.2.4)
$\sum_{i=1}^{\infty}\frac{cap_{p}(E\cap\omega_{i}(x_{0}),\Omega_{i}(x_{0}))}{cap_{p}(\partial
B(x_{0},2^{-i}),B(x_{0},2^{-i+1}))}<+\infty.$

Meanwhile a set $E$ is said to be $n$-thin at $x_{0}\in\mathbb{R}^{n}$ if

(3.2.5)
$\sum_{i=1}^{\infty}i^{n-1}\,cap_{n}(E\cap\omega_{i}(x_{0}),\Omega_{i}(x_{0}))<+\infty.$

To compare, for the thinness used in earlier works, for instance, [KM94,
Theorem 1.3], which has been confirmed to be equivalent to the Cartan property
(cf. [KM94, (1.4)]), we recall that

$W_{p}(E,x_{0})=\int_{0}^{1}(\frac{cap_{p}(B(x_{0},t)\cap
E,B(x_{0},2t))}{cap_{p}(B(x_{0},t),B(x_{0},2t))})^{\frac{1}{p-1}}\frac{dt}{t}<+\infty.$

or equivalently (cf. [HKM93, Lemma 12.10])

(3.2.6) $\sum_{i=1}^{\infty}(\frac{cap_{p}(B(x_{0},2^{-i})\cap
E,B(x_{0},2^{-i+1}))}{cap_{p}(B(x_{0},2^{-i}),B(x_{0},2^{-i+1}))})^{\frac{1}{p-1}}<+\infty.$

Clearly, a set $E\subset\Omega\subset\mathbb{R}^{n}$ is $p$-thin at
$x_{0}\in\Omega$ by Definition 3.2.2 if it is $p$-thin at $x_{0}$ by (3.2.6)
when $p\in(2,n)$ (cf. [Mi96, Remark 5.1 in Chapter 2]). The most important
fact about $p$-thin to us in this paper is the following (cf. [MQ22, Theorem
2.1])

######  Theorem 3.2.1.

Let $E$ be a subset in the Euclidean space $R^{n}$ and $x_{0}\in R^{n}$ be a
point. Suppose that $E$ is $p$-thin at the point $x_{0}$ for $p\in(1,n]$. Then
there is a ray from $x_{0}$ that avoids $E$ at least within some small ball at
$x_{0}$.

###### Proof.

In the light of properties in Lemma 3.2.1 the proof goes exactly as the proof
of [MQ22, Theorem 2.1]. Applying the rescaling from $\Omega_{i}$ to
$\Omega_{-2}$ and the projection from $\Omega_{-2}$ to the unit sphere, it is
then easily seen that the $p$-capacity of the union of all the projections of
rescaled $\\{E\cap\omega_{i}\\}_{i=k}^{\infty}$ is smaller than the capacity
of the unit sphere in $\Omega_{0}$ when $k$ is sufficiently large, which
guarantees that there are rays from $x_{0}$ that avoids $E$ in some small
ball. ∎

###  3.3. The key estimates of the Wolff potential

In this subsection, we want to state and prove the following estimates on the
Wolff potentials, which is essential for our investigation of singularities of
$p$-superharmonic functions in Section 4.

######  Theorem 3.3.1.

Suppose $\mu$ is a nonnegative finite Radon measure in $\Omega$. Assume that,
for a point $x_{0}\in\Omega$ and some number $m\in(0,n-p)$,

(3.3.1) $\mu(B(x_{0},t))\leq Ct^{m}$

for all $t\in(0,3r_{0})$ with $B(x_{0},3r_{0})\subset\Omega$. Then, for
$\varepsilon>0$, there are a subset $E\subset\Omega$, which is $p$-thin at
$x_{0}$, and a constant $C>0$ such that

(3.3.2) $W_{1,p}^{\mu}(x,\,r_{0})\leq
C|x-x_{0}|^{-\frac{n-p-m+\varepsilon}{p-1}}\text{ for all $x\in\Omega\setminus
E$}$

for $p\in[2,n)$.

###### Proof.

The proof of this theorem is in principle similar to the proof of [MQ22,
Theorem 2.2]. For $x\in\omega_{i}$, let us write

(3.3.3) $\displaystyle W_{1,p}^{\mu}$
$\displaystyle(x,r_{0})=\int_{0}^{r_{0}}(\frac{\mu(B(x,t))}{t^{n-p}})^{\frac{1}{p-1}}\frac{dt}{t}$
$\displaystyle=\left\\{\int_{2^{-i_{0}}}^{r_{0}}+\int_{2^{-i-1}}^{2^{-i_{0}}}+\int_{2^{-i-2}}^{2^{-i-1}}+\int_{0}^{2^{-i-1}}\right\\}(\frac{\mu(B(x,t))}{t^{n-p}})^{\frac{1}{p-1}}\frac{dt}{t}.$

for some large $i_{0}$ to be fixed by $r_{0}$. For the first term in the right
side of (3.3.3)

$\displaystyle\int_{2^{-i_{0}}}^{r_{0}}$
$\displaystyle(\frac{\mu(B(x,t))}{t^{n-p}})^{\frac{1}{p-1}}\frac{dt}{t}\leq\frac{p-1}{n-p}\mu(\Omega)^{\frac{1}{p-1}}(2^{i_{0}})^{\frac{n-p}{p-1}}$
$\displaystyle\leq
C(n,p,\varepsilon,i_{0})|x-x_{0}|^{-\frac{n-p-m+\varepsilon}{p-1}}.$

For the second term in the right side of (3.3.3)

$\displaystyle\int_{2^{-i-1}}^{2^{-i_{0}}}$
$\displaystyle(\frac{\mu(B(x,t))}{t^{n-p}})^{\frac{1}{p-1}}\frac{dt}{t}=\sum_{k=-i-1}^{-i_{0}-1}\int_{2^{k}}^{2^{k+1}}(\frac{\mu(B(x,t))}{t^{n-p}})^{\frac{1}{p-1}}\frac{dt}{t}$
$\displaystyle\leq$
$\displaystyle\sum_{k=-i-1}^{-i_{0}}\frac{p-1}{n-p}(\mu(B(x_{0},2^{k+2}))^{\frac{1}{p-1}}(2^{k})^{\frac{n-p}{p-1}}$
$\displaystyle\leq$ $\displaystyle
c(n,p)(\sum_{l=0}^{i-i_{0}}((2^{\frac{n-p-m}{p-1}})^{-l})|x-x_{0}|^{-\frac{n-p-m}{p-1}}$
$\displaystyle\leq$ $\displaystyle
C(n,p,m,\varepsilon)|x-x_{0}|^{-\frac{n-p-m+\varepsilon}{p-1}}.$

For the third term in the right side of (3.3.3)

$\displaystyle\int_{2^{-i-2}}^{2^{-i-1}}$
$\displaystyle(\frac{\mu(B(x,t))}{t^{n-p}})^{\frac{1}{p-1}}\frac{dt}{t}\leq\frac{p-1}{n-p}\mu(B(x_{0},2^{-i+1}))^{\frac{1}{p-1}}(2^{-i-2})^{-\frac{n-p}{p-1}}$
$\displaystyle\leq
C(n,p,\varepsilon)|x-x_{0}|^{-\frac{n-p-m+\varepsilon}{p-1}}.$

For the fourth term in the right side of (3.3.3), we now switch the gear and
estimate $cap_{p}(E_{i},\Omega_{i})$, where

$E_{i}=\\{x\in\omega_{i}:W_{1,p}^{\mu}(x,2^{-i-2})\geq
C|x-x_{0}|^{-\frac{n-p-m+\varepsilon}{p-1}}\\}.$

For this purpose, we let $u_{i}$ be the nonnegative $p$-superharmonic function
that solves

$\left\\{\begin{aligned} -\Delta_{p}u_{i}&=\mu\text{ in $\Omega_{i}$}\\\
u_{i}&=0\text{ on $\partial\Omega_{i}$}.\end{aligned}\right.$

The existence of such $u_{i}$ is given by Lemma 3.1.1 (cf. [KM92, Theorem
2.4]). Applying Theorem 3.1.1 (cf. [KM94, Theorem 1.6]), we have

$c(n,p)W_{1,p}^{\mu}(x,2^{-i-2})\leq u_{i}(x)\text{ for $x\in\omega_{i}$.}$

Therefore, in the light of Lemma 3.2.2 (cf. [KM94, Lemma 3.9]), we have

$cap_{p}(E_{i},\Omega_{i})\leq
c(n,p)\mu(\Omega_{i})|x-x_{0}|^{n-p-m+\varepsilon}\leq
c(n,p)2^{-(n-p)i}2^{-\varepsilon i}.$

Thus, if let $E=\bigcup_{i}E_{i}$, we have

$\sum_{i}2^{(n-p)i}cap_{p}(E\bigcap\omega_{i},\Omega_{i})\leq
c(n,p)\sum_{i}2^{-\varepsilon i}<\infty,$

which implies that $E$ is $p$-thin according to Definition 3.2.2. Here we have
used the facts that are stated as (2) and (4) in Lemma 3.2.1. So the proof is
completed. ∎

##  4\. Applications in conformal geometry

In this section we will use the $p$-Laplace equation (2.1.4) and Theorem 3.3.1
to derive the consequence of the curvature condition $A^{(p)}\geq 0$ and prove
Theorem 1.1.2. Let us first restate it as follows:

######  Theorem 4.0.1.

Suppose that $S$ is a compact subset of a bounded domain
$\Omega\subset\mathbb{R}^{n}$. And suppose that there is a metric $\bar{g}$ on
$\Omega\setminus S$ such that

  * •

it is conformal to the Euclidean metric $g_{\mathbb{E}}$;

  * •

it is geodesically complete near $S$.

Assume that $A^{(p)}[\bar{g}]\geq 0$ for some $p\in[2,n)$. Then

$dim_{\mathcal{H}}(S)\leq\frac{n-p}{2}$

for $p\in[2,n)$.

###### Proof.

Let $\bar{g}=u^{\frac{4(p-1)}{n-p}}\,g_{\mathbb{E}}$ for $u>0$ in
$\Omega\setminus S$. First, in the light of (2.1.4) and the assumption on the
curvature $A^{(p)}[\bar{g}]$, we know

(4.0.1) $-\Delta_{p}u=\frac{n-p}{2(p-1)}|\nabla u|^{p-2}A^{(p)}(\nabla
u)u^{q}=f\geq 0\text{ in $\Omega\setminus S$.}$

Step 1: We claim that $u$ is actually a $p$-superharmonic function such that

(4.0.2) $-\Delta_{p}u=\mu\text{ in $\Omega$}$

for some nonnegative finite Radon measure $\mu$ on $\Omega$. Due to [MQ21b,
Lemma 3.1] (see also [CHY04, Proposition 8.1]), we know

$u(x)\to\infty\text{ as $x\to S$}$

since the scalar curvature of the conformal metric $\bar{g}$ is nonnegative
from $A^{(p)}[\bar{g}]\geq 0$. In fact, from [MQ22, Theorem 3.1], we also know

$dim_{\mathcal{H}}(S)\leq\frac{n-2}{2}.$

To derive (4.0.2) for some nonnegative finite Radon measure $\mu$, as in the
proof of [MQ22, Lemma 3.2] (see also [DHM97, MQ21b]), we use the cut-off
function based on the level sets of $u$. Let

$\alpha_{s}(t)=\left\\{\begin{aligned} t\quad&t\in[0,s];\\\
\text{increasing}\quad&t\in[s,10s];\\\
2s\quad&t\in[10s,\infty)\end{aligned}\right.$

be a function satisfying $\alpha_{s}^{\prime}=\frac{d}{dt}\alpha_{s}\in[0,1]$
and $\alpha_{s}^{\prime\prime}=\frac{d^{2}}{dt^{2}}\alpha_{s}\leq 0$. The
proof from here goes exactly the same line by line of the proof of [MQ22,
Lemma 3.2] with little changes. And the non-negativity of the finite Radon
measure $(-\Delta_{p}u)$ is readily verified due to (4.0.1).

Step 2: Next we apply Theorem 3.3.1 to derive the Hausdorff dimension
estimate. The approach runs similarly to the proof of [MQ22, Theorem 3.1].
Assume otherwise

$dim_{\mathcal{H}}(S)>\frac{n-p}{2}.$

Therefore, there is $\varepsilon>0$ such that
$\mathcal{H}_{\frac{n-p+3\varepsilon}{2}}(S)=\infty$. Applying [Kp19,
Proposition 1.4] (see also [MQ22, Lemma 2.7]), there is a point $x_{0}\in S$
such that

$\mu(B(x_{0},t))\leq Ct^{\frac{n-p+3\varepsilon}{2}}$

for all $t\in(0,r_{0})$ and $B(x_{0},3r_{0})\subset\Omega$. In the light of
Theorem 3.3.1, we have

(4.0.3) $W_{1,p}^{\mu}(x,\,r_{0})\leq
C|x-x_{0}|^{-\frac{n-p-\varepsilon}{2(p-1)}}\text{ for all
$x\in\Omega\setminus E$}$

for some $E$ that is $p$-thin at $x_{0}$. To get the upper bound for the
$p$-superharmonic function $u$, we have, from Theorem 3.1.1 (see also [KM94,
Theorem 1.6]), we have

$u(x)\leq C(\inf_{B(x,r_{0})}u(y)+W^{\mu}_{1,p}(x,2r_{0}))$

for $x\in B(x_{0},r_{0})$. In the light of (4.0.3), we have

(4.0.4) $u(x)\leq C\frac{1}{|x-x_{0}|^{\frac{n-p-\varepsilon}{2(p-1)}}}$

for $x\in B(x_{0},r_{0})\setminus(E\cup\\{x_{0}\\})$ for a subset $E$ that is
$p$-thin at $x_{0}$.

Step 3: In this final step we use the geodesic completeness of the conformal
metric $\bar{g}=u^{\frac{4(p-1)}{n-p}}g_{\mathbb{E}}$ and the fact that there
is a ray $\Gamma$ from $x_{0}$ that avoids $E$ in some small ball around
$x_{0}$ because $E$ is $p$-thin at $x_{0}$ according to Theorem 3.2.1. On this
ray $\Gamma$, we calculate the length with respect to the conformal metric
$\bar{g}$

$\int_{0}^{l_{0}}u^{\frac{2(p-1)}{n-p}}dr\leq
C\int_{0}^{l_{0}}\frac{1}{r^{\frac{n-p-\varepsilon}{n-p}}}dr<\infty$

for some $l_{0}>0$ based on (4.0.4), which contradicts with the geodesic
completeness of $\bar{g}$. Therefore

$dim_{\mathcal{H}}(S)\leq\frac{n-p}{2}.$

Thus the proof is complete. ∎

##  5\. Vanishing of Topology

In this section we discuss topological consequences for locally conformally
flat manifolds to have $A^{(p)}\geq 0$. On homotopy groups, we take the
approach based on the work in Section 4 and [SY88]. On cohomology spaces, we
derive some consequences from the comparison of positive cones with the help
from [Nay97, GLW05].

###  5.1. On homotopy groups

In this subsection we follow the set-up from [SY88] and use the development
map to unfold a compact locally conformally flat manifold in $\mathbb{S}^{n}$.
We always assume the dimension is greater than $2$. Let us first restate
Theorem 4.0.1 for subsets in sphere $\mathbb{S}^{n}$.

######  Theorem 5.1.1.

Suppose that $S$ is a closed subset of the sphere $\mathbb{S}^{n}$. And
suppose that there is a metric $\bar{g}$ on $\mathbb{S}^{n}\setminus S$ that
is conformal to the standard round metric $g_{\mathbb{S}}$. Assume that it is
geodesically complete near $S$ and that $A^{(p)}[\bar{g}]\geq 0$ for some
$p\in[2,n)$. Then

$dim_{\mathcal{H}}(S)\leq\frac{n-p}{2}.$

###### Proof.

Pick a point $x_{\infty}$ outside $S$ in $\mathbb{S}^{n}$ and take the
stereographic projection from $\mathbb{S}^{n}\setminus x_{\infty}$ to
$\mathbb{R}^{n}$. Let $\Omega$ be a bounded and open neighborhood of $S$ in
$\mathbb{R}^{n}$. Now this theorem simply follows from Theorem 4.0.1. ∎

Next let us consider compact locally conformally flat manifolds $(M^{n},g)$
with $A^{(p)}\geq 0$ for $p\in[2,n)$. By [SY88], we know the development map,
the conformal immersion from the universal covering $\tilde{M}^{n}$ into
$\mathbb{S}^{n}$,

$\tilde{M}^{n}$$\mathbb{S}^{n}$$M^{n}$$\Phi$$\pi$

is injective when the scalar curvature $R[g]$ is nonnegative. Then the image
$\Omega=\Phi(\tilde{M}^{n})$ in $\mathbb{S}^{n}$ comes with the metric
$(\Phi^{-1})^{*}(\tilde{g})=\bar{g}$ that is conformal to the standard round
metric $g_{\mathbb{S}}$, where $\tilde{g}$ is the lift of $g$ on
$\tilde{M}^{n}$.

######  Corollary 5.1.1.

Suppose that $(M^{n},g)$ is a compact locally conformally flat manifold with
$A^{(p)}\geq 0$ for some $p\in(2,n)$. Then the homotopy groups
$\pi_{k}(M^{n})$ vanishes for all $1<k<\frac{n+p-2}{2}$.

###### Proof.

The proof goes exactly like that in [SY88]. First, by Theorem 5.1.1 in the
above, we conclude

$dim_{\mathcal{H}}(\mathbb{S}^{n}\setminus\Omega)\leq\frac{n-p}{2}.$

This is to say that the co-dimension of $\mathbb{S}^{n}\setminus\Omega$ is at
least $\frac{n+p}{2}$, which implies any map

$h:\mathbb{S}^{k}\to\Omega\subset\mathbb{S}^{n}$

is homotopic to a trivial map without across $\mathbb{S}^{n}\setminus\Omega$
if $0<k<\frac{n+p}{2}-1$. Therefore the homotopy groups $\pi_{k}(\Omega)$
vanishes and therefore the homotopy groups $\pi_{k}(M^{n})$ vanishes when
$1<k<\frac{n+p}{2}-1$ by the property of covering maps. The proof is complete.
∎

###  5.2. On Betti numbers

In this subsection we draw corollaries from the vanishing theorem in [Nay97,
GLW05] and the comparison of positive cones in Section 2.3. Let us recall from
[Nay97, GLW05] the Böchner formula for harmonic forms

(5.2.1) $\Delta\omega=\nabla^{*}\nabla\omega+\mathcal{R}\omega$

where $\Delta=d^{*}d+dd^{*}$ is the Hodge Laplace, $\nabla^{*}\nabla$ is on
the other hand the rough Laplace, and the curvature operator

$\mathcal{R}\omega=\sum_{i,j=1}^{n}\omega_{i}\wedge\iota_{e_{j}}(R(e_{i},e_{j})\omega)$

where $\\{e_{k}\\}$ is a basis, $\\{\omega_{k}\\}$ is the dual basis, and
$R(e_{i},e_{j})$ is the Riemann curvature operator on $r$-forms. Recall

$R_{ijkl}=W_{ijkl}+A_{ik}g_{jl}-A_{il}g_{jk}+A_{jl}g_{ik}-A_{jk}g_{il}.$

Particularly, when the metric is locally conformally flat, i.e. $W=0$, one has

$R_{ijkl}=A_{ik}g_{jl}-A_{il}g_{jk}+A_{jl}g_{ik}-A_{jk}g_{il}.$

So, for an orthonormal basis $\\{e_{j}\\}$ under which $A$ is diagonalized
with eigenvalues $\\{\lambda_{j}\\}$, we have

$R_{ijkl}=\lambda_{i}(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})+\lambda_{j}(\delta_{jl}\delta_{ik}-\delta_{jk}\delta_{il})$

and therefore

$\mathcal{R}\omega=((n-r)\sum_{i=1}^{r}\lambda_{i}+r\sum_{i=r+1}^{n}\lambda_{i})\omega$

for $\omega=\omega_{1}\wedge\omega_{2}\cdots\wedge\omega_{r}$. Thus

$<\mathcal{R}\omega,\omega>\geq 0\text{ when }\lambda(A)\in\mathcal{R}^{(r)}$

for any $r$-form $\omega$ according to (2.3.2). Similar to the proof of
[GLW05, Proposition 2.1] , using Lemma 2.3.2 in Section 2.3, we have the
following restatement of Theorem 1.1.1:

######  Theorem 5.2.1.

Suppose that $(M^{n},g)$ is a compact locally conformally flat manifold with
$A^{(p)}\geq 0$ for some $p\in[2,n)$. And suppose that the scalar curvature is
positive at some points. Then, for $\frac{n-p}{2}+1\leq r\leq\frac{n+p}{2}-1$,
the cohomology spaces $H^{r}(M^{n},R)=\\{0\\}$, unless $(M^{n},g)$ is
isometric to a quotient of $\mathbb{H}^{r}\times\mathbb{S}^{n-r}$ for
$r=\frac{n-p}{2}+1<\frac{n}{2}$.

###### Proof.

We focus on the cases $p\in(2,n)$. Recall that, if $\mathcal{R}\geq 0$ and
$\mathcal{R}>0$ at some point on $r$-forms, then $H^{q}(M^{n},R)=\\{0\\}$ for
$r\leq q\leq n-r$, which follows from the Böchner formula (5.2.1). In the
light of (2.3.4) in Lemma 2.3.2, $\lambda(A)\in\mathcal{R}^{(r)}$ when
$\frac{n-p}{2}+1\leq r\leq\frac{n}{2}$ and $A^{(p)}\geq 0$. It is also easily
seen that $\lambda(A)$ is in the interior of $\mathcal{R}^{(r)}$ if
$\frac{1}{2(n-1)}R>0$, unless $\frac{n-p}{2}$ is an integer and the first
$r=\frac{n-p}{2}+1$ eigenvalues of the Schouten tensor are the same in the
light of (2.3.5) and (2.3.6), and in these cases, one has to skip
$\frac{n-p}{2}+1$ and $\frac{n+p}{2}-1$. We claim those exceptional cases one
skips are very specific, and in fact, based on the rigidity result in the
proof of [GLW05, Proposition 2.1 (iii)] following [Lich63] (see also [Laf88]),
they are simply the cases where $(M^{n},g)$ is isometric to a quotient of
$\mathbb{H}^{r}\times\mathbb{S}^{n-r}$. ∎

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Huajie Liu: Department of Mathematics, Nankai University, Tianjin, China;
e-mail<EMAIL_ADDRESS>
Shiguang Ma: Department of Mathematics, Nankai University, Tianjin, China;
e-mail<EMAIL_ADDRESS>
Jie Qing: Department of Mathematics, University of California, Santa Cruz, CA
95064;
e-mail<EMAIL_ADDRESS>
Research of this author is partially supported by Simons Foundation.

Shuhui Zhong: School of Mathematics, Tianjin University, Tianjin, China;
e-mail<EMAIL_ADDRESS>


# Efficient similarity-based passive filter pruning for compressing CNNs

###### Abstract

Convolution neural networks (CNNs) have shown great success in various
applications. However, the computational complexity and memory storage of CNNs
is a bottleneck for their deployment on resource-constrained devices. Recent
efforts towards reducing the computation cost and the memory overhead of CNNs
involve similarity-based passive filter pruning methods. Similarity-based
passive filter pruning methods compute a pairwise similarity matrix for the
filters and eliminate a few similar filters to obtain a small pruned CNN.
However, the computational complexity of computing the pairwise similarity
matrix is high, particularly when a convolutional layer has many filters. To
reduce the computational complexity in obtaining the pairwise similarity
matrix, we propose to use an efficient method where the complete pairwise
similarity matrix is approximated from only a few of its columns by using a
Nyström approximation method. The proposed efficient similarity-based passive
filter pruning method is 3 times faster and gives same accuracy at the same
reduction in computations for CNNs compared to that of the similarity-based
pruning method that computes a complete pairwise similarity matrix. Apart from
this, the proposed efficient similarity-based pruning method performs
similarly or better than the existing norm-based pruning methods. The efficacy
of the proposed pruning method is evaluated on CNNs such as DCASE 2021 Task 1A
baseline network and a VGGish network designed for acoustic scene
classification.

Index Terms—  Acoustic scene classification, pruning, VGGish, DCASE.

##  1 Introduction

Compressing convolutional neural networks (CNNs) is crucial to reduce their
computational complexity and memory storage for efficient deployment on
resource-constrained devices [1], despite state-of-the-art performances of
CNNs in various applications [2]. Typically, CNNs have redundant parameters
such as weights or filters, which yield only extra computations and storage
without contributing much to the performance of the underlying task [3, 4].
For example, Singh et al. [5, 6] found that 73% of the filters in SoundNet
that do not provide discriminative information across different acoustic scene
classes, and eliminating such filters gives similar performance compared to
that of using all filters in SoundNet. Thus, the compression of CNNs has
recently drawn significant attention from the research community.

Recent efforts towards compressing CNNs involve filter pruning methods [7, 8]
that eliminate some of the filters in CNNs based on their importance. The
importance of the CNN filters is measured in an active or in a passive manner.
Active filter pruning methods involve a dataset. For example, some methods [9,
10, 11] use feature maps which are outputs produced by the filters
corresponding to a set of examples, and apply metrics such as entropy or the
average percentage of zeros on the feature maps to quantify the filter
importance. On the other hand, passive filter pruning methods [12, 13] use
only parameters of the filters, such as an absolute sum of the weights in the
filters, to quantify the filter importance. The passive filter pruning methods
do not involve a dataset to measure filter importance and therefore are easier
to apply compared to active filter pruning methods. After eliminating filters
from the CNNs, the pruned network is fine-tuned to regain some of the
performance lost due to the filter elimination.

Fig. 1: An illustration of output produced in a convolution layer by three CNN
filters, $\mathbf{F}^{1}$, $\mathbf{F}^{2}$ and $\mathbf{F}^{3}$, with a
convolution operation on randomly generated data points,
$\mathbf{X}\in\mathbb{R}^{2\times 1000}$.

Previously, passive filter pruning methods used norm-based metrics such as
$l_{1}$-norm [12], which is a sum of the absolute values of each weight in the
filter, or $l_{2}$-distance of the filters from a geometric median of all
filters [13] to quantify the importance of the filters. These norm-based
methods use a “smaller-norm-less-important” criterion to eliminate filters.
For example, a filter having a relatively high $l_{1}$-norm is considered more
important than others. However, while selecting relatively high-norm filters
as important, norm-based methods may ignore the redundancy among the high-norm
filters. To illustrate this, we show outputs produced by three filters in
Figure 1. Filters $\mathbf{F}^{1}$ and $\mathbf{F}^{3}$ have similar
$l_{1}$-norm and produce similar outputs. However, selecting two important
filters out of the three filters shown in Figure 1, the norm-based method
selects filters $\mathbf{F}^{1}$ and $\mathbf{F}^{3}$ as important due to
their relatively high norm, despite producing similar outputs, while it
eliminates filter $\mathbf{F}^{2}$ that produces significantly different
output than the other filters. Thus the diversity learned in the network may
be ignored.

To capture diversity in the network, similarity-based methods are employed
that eliminate similar filters with an assumption that the similar filters
produce similar or redundant outputs. For example, Kim et al. [14] perform
clustering on filters and selects a filter from each cluster as important and
eliminates the other filters. Singh et al. [15] measure similarity between
filters by computing a pairwise cosine distance for all filters and then
eliminating a filter from a pair of similar filters. Such similarity-based
methods give better performance compared to norm-based methods. However,
similarity-based pruning methods involve a similarity matrix that takes
$\mathcal{O}(n^{2}d)$ computations to compute for $n$ filters having $d$
parameters. Due to this, the computational complexity is high, particularly
when there is a large number of filters in the convolutional layer.

In this work, we propose passive filter pruning method for CNNs to reduce
their computational complexity and memory storage by using a Nyström
approximation [16] to approximate the similarity matrix using only a few
columns of the complete similarity matrix. We evaluate the proposed pruning
framework on acoustic scene classification using two CNNs, DCASE 2021 Task 1A
baseline network [1] and VGGish network [17].

The rest of this paper is organised as follows. Section 2 explains efficient
similarity-based passive filter pruning method. Experimental setup is included
in Section 3. Section 4 presents results and analysis. Finally, conclusion is
included in Section 5.

##  2 Efficient Similarity-based passive filter pruning method

Consider a set of $n$ filters, $\mathbf{F}^{l},1\leq l\leq n$ each of size
($w\times h\times c$) with $w$ is a width, $h$ is a height and $c$ is the
number of channels, in a convolution layer of a CNN. Each filter is
transformed to a 2D matrix of size ($d\times c$) without loss of generality
with $d=wh$. Next, we compute a Rank-1 approximation of the filter by
performing singular value decomposition (SVD) on the transformed 2D filter.
Next, a column $\in\mathbb{R}^{d}$ with unit norm from the Rank-1
approximation of $\mathbf{F}^{l}$ is chosen as a representative of the
corresponding filter. Let $\mathbf{R}\in\mathbb{R}^{d\times n}$ denotes the
filter representative matrix which is constructed by stacking the filter
representatives of the $n$ filters.

Data: Pair-wise similarity matrix of $n$ filters with $m$ filters,
$\mathbf{C}$ $=[\mathbf{W}\hskip
13.37282pt\mathbf{A}]^{\text{T}}\in\mathbb{R}^{n\times m}$, $m<<n$.

Result: Indices of important filters (Imp_list).

(Step 1): Obtaining distance matrix via approximating $\mathbf{S}$

$\mathbf{W}=\mathbf{U}\Sigma\mathbf{U}^{\text{T}}$,

$\mathbf{W}_{k}^{+}$ = $\sum_{j=1}^{k}$
$\sigma_{j}^{-1}\mathbf{U}^{j}\mathbf{U}^{j^{\text{T}}}$,

$\tilde{\mathbf{S}}=\mathbf{C}\mathbf{W}_{k}^{+}\mathbf{C}^{\text{T}}$,

$\tilde{\mathbf{Z}}=\mathbf{1}-\tilde{\mathbf{S}}.$ %Distance matrix

(Step 2): Identify important filter indices

Q= [ ], Imp_list =[ ], Red_list = [ ]

for  _$l\leq n$_ do

[$q$, $D$] = argmin{ $\tilde{\mathbf{Z}}$[ $l$, : - {$l$}] }  %Identify the
closet filter with index $q$ to $l^{\text{th}}$ filter with their distance $D$

Q.append($(l,q)$, $D$)

end for

Q_sort = Sort(Q)  %Sort Q based on the distance $D$

for  _$i\leq len(\text{{Q}})$_ do

id_imp = Q_sort[$i$][0]  %important filter index

id_red = Q_sort[$i$][1]  %redundant filter index

if  _id_imp $\notin$ Red_list_ then

Imp_list.append( id_imp )

Red_list.append( id_red )

end if

end for

Algorithm 1 Efficient similarity-based pruning algorithm to identify important
filters in a convolution layer.

Given $\mathbf{R}$, we identify a small set of important filters out of total
$n$ filters in a given convolutional layer based on the similarity between the
filters using the following two steps:

(Step 1) Approximating distance matrix: In the first step, we approximate the
pairwise cosine distance matrix $\mathbf{Z}=\mathbf{1}-\mathbf{S}$, where
$\mathbf{S}=\mathbf{R}^{\text{T}}\mathbf{R}\in\mathbb{R}^{n\times n}$ denotes
a pairwise similarity matrix for $n$ filters.

We take a few columns of $\mathbf{S}$ to approximate the rest of the entries
of $\mathbf{S}$ by using a Nyström approximation method [16]. Without loss of
generality, the matrix $\mathbf{S}$ can be written as follows:

$\centering\mathbf{S}=\begin{bmatrix}\mathbf{W}&\mathbf{A}^{\text{T}}\\\
\mathbf{A}&\mathbf{B}\\\ \end{bmatrix}\textrm{and
}\mathbf{C}=\begin{bmatrix}\mathbf{W}\\\
\mathbf{A}\end{bmatrix}\@add@centering$ (1)

where $\mathbf{W}\in\mathbb{R}^{m\times m}$, $\mathbf{C}\in\mathbb{R}^{n\times
m}$, $\mathbf{A}\in\mathbb{R}^{(n-m)\times m}$, and $m<<n$.

The Nyström method approximates $\mathbf{S}$ by taking $\mathbf{C}$, $m$
columns from $\mathbf{S}$, generating a rank-$k$ approximation
$\tilde{\mathbf{S}}$ of $\mathbf{S}$ given by,

$\tilde{\mathbf{S}}=\mathbf{C}\mathbf{W}_{k}^{+}\mathbf{C}^{\text{T}},$ (2)

where $\mathbf{W}_{k}$ is the best rank-$k$ approximation of $\mathbf{W}$ for
the Frobenius norm with $k\leq\text{rank}(\mathbf{W})$ and
$\mathbf{W}_{k}^{+}$ = $\sum_{j=1}^{k}$
$\sigma_{j}^{-1}\mathbf{U}^{j}\mathbf{U}^{j^{\text{T}}}$ denotes the pseudo-
inverse of $\mathbf{W}_{k}$. $\mathbf{W}_{k}^{+}$ is obtained by performing
SVD on $\mathbf{W}=\mathbf{U}\Sigma\mathbf{U}^{\text{T}}$, where $\mathbf{U}$
is an orthonormal matrix, $\mathbf{U}^{j}$ is an $j^{\text{th}}$ column of
$\mathbf{U}$ and
$\Sigma=\text{diag}\\{\sigma_{1},\sigma_{2},\dots,\sigma_{m}\\}$ is a real
diagonal matrix with $\sigma_{1}\geq\sigma_{2},\dots,\sigma_{m}\geq 0$. The
computational complexity needed to obtain $\tilde{\mathbf{S}}$ is
$\mathcal{O}(m^{3}+nmk)$. After obtaining $\tilde{\mathbf{S}}$, we compute
$\tilde{\mathbf{Z}}=\mathbf{1}-\tilde{\mathbf{S}}$, as an approximation of
$\mathbf{Z}$.

(Step 2) Obtaining important filters: Given $\tilde{\mathbf{Z}}$, we identify
the closet filter corresponding to each filter. A filter from the closest
filter pairs is then considered redundant and eliminated from the underlying
convolution layer.

A summary of the overall framework is given in Algorithm 1.

Obtaining pruned network and performing fine-tuning: After obtaining the
important filters across different convolution layers using Algorithm 1, we
retain the set of important filters and eliminate the other filters from the
unpruned CNN to obtain a pruned network. Eliminating a filter from a given
convolutional layer also removes the corresponding feature map produced by the
filter and the associated channel of the filter in the following convolutional
layer. Therefore, the computations in the next convolutional layer are also
reduced in the pruned network.

After removing filters, we perform fine-tuning which involves re-training of
the pruned network to regain some of the lost performance due to the removal
of the connection from the unpruned CNN. The codes for the proposed efficient
pruning framework can be found at the link111https://github.com/Arshdeep-
Singh-Boparai/Efficient_similarity_Pruning_Algo.git.

##  3 Experimental setup

We evaluate the proposed pruning framework on CNNs designed for acoustic scene
classification (ASC). An overview of the unpruned CNNs is given below,

Fig. 2: Approximation error ($\delta$) when $m$ columns are selected out of
$n$ columns from the similarity matrix for different convolutional layers in
(a) DCASE21_Net and (b) VGGish_Net . Here, the similarity matrix is computed
using rank-$k$ approximation with $k=m$.

(a) DCASE21_Net: DCASE21_Net is a publicly available pre-trained network
designed for DCASE 2021 Task 1A that is trained using TAU Urban Acoustic
Scenes 2020 Mobile development dataset (we denote “DCASE-20”) to classify 10
different acoustic scenes [1]. The input to the network is a log-mel
spectrogram of size (40 × 500) corresponding to a 10s audio clip. DCASE21_Net
is trained using the Adam optimizer with cross-entropy loss function for 200
epochs. The trained network has 46,246 parameters and requires approximately
287M multiply-accumulate operations (MACs) during inference corresponding to
10-second-length audio clip, and gives 48.58% accuracy on the DCASE-20
development validation dataset. DCASE21_Net consists of three convolutional
layers (termed as C1 to C3) and one fully connected layer. C1 has $n_{1}=16$,
C2 has $n_{2}=16$ and C3 has $n_{3}=32$ filters.

(b) VGGish_Net: This is built using a publicly available pre-trained VGGish
network [17] followed by a dense and a classification layer. We train
VGGish_Net on the TUT Urban Acoustic Scenes 2018 development (“DCASE-18”)
training dataset [18] to classify 10 different acoustic scenes using Adam
optimizer with cross-entropy loss function for 200 epochs. The input to the
VGGish_Net is a log-mel spectrogram of size (96 $\times$ 64) computed
corresponding to a 960ms audio segment from a whole 10s audio scene. The
VGGish_Net has approximately 55.361M parameters and requires 903M MACs during
inference corresponding to an audio clip of 960ms and gives 64.69% accuracy on
10s audio scene for DCASE-18 development validation dataset. VGGish_Net has
six convolution layers (termed as C1 to C6). The number of filters in each
convolutional layers are {64, 128, 256, 256, 512, 512} respectively.

For $i^{\text{th}}$ convolutional layer, we approximate the distance matrix
$\tilde{\mathbf{Z}}_{m_{i},k_{i}}$ using first 1 to $m_{i}$ columns of the
similarity matrix $\mathbf{S}_{i}$ and approximating the similarity matrix by
rank-$k_{i}$ approximation where 1 $\leq k_{i}\leq m_{i}$. To measure the
effectiveness of the approximation, we compute an approximation error
$\delta_{i}=||\mathbf{Z}_{i}-\tilde{\mathbf{Z}}_{m_{i},k_{i}}||_{2}$ at
different values of $m_{i}$ and $k_{i}$.

To obtain the pruned network, we identify a set of important filters by
computing $\tilde{\mathbf{Z}}_{m_{i},k_{i}}$ at $m_{i}$ and $k_{i}$, where
$\delta_{i}$ $<$ 1\. Fine-tuning of the pruned network is performed with
similar conditions such as loss function, optimizer as used for training the
unpruned network except for 100 epochs.

Performance metrics: We analyse a total time required to obtain the set of
important filters for all convolutional layer. The total pruning time is
computed after running the pruning algorithm for 10K times and an average of
the total pruning time is reported. To measure the performance of the pruned
network, we compute accuracy, the number of MACs per inference and the number
of parameters. The accuracy of the pruned network is computed after fine-
tuning the pruned network independently for 5 times and we report the average
accuracy.

Other methods for comparison: We compare the proposed Algorithm 1 with
existing norm-based pruning methods such as an $l_{1}$-norm [12] method and a
geometric median (GM) method [13], and a similarity-based pruning method [15]
that first computes complete cosine distance matrix, and then uses Step 2 of
the Algorithm 1 to compute important set of filters for a given convolutional
layer.

Fig. 3: Approximation error ($\delta$) when the similarity matrix is generated
with rank-$k$ approximation by varying $k$ using fixed number of columns ($m$)
across various convolutional layers for (a) DCASE21_Net having $m_{1}=$12 for
C1, $m_{2}=$ 6 for C2 and $m_{3}=$21 for C3 and (b) VGGish_Net where $m$ = 9
for all C1 to C6 layers.

##  4 Results and Analysis

Table 1: Comparison of accuracy, MACs and parameters among the (a) unpruned,
(b) the pruned networks obtained using various pruning methods. Total pruning
time to obtain important set of filters across all convolutional layers is
also shown.

Network | Pruning Method | Accuracy (%) | MACs | Parameters | Total pruning time (s)
---|---|---|---|---|---
DCASE21_Net | No pruning | 48.58 | 287M | 46246 | -
on DCASE-20 dataset |  $l_{1}$-norm [12] | 44.42 | 139M | 24056 | 0.0072
|  GM [13] | 45.84 |  —"— |  —"— | 0.010
|  Similarity-based [15] | 45.54 |  —"— |  —"— | 0.063
| (Proposed) Efficient similarity-based | 45.54 |  —"— |  —"— | 0.011
VGGish_Net | No pruning | 64.69 | 903M | 55M | -
on DCASE-18 dataset |  $l_{1}$-norm [12] | 60.02 | 595M | 42.89M | 0.21
|  GM [13] | 59.71 |  —"— |  —"— | 0.42
|  Similarity-based [15] | 62.00 |  —"— |  —"— | 34.80
| (Proposed) Efficient similarity-based | 62.00 |  —"— |  —"— | 11.70

Fig. 4: Total pruning time to obtain important set of filters using Algorithm
1 for various convolutional layers, with and without approximating the
distance matrix.

Figure 2 shows the approximation error when $m_{i}$ columns are selected out
of $n_{i}$ columns from the similarity matrix and the similarity matrix is
approximated using rank-$k_{i}$ approximation with $k_{i}=m_{i}$ for different
convolutional layers in DCASE21_Net and VGGish_Net. We observe that selecting
few columns from the similarity matrix are sufficient to approximate the
complete similarity matrix with $\delta_{i}<1$. Also, the distance matrix
($\tilde{\mathbf{Z}}_{m_{i},m_{i}}$) approximated by choosing $m_{i}$ columns
of the similarity matrix that gives $\delta_{i}<1$ results in a same set of
important filters as obtained using the complete distance matrix
($\mathbf{Z}_{i}$).

For DCASE21_Net as shown in Figure 2(a), we find that choosing $m_{1}=$ 12 out
of $n_{1}=$ 16 columns for C1, $m_{2}=$ 6 out of $n_{2}=$ 16 columns for C2
and $m_{3}=$ 21 out of $n_{3}=$ 32 columns for C3 gives
$\tilde{\mathbf{Z}}_{m_{i},m_{i}}\approx\mathbf{Z}_{i}$. For VGGish_Net as
shown in Figure 2(b), we find that choosing $m_{i}=$ 9 out of $n_{i}$ columns,
where $n_{i}\in$ {64, 128, 256, 256, 512, 512} for C1 to C6 layers
respectively, gives $\tilde{\mathbf{Z}}_{m_{i},m_{i}}\approx\mathbf{Z}_{i}$.

Next, Figure 3 shows the approximation error as $k_{i}$ varies from 1 to
$m_{i}$ across different convolutional layers of (a) DCASE21_Net, where
$m_{1}=$ 12, $m_{2}=$ 6 and $m_{3}=$ 21 for C1, C2 and C3 layers respectively,
and (b) VGGish_Net, where $m_{i}=$ 9 across various convolutional layers. We
obtain the same set of important filters using
$\tilde{\mathbf{Z}}_{m_{i},k_{i}}$ as that of $\mathbf{Z}_{i}$ with
$\delta_{i}<1$ when $k_{1}=$ 9 for C1, $k_{2}=$ 6 for C2 and $k_{3}=$ 13 for
C3 layer in DCASE21_Net, and $k_{i}=$ 9 for $1\leq i\leq 6$ convolutional
layers of VGGish_Net.

Figure 4 compares the total pruning time computed for each convolutional
layer, when the distance matrix ($\tilde{\mathbf{Z}}_{m_{i},k_{i}}\text{with
}\delta_{i}<1$) is approximated using Algorithm 1: Step 1, and when the
complete pairwise distance matrix ($\mathbf{Z}_{i}$) is computed without any
approximation for (a) DCASE21_Net and (b) VGGish_Net.

The total pruning time is reduced by approximating the distance matrix
compared to computing the complete pairwise distance matrix for various
convolution layers. When the number of filters is large, for example, the C6
layer in VGGish_Net has 512 filters, the total pruning time reduces
significantly with the distance matrix approximation to that of computing the
complete distance matrix. On the other hand, when the number of filters is
smaller, for example C1 layer of VGGish_Net has 64 filters or all
convolutional layers in DCASE21_Net has $n_{i}\leq$ 32, the total pruning time
reduces marginally by approximating the distance matrix compared to that of
computing the complete distance matrix.

Table 1 compares the performance metrics with the other methods. For
DCASE21_Net, the pruned network obtained using the proposed pruning method
reduces both the MACs and the parameters by approximately 50% at 3 percentage
points drop in accuracy compared to the unpruned network. The total pruning
time for $l_{1}$-norm method [12] is the smallest among other methods.
However, the accuracy obtained using the $l_{1}$-norm method is 1 percentage
points lesser than that of the proposed pruning method. The accuracy and the
total pruning time for the geometrical median (GM) pruning method [13] is
marginally better than that of the proposed pruning method. In contrast to the
similarity-based pruning method [15], the proposed efficient similarity-based
pruning method takes less total pruning time and gives similar accuracy.

For VGGish_Net, the pruned network obtained using the proposed pruning method
reduces the MACs by 34%, and the parameters are reduced by 23% at 2.7
percentage points drop in the accuracy compared to that of the unpruned
network. Even though the $l_{1}$-norm and the GM pruning methods take
significantly less computations than the proposed pruning method, the proposed
pruning method improves the accuracy of the pruned network by 2 percentage
points compared to that of the $l_{1}$-norm and the GM pruning methods. In
contrast to the similarity-based pruning method, the proposed efficient
similarity-based pruning method is 3 times faster and gives the same accuracy.

##  5 Conclusion

This paper presents an efficient similarity-based passive filter pruning
framework to reduce computational complexity and memory storage in CNNs. We
show that using only a few columns of the similarity matrix is sufficient to
approximate similarity matrix and is 3 times faster than computing the
complete pairwise similarity matrix with no loss in accuracy. The proposed
pruning method yields a pruned network that performs similarly or better than
the existing norm-based pruning methods.

In future, we would like to improve the performance of the pruned network
obtained using the proposed pruning framework to achieve a similar performance
as that of the unpruned network by using better distance measures such as
graph-based similarity between the filters. Also, reducing the number of fine-
tuning epochs (e.g. $<100$) to recover some of the performance lost due to
filter elimination is a future goal to reduce overall computations.

##  6 Acknowledgements

This work was partly supported by Engineering and Physical Sciences Research
Council (EPSRC) Grant EP/T019751/1 “AI for Sound (AI4S)”. For the purpose of
open access, the authors have applied a Creative Commons Attribution (CC BY)
licence to any Author Accepted Manuscript version arising.

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