# Hardy inequalities on metric measure spaces, IV: The case $p=1$

Michael Ruzhansky  Michael Ruzhansky: Department of Mathematics: Analysis,
Logic and Discrete Mathematics Ghent University, Belgium and School of
Mathematical Sciences Queen Mary University of London United Kingdom E-mail-
<EMAIL_ADDRESS>,  Anjali Shriwastawa  Anjali Shriwastawa: DST-
Centre for Interdisciplinary Mathematical Sciences Banaras Hindu University,
Varanasi-221005, India E-mail-<EMAIL_ADDRESS>and
Bankteshwar Tiwari  Bankteshwar Tiwari : DST-Centre for Interdisciplinary
Mathematical Sciences Banaras Hindu University, Varanasi-221005, India E-mail-
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we investigate the two-weight Hardy inequalities on metric
measure space possessing polar decompositions for the case $p=1$ and $1\leq
q<\infty.$ This result complements the Hardy inequalities obtained in [21] in
the case $1<p\leq q<\infty.$ The case $p=1$ requires a different argument and
does not follow as the limit of known inequalities for $p>1.$ As a byproduct,
we also obtain the best constant in the established inequality. We give
examples obtaining new weighted Hardy inequalities on homogeneous Lie groups,
on hyperbolic spaces and on Cartan-Hadamard manifolds for the case $p=1$ and
$1\leq q<\infty.$

###### Key words and phrases:

Integral Hardy inequalities, homogeneous Lie groups, metric measure spaces,
quasi-norm, Riemannian manifold with negative curvature, hyperbolic spaces

###### 2010 Mathematics Subject Classification:

26D10, 22E30, 45J05.

##  1\. Introduction

The main purpose of this article is to study the Hardy type inequalities on
metric measure spaces for the case $p=1$ and $1\leq q<\infty$. We begin with
the brief overview of Hardy inequalities in the literature.
Let us recall the following celebrated Hardy inequality [11] which plays an
important role in many areas such as analysis, probability and partial
differential equations:

(1.1)
$\displaystyle\int_{a}^{\infty}\left(\frac{\int_{b}^{x}f(t)\,\text{d}t}{x}\right)^{p}\text{d}x\leq\left(\frac{p}{p-1}\right)^{p}\int_{b}^{\infty}f^{p}(x)\,\text{d}x,$

where $p>1,\,b>0,$ and $f\geq 0$ is a non-negative function. Several forms of
the Hardy inequalities (1.1) have been known.The original discrete version of
this inequality goes back in [10]. Since a lot of work has been done on Hardy
inequalities in different forms and different settings, it is difficult to
give a complete overview of the literature, let us refer to books and surveys
by Opic and Kufner [18], Davies [3], Kufner, Persson and Samko [14]. We also
refer [1, 4, 6, 8, 9, 10, 11, 15, 19, 20, 24, 22] and references therein for a
historic overview as well as some recent developments on this subject.

The first author and Verma [21] obtained several characterizations of weights
for two-weight Hardy inequalities to hold on general metric measure spaces
possessing polar decompositions for the range $1<p\leq q<\infty$ (see, [22]
for the case $0<q<p$ and $1<p<\infty$). As a consequence, one deduced new
weighted Hardy inequalities on $\mathbb{R}^{n}$, on homogeneous groups, on
hyperbolic spaces and on Cartan–Hadamard manifolds. The first author with
Kassymov and Suragan established a reverse version of the integral Hardy
inequality on metric measure spaces with two negative exponents for the range
$q\leq p<0$ (see [13]). Recently, the authors of this paper established sharp
weighted integral Hardy inequality and conjugate integral Hardy inequality on
homogeneous Lie groups with any quasi-norm for the range $1<p\leq q<\infty$
(see [20]). We also calculated the precise value of sharp constants in
respective inequalities, improving the result of [21] in the case of
homogeneous groups. Let us recall the following Hardy inequality on metric
measure spaces with a polar decomposition from [21]. A metric space
$(\mathbb{X},d)$ with a Borel measure $dx$ is said to have a polar
decomposition at $a\in{\mathbb{X}}$ if there is a locally integrable function
$\lambda\in L^{1}_{loc}$ such that for all $f\in L^{1}(\mathbb{X})$ we have

(1.2)
$\int_{\mathbb{X}}f(x)dx=\int_{0}^{\infty}\int_{\Sigma_{r}}f(r,\omega)\lambda(r,\omega)d\omega_{r}dr,$

for the set $\Sigma_{r}=\\{x\in\mathbb{X}:d(x,a)=r\\}\subset\mathbb{X}$ with a
measure on it denoted by $d\omega_{r}$, and $(r,\omega)\rightarrow a$ as
$r\rightarrow 0.$ We denote $|x|_{a}:=d(x,a)$.

######  Theorem 1.1 ([21]).

Let $1<p\leq q<\infty$ and let $s>0$. Let $\mathbb{X}$ be a metric measure
space with a polar decomposition at $a$. Let $u,v>0$ be measurable functions
positive a.e in $\mathbb{X}$ such that $u\in
L^{1}(\mathbb{X}\backslash\\{a\\})$ and $v^{1-p^{\prime}}\in
L^{1}_{loc}(\mathbb{X})$. Denote

$\displaystyle
U(x):={\int_{\mathbb{X}\backslash{\mathbb{B}(a,|x|_{a})}}u(y)dy}\quad\text{and}\quad
V(x):=\int_{\mathbb{B}(a,|x|_{a})}v^{1-p^{\prime}}(y)dy.$

Then the inequality

(1.3)
$\bigg{(}\int_{\mathbb{X}}\bigg{(}\int_{\mathbb{B}(a,|x|_{a})}|f(y)|dy\bigg{)}^{q}u(x)dx\bigg{)}^{\frac{1}{q}}\leq
C\bigg{\\{}\int_{\mathbb{X}}{|f(x)|}^{p}v(x)dx\bigg{\\}}^{\frac{1}{p}}$

holds for all measurable functions $f:\mathbb{X}\to{\mathbb{C}}$ if and only
if the following condition holds:

(1.4)
$\sup\limits_{x\not=a}\bigg{(}U^{\frac{1}{q}}(x)V^{\frac{1}{p^{\prime}}}(x)\bigg{)}<\infty.$

In this paper we prove two weight Hardy inequalities on general metric measure
spaces with a polar decomposition for the case $p=1$ and $1\leq q<\infty.$ For
the Euclidean space, this was established in [16]. It is worth noting that we
do not assume any doubling condition on the measure. This inequality will
serve as the complementary Hardy inequality of (1.1) for the limiting case. We
state the main result of this paper as follows:

######  Theorem 1.2.

Let $1\leq q<\infty$ and let $\mathbb{X}$ be a metric measure space with a
polar decomposition satisfying (1.2) at $a$. Let $u,v>0$ be measurable
functions positive a.e. in $\mathbb{X}$ such that $u\in
L^{1}(\mathbb{X}\backslash\\{a\\})$ and $v\in L^{1}_{loc}(\mathbb{X}).$ Then
the inequality

(1.5)
$\displaystyle\left[\int_{\mathbb{X}}\left(\int_{\mathbb{B}(a,|x|_{a})}|f(z)|dz\right)^{q}u(x)\,dx\right]^{\frac{1}{q}}\leq
C\left\\{\int_{\mathbb{X}}|f(x)|\,v(x)\,dx\right\\}$

holds for all measurable functions $f:\mathbb{X}\rightarrow\mathbb{C}$ if and
only if the following condition holds:

(1.6) $\displaystyle
A:=\sup_{R>0}\left(\int_{\mathbb{X}\backslash\mathbb{B}(a,R)}u(y)\,dy\right)^{\frac{1}{q}}\left(\sup_{y\in\mathbb{B}(a,R)}v^{-1}(y)\right)<\infty.$

Theorem 1.2 enables us to obtain interesting and new weighted integral Hardy
inequality on homogeneous groups, on hyperbolic spaces and on Cartan-Hadamard
manifolds for the case $p=1$ and $1\leq q<\infty$ (see Corollary 3.3 and
Corollary 3.4).

Apart from Section 1, this manuscript is divided in two sections. In the next
section, we will recall the basics of metric measure space and homogeneous Lie
groups with some other useful concepts which we will use in our main results.
The last section is devoted to presenting important proofs of the main results
of this paper.
Throughout this paper, the symbol $A\asymp B$ means that
$\exists\,C_{1},C_{2}>0$ such that $C_{1}A\leq B\leq C_{2}A$.

##  2\. Preliminaries

In this section, we present a brief overview on the basics of metric measure
spaces and homogeneous Lie groups. For more detail on metric measure space and
on homogeneous Lie group as well of several functional inequalities on metric
measure space and on homogeneous Lie groups, we refer to monographs [5, 6, 26,
24, 21, 22, 20, 25, 23] and references therein.
Let $(\mathbb{X},d)$ be a metric space with a Borel measure d$x$ allowing for
the following polar decomposition at some fixed point $a\in\mathbb{X}$ and let
us assume that there is a locally integrable function $\lambda\in L^{1}_{loc}$
such that for all $f\in L^{1}(\mathbb{X})$ we have

(2.1)
$\displaystyle\int_{\mathbb{X}}f(x)dx=\int_{0}^{\infty}\int_{\sum_{r}}f(r,\omega)\,\lambda(r,\omega)\,\text{d}\omega_{r}\,\text{d}r$

for the sets $\sum_{r}=\\{x\in\mathbb{X};d(x,a)=r\\}$ with a measure on it
denoted by $d\omega=d\omega_{r}.$ The condition (2.1) is general and we allow
the function $\lambda$ to depend on the whole variable $x=(r,\omega).$
Let us briefly provide the motivation behind the condition (2.1) above. We
first note that, when a metric measure space has a differential structure, one
can easily deduce the function $\lambda(r,\omega)$ as the Jacobian of the
polar change of coordinates or by polar decomposition formula on the
corresponding differential metric measure space. The situation we are dealing
with is very general and we do not assume any differential structure on the
metric measure space $\mathbb{X}.$ So it seems natural to assume the
decomposition formula (2.1) on $\mathbb{X}.$ It is worth noting that we do not
impose any doubling condition of the metric measure space $\mathbb{X}.$
Indeed, we provide several examples of $\mathbb{X}$ for which the condition
(2.1) is satisfied with different expressions for $\lambda(r,\omega)$ and some
of them are of non-doubling volume growth such as hyperbolic spaces:

  * (i)

Euclidean Spaces $\mathbb{R}^{n}$: We have

$\displaystyle\lambda(r,\omega)=r^{n-1}.$

  * (ii)

Homogeneous Lie groups: If $Q$ is the homogeneous dimension of the homogeneous
Lie group $\mathbb{G},$ then

$\displaystyle\lambda(r,\omega)=r^{Q-1}.$

For more detail on homogeneous Lie groups, we refer to [24, 6].

  * (iii)

Hyperbolic Spaces: The hyperbolic space of dimension $n$ denoted by
$\mathbb{H}^{n},$ is the simply connected, $n$-dimensional Riemannian manifold
of constant sectional curvature equal to $-1.$ It satisfies the stronger
property of being a symmetric space. In the case of hyperbolic spaces, we have

$\displaystyle\lambda(r,\omega)=(\sinh{r})^{n-1}.$

  * (iv)

Cartan-Hadamard manifolds: Let $(M,g)$ be a Riemannian manifold. A Riemannian
manifold is said to be a Cartan-Hadamard manifold if it is complete, simply
connected and has non-positive sectional curvature. If $\mathbb{K}_{M}$ be the
sectional curvature of the Riemannian manifold, then $\mathbb{K}_{M}\leq 0$
along each plane section at each point of $M.$ The exponential map
$\exp_{a}:T_{a}M\rightarrow M$ is a diffeomorphism (Helgason [12]). Let
$J(\rho,\omega)$ be the density function on manifold $M,$ then we have the
following polar decomposition:

$\displaystyle\int_{M}f(x)\,dx=\int_{0}^{\infty}\int_{\mathbb{S}^{n-1}}f(\exp_{a}(\rho\omega))J(\rho,\omega)\rho^{n-1}\text{d}\rho\,\text{d}\omega,$

with

$\lambda(\rho,\omega)=J(\rho,\omega)\rho^{n-1}.$

For more details on Cartan-Hadamard manifolds, we refer to [7, 2, 12].

###  2.1. Basics on homogeneous Lie groups

A Lie group $\mathbb{G}$ (identified with $(\mathbb{R}^{N},\circ)$) is called
a homogeneous Lie group if it is equipped with a dilation mapping

$D_{\lambda}:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N},\quad\lambda>0,$

defined as

(2.2)
$D_{\lambda}(x)=(\lambda^{v_{1}}x_{1},\lambda^{v_{2}}x_{2},\ldots,\lambda^{v_{N}}x_{N}),\quad
v_{1},v_{2},\dots,v_{N}>0,$

which is an automorphism of the group $\mathbb{G}$ for each $\lambda>0$. At
times, we will denote the image of $x\in\mathbb{G}$ under $D_{\lambda}$ by
$\lambda(x)$ or, simply $\lambda x$. The homogeneous dimension $Q$ of a
homogeneous Lie group $\mathbb{G}$ is defined by

$Q=v_{1}+v_{2}+\dots+v_{N}.$

It is well known that a homogeneous Lie group is necessarily nilpotent and
unimodular. The Haar measure $dx$ on $\mathbb{G}$ is nothing but the Lebesgue
measure $\mathbb{R}^{N}$.

Let us denote the volume of a measurable set $\omega\subset\mathbb{G}$ by
$|\omega|$. Then we have the following consequences: for $\lambda>0$

(2.3)
$|D_{\lambda}(\omega)|=\lambda^{Q}|\omega|\quad\text{and}\quad\int_{\mathbb{G}}f(\lambda
x)dx=\lambda^{-Q}\int_{\mathbb{G}}f(x)dx.$

A quasi-norm on $\mathbb{G}$ is any continuous non-negative function
$|\cdot|:\mathbb{G}\rightarrow[0,\infty)$ satisfying the following conditions:

  * (i)

$|x|=|x^{-1}|$ for all $x\in\mathbb{G},$

  * (ii)

$|\lambda x|=\lambda|x|$ for all $x\in\mathbb{G}$ and $\lambda>0,$

  * (iii)

$|x|=0\iff x=0.$

If $\mathfrak{S}=\\{x\in\mathbb{G}:|x|=1\\}\subset\mathbb{G}$ is the unit
sphere with respect to the quasi-norm, then there is a unique Radon measure
$\sigma$ on $\mathfrak{S}$ such that for all $f\in L^{1}(\mathbb{G})$, we have
the following polar decomposition

(2.4)
$\int_{\mathbb{G}}f(x)dx=\int_{0}^{\infty}\int_{\mathfrak{S}}f(ry)r^{Q-1}d\sigma(y)dr.$

##  3\. Main Results

###  3.1. Hardy inequality in metric measure space for the case $\mathbf{p=1}$
and $\mathbf{1\leq q<\infty}$

We will denote the closed balls by $\mathbb{B}(a,r)$ in the metric measure
space $\mathbb{X},$ where $a$ and $r$ are the centre and the radius of the
ball, respectively. If $d$ is the metric on $\mathbb{X}$ then we can express
$\mathbb{B}(a,r)$ by

$\displaystyle\mathbb{B}(a,r):=\\{x\in\mathbb{X};d(x,a)\leq r\\},$

and we write $|x|_{a}:=d(x,a)$ for the fixed point $a\in\mathbb{X}.$
Our first result is the characterization of weights $u$ and $v$ for the
$L^{1}-$ integral Hardy type inequality on metric measure spaces. We will also
discuss the conjugate $L^{1}-$ integral Hardy on metric measure spaces which
is given in Theorem 3.2.

######  Theorem 3.1.

Let $1\leq q<\infty$ and let $\mathbb{X}$ be a metric measure space with a
polar decomposition satisfying (2.1) at $a$. Let $u,v>0$ be measurable
functions positive a.e. in $\mathbb{X}$ such that $u\in
L^{1}(\mathbb{X}\backslash\\{a\\})$ and $v\in L^{1}_{loc}(\mathbb{X}).$ Then
the inequality

(3.1)
$\displaystyle\left[\int_{\mathbb{X}}\left(\int_{\mathbb{B}(a,|x|_{a})}|f(z)|dz\right)^{q}u(x)\,dx\right]^{\frac{1}{q}}\leq
C\left\\{\int_{\mathbb{X}}|f(x)|\,v(x)\,dx\right\\}$

holds for all measurable functions $f:\mathbb{X}\rightarrow\mathbb{C}$ if and
only if the following condition holds:

(3.2) $\displaystyle
A:=\sup_{R>0}\left(\int_{\mathbb{X}\backslash\mathbb{B}(a,R)}u(y)\,dy\right)^{\frac{1}{q}}\left(\sup_{y\in\mathbb{B}(a,R)}v^{-1}(y)\right)<\infty.$

The best constant $C$ in (3.1) is given by $C=A.$

###### Proof.

For the proof of Theorem 3.1, first we will prove that (3.2) implies (3.1). We
have

(3.3)
$\displaystyle\left[\int_{\mathbb{X}}\left(\int_{\mathbb{B}(a,|x|_{a})}|f(z)|dz\right)^{q}u(x)\,dx\right]^{\frac{1}{q}}=\left[\int_{\mathbb{X}}\left(\int_{\mathbb{X}}|f(z)|\mathbb{\chi}_{|z|_{a}\leq{|x|_{a}}}(z,x)\,dz\right)^{q}u(x)\,dx\right]^{\frac{1}{q}},$

and by using Minkowski integral inequality in (3.3), we get

$\displaystyle\left[\int_{\mathbb{X}}\left(\int_{\mathbb{X}}|f(z)|\mathbb{\chi}_{|z|_{a}\leq{|x|_{a}}}(z,x)\,dz\right)^{q}u(x)\,dx\right]^{\frac{1}{q}}\leq\int_{\mathbb{X}}\left(\int_{\mathbb{X}}|f(z)|^{q}\mathbb{\chi}_{|z|_{a}\leq{|x|_{a}}}(z,x)\,u(x)dx\right)^{\frac{1}{q}}dz$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad=\int_{\mathbb{X}}|f(z)|\left(\int_{\mathbb{X}}\mathbb{\chi}_{|z|_{a}\leq{|x|_{a}}}(z,x)\,u(x)dx\right)^{\frac{1}{q}}dz$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\leq\int_{\mathbb{X}}|f(z)|\left(\int_{\mathbb{X}\backslash\mathbb{B}(a,|z|_{a})}u(x)dx\right)^{\frac{1}{q}}dz$
(3.4)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad=\int_{\mathbb{X}}|f(z)|\,v(z)\left(\int_{\mathbb{X}\backslash\mathbb{B}(a,|z|_{a})}u(x)dx\right)^{\frac{1}{q}}v^{-1}(z)dz.$

By simple calculation in (3.1) and applying (3.2), we get

(3.5)
$\displaystyle\left(\int_{\mathbb{X}\backslash\mathbb{B}(a,|z|_{a})}u(x)dx\right)^{\frac{1}{q}}v^{-1}(z)\leq\left(\int_{\mathbb{X}\backslash\mathbb{B}(a,|z|_{a})}u(x)dx\right)^{\frac{1}{q}}\left(\sup_{y\in\mathbb{B}(a,|z|_{a})}v^{-1}(y)\right)\leq
A.$

Again using (3.5) in (3.1), we obtain

(3.6)
$\displaystyle\left[\int_{\mathbb{X}}\left(\int_{\mathbb{B}(a,|x|_{a})}|f(z)|dz\right)^{q}u(x)\,dx\right]^{\frac{1}{q}}\leq
A\int_{\mathbb{X}}|f(z)|\,v(z)\,dz.$

This implies that (3.1) holds with $C=A.$ Then, the best constant $C$ in (3.1)
satisfies $C\leq A.$

Conversely, denote $S(R):=\sup_{z\in\mathbb{B}(a,R)}v^{-1}(z)$ for $R>0$ and
then, for each $n\in\mathbb{N},$ define the following set

$\widetilde{P}_{n}:=\bigg{\\{}z\in\mathbb{B}(a,R);v^{-1}(z)>S(R)-\frac{1}{n}\bigg{\\}}.$

We note that $|\widetilde{P}_{n}|>0$ by the definition of $S(R)$ and
$\widetilde{P_{n}}\subset\mathbb{B}(a,R)$ implies that
$|\widetilde{P}_{n}|<\infty$ for every $n\in\mathbb{N}.$ Now, for $R>0,$ we
have

$\displaystyle\bigg{[}\int_{\mathbb{X}\backslash\mathbb{B}(a,R)}|\widetilde{P}_{n}|^{q}\,u(x)\,dx\bigg{]}^{\frac{1}{q}}$
$\displaystyle=\bigg{[}\int_{\mathbb{X}\backslash\mathbb{B}(a,R)}\bigg{(}\int_{\mathbb{B}(a,R)}\chi_{\widetilde{P}_{n}}(y)dy\bigg{)}^{q}\,u(x)\,dx\bigg{]}^{\frac{1}{q}}$
$\displaystyle\leq\bigg{[}\int_{\mathbb{X}\backslash\mathbb{B}(a,R)}\bigg{(}\int_{\mathbb{B}(a,|x|_{a})}\chi_{\widetilde{P}_{n}}(y)dy\bigg{)}^{q}\,u(x)\,dx\bigg{]}^{\frac{1}{q}}$
(3.7)
$\displaystyle\leq\bigg{[}\int_{\mathbb{X}}\bigg{(}\int_{\mathbb{B}(a,|x|_{a})}\chi_{\widetilde{P}_{n}}(y)dy\bigg{)}^{q}\,u(x)\,dx\bigg{]}^{\frac{1}{q}},$

where in the second inequality we used that $|x|_{a}>R.$

Using (3.1) in (3.1), we obtain

(3.8)
$\displaystyle\bigg{[}\int_{\mathbb{X}\backslash\mathbb{B}(a,R)}|\widetilde{P}_{n}|^{q}\,u(x)\,dx\bigg{]}^{\frac{1}{q}}\leq
C\int_{\mathbb{X}}\chi_{\widetilde{P}_{n}}(x)\,v(x)\,dx=C\int_{\widetilde{P}_{n}}v(x)dx,$

and we have

(3.9)
$\displaystyle\int_{\widetilde{P}_{n}}v(x)\,dx\leq\bigg{(}S(R)-\frac{1}{n}\bigg{)}^{-1}\,|\widetilde{P}_{n}|.$

Again using (3.9) in (3.8), we get

(3.10)
$\displaystyle\bigg{[}\int_{\mathbb{X}\backslash\mathbb{B}(a,R)}|\widetilde{P}_{n}|^{q}\,u(x)\,dx\bigg{]}^{\frac{1}{q}}\leq
C\bigg{(}S(R)-\frac{1}{n}\bigg{)}^{-1}\,|\widetilde{P}_{n}|.$

As $n\rightarrow\infty,$ we have

$\displaystyle\bigg{[}\int_{\mathbb{X}\backslash\mathbb{B}(a,R)}u(x)\,dx\bigg{]}^{\frac{1}{q}}\leq
C\bigg{(}S(R)\bigg{)}^{-1}$

which says that

(3.11)
$\displaystyle\bigg{[}\int_{\mathbb{X}\backslash\mathbb{B}(a,R)}u(x)\,dx\bigg{]}^{\frac{1}{q}}S(R)\leq
C$

for all $R>0,$ which is our required result (3.2). Indeed, this shows that

$A:=\sup_{R>0}\left(\int_{\mathbb{X}\backslash\mathbb{B}(a,R)}u(y)\,dy\right)^{\frac{1}{q}}\left(\sup_{y\in\mathbb{B}(a,R)}v^{-1}(y)\right)\leq
C$

as $C$ is independent of $R.$ Therefore, the best constant $C$ in (3.1) is
given by $C=A.$ This completes the proof of this theorem. ∎

Let us also briefly discuss the conjugate of the $L^{1}$\- Hardy inequality in
Theorem 3.1:

######  Theorem 3.2.

Let $1\leq q<\infty$ and let $\mathbb{X}$ be a metric measure space with a
polar decomposition satisfying (2.1) at $a$. Let $u,v>0$ be measurable
functions positive a.e. in $\mathbb{X}$ such that $u\in
L_{loc}^{1}(\mathbb{X})$ and $v\in L^{1}_{loc}(\mathbb{X}\backslash\\{a\\}).$
Then the inequality

(3.12)
$\displaystyle\left[\int_{\mathbb{X}}\left(\int_{\mathbb{X}\backslash\mathbb{B}(a,|x|_{a})}|f(z)|dz\right)^{q}u(x)\,dx\right]^{\frac{1}{q}}\leq
C\left\\{\int_{\mathbb{X}}|f(x)|\,v(x)\,dx\right\\},$

holds for all measurable functions $f:\mathbb{X}\rightarrow\mathbb{C}$ if and
only if the following condition holds:

(3.13) $\displaystyle
A:=\sup_{R>0}\left(\int_{\mathbb{B}(a,R)}u(y)\,dy\right)^{\frac{1}{q}}\left(\sup_{y\in\mathbb{X}\backslash\mathbb{B}(a,R)}v^{-1}(y)\right)<\infty.$

The best constant $C$ in (3.12) is given by $C=A.$

###### Proof.

The proof of the theorem is similar to the proof of Theorem 3.1, and is
omitted.∎

###  3.2. Applications and examples

In this section, we will give examples of the application of Theorem 3.1 in
the settings of homogeneous Lie groups, hyperbolic spaces and Cartan-Hadamard
manifolds.

  * (a)

Homogeneous Lie groups : Let $\mathbb{G}$ be a homogeneous Lie group of
homogeneous dimension $Q,$ equipped with a quasi norm $|\cdot|$. We refer to
[24, 5] for more information on the set-up of homogeneous groups. Particular
examples of homogeneous Lie groups are the Euclidean space $\mathbb{R}^{n}$
(in which case $Q=n$), the Heisenberg group, as well as general stratified Lie
groups (homogeneous Carnot groups) and graded groups. Here we use notation for
the quasi norm $|\cdot|,$ and denote $|x|_{a}$ by $|x|$ with $a=0.$
Let us suppose that the power weights are given by

$u(x)=|x|^{-\beta q}\quad\text{and}\quad v(x)=|x|^{\alpha}.$

For the case $1\leq q<\infty$ the inequality (3.1) holds if and only if the
following condition

(3.14) $\displaystyle
A=\sup_{R>0}\bigg{(}|\mathfrak{S}|\int_{R}^{\infty}\rho^{-\beta
q}\rho^{Q-1}\,d\rho\bigg{)}^{\frac{1}{q}}\bigg{(}\sup_{s\in(0,R)}s^{-\alpha}\bigg{)}<\infty,$

holds, where $\mathfrak{S}$ is the area of the unit sphere in homogeneous Lie
group $\mathbb{G}$ equipped with quasi norm $|\cdot|.$ For this supremum to be
well defined, we need to have $\beta q>Q$ and $\alpha\leq 0.$ Again from
(3.14), we have

(3.15) $\displaystyle
A=|\mathfrak{S}|^{\frac{1}{q}}\sup_{R>0}\bigg{(}\int_{R}^{\infty}\rho^{-\beta
q+Q-1}d\rho\bigg{)}^{\frac{1}{q}}\bigg{(}R^{-\alpha}\bigg{)}=|\mathfrak{S}|^{\frac{1}{q}}\sup_{R>0}\bigg{(}\frac{R^{\frac{-\beta
q+Q}{q}}}{|-\beta q+Q|^{\frac{1}{q}}}\bigg{)}\bigg{(}R^{-\alpha}\bigg{)},$

which is finite if and only if $\alpha+\beta=\frac{Q}{q}.$
After the above conclusion, we obtain the following result:

######  Corollary 3.3.

Let $\mathbb{G}$ be a homogeneous Lie group of homogeneous dimension $Q,$
equipped with a quasi norm $|\cdot|.$ Let $1\leq q<\infty$ and let
$\alpha,\beta\in\mathbb{R}.$ Then the inequality

(3.16)
$\displaystyle\bigg{(}\int_{\mathbb{G}}\bigg{(}\int_{\mathbb{B}(0,|x|)}f(y)\,dy\bigg{)}^{q}|x|^{-\beta
q}dx\bigg{)}^{\frac{1}{q}}\leq
C\bigg{\\{}\int_{\mathbb{G}}|f(x)|\,|x|^{\alpha}dx\bigg{\\}}$

holds for all measurable functions $f:\mathbb{G}\rightarrow\mathbb{C}$ if and
only if $\beta q>Q,\,\alpha<0$ and $\alpha+\beta=\frac{Q}{q}.$ Moreover, the
best constant $C$ is given by

(3.17) $\displaystyle C=|\mathfrak{S}|^{\frac{1}{q}}\left(\frac{1}{|\beta
q-Q|^{\frac{1}{q}}}\right),$

where $|\mathfrak{S}|$ is the area of the unit sphere in the homogeneous Lie
group $\mathbb{G}$ with respect to the quasi norm $|\cdot|.$

  * (b)

Hyperbolic space: Let $\mathbb{H}^{n}$ be the hyperbolic space of dimension
$n$ and let $a\in\mathbb{H}^{n}.$ Let us take the weights

$u(y)=\bigg{(}\sinh{|y|_{a}}\bigg{)}^{-\beta q}\quad\text{and}\quad
v(y)=\bigg{(}\sinh{|y|_{a}}\bigg{)}^{\alpha}.$

Using the polar decomposition in the first term of (3.2), we get

(3.18) $\displaystyle
A\simeq\sup_{R>0}\bigg{(}\int_{R}^{\infty}\left(\sinh{\rho}\right)^{-\beta
q+n-1}d\rho\bigg{)}^{\frac{1}{q}}\bigg{(}\sup_{y\in\mathbb{B}(0,R)}\left(\sinh{|y|_{a}}\right)^{-\alpha}\bigg{)}.$

Next, we want to show that the supremum defined in (3.18) is finite. For $R\gg
1$, we have $\sinh{z}\approx\exp{z}.$ Then (3.18) can be written as

$\displaystyle A$ $\displaystyle\simeq\begin{cases}\sup_{R\gg
1}\bigg{(}\int_{R}^{\infty}(\exp{\rho})^{-\beta
q+n-1}d\rho\bigg{)}^{\frac{1}{q}}&\quad\text{if}\quad\alpha\geq 0\\\
\sup_{R\gg 1}\bigg{(}\int_{R}^{\infty}(\exp{\rho})^{-\beta
q+n-1}d\rho\bigg{)}^{\frac{1}{q}}\left(\exp{R}\right)^{-\alpha}&\quad\text{if}\quad\alpha\leq
0\end{cases}$ (3.19) $\displaystyle\simeq\begin{cases}\sup_{R\gg
1}\bigg{(}\exp{R\bigg{)}^{{\frac{-\beta
q+n-1}{q}}}}&\quad\text{if}\quad\alpha\geq 0\\\ \sup_{R\gg
1}\bigg{(}\exp{R\bigg{)}^{{\frac{-\beta
q+n-1}{q}}-\alpha}}&\quad\text{if}\quad\alpha\leq 0,\end{cases}$

which are finite if and only if $\beta\geq\frac{n-1}{q}$ and
$\alpha+\beta\geq\frac{n-1}{q},$ respectively.
Similarly, if $R\ll 1$ and for some small $L$ such as
$\sinh{\rho}_{{R}\leq\rho<L}\approx\rho,$ (3.18) can be written as

$\displaystyle A\simeq\sup_{R\ll
1}\bigg{(}\int_{R}^{\infty}(\sinh{\rho})^{-\beta
q+n-1}d\rho\bigg{)}^{\frac{1}{q}}\bigg{(}\sup_{y\in\mathbb{B}(0,R)}(\sinh{|y|_{a}})^{-\alpha}\bigg{)}$
$\displaystyle\simeq\sup_{R\ll 1}\bigg{(}\int_{R}^{L}(\sinh{\rho})^{-\beta
q+n-1}d\rho+\int_{L}^{\infty}(\sinh{\rho})^{-\beta
q+n-1}d\rho\bigg{)}^{\frac{1}{q}}\bigg{(}\sup_{y\in\mathbb{B}(0,R)}(\sinh{|y|_{a}})^{-\alpha}\bigg{)}$
$\displaystyle\ \simeq\sup_{R\ll 1}\bigg{(}\int_{R}^{L}({\rho})^{-\beta
q+n-1}d\rho+\int_{L}^{\infty}(\exp{\rho})^{-\beta
q+n-1}d\rho\bigg{)}^{\frac{1}{q}}\bigg{(}{R}^{-\alpha}\bigg{)}$
$\displaystyle\approx\sup_{R\ll 1}\bigg{(}\frac{1}{-\beta q+n}\rho^{-\beta
q+n}\bigg{|}_{R}^{L}+\frac{1}{(-\beta q+n-1)}\left(\exp{\rho}\right)^{-\beta
q+n-1}\big{|}_{L}^{\infty}\bigg{)}^{\frac{1}{q}}\bigg{(}{R}^{-\alpha}\bigg{)}$
(3.20) $\displaystyle\approx\sup_{R\ll 1}\bigg{(}{R}^{-\beta
q+n}+C_{L}\bigg{)}^{\frac{1}{q}}\left(R^{-\alpha}\right),$

where $C_{L}$ is the remainder term and the condition $\alpha\leq 0$ is used
in third inequality. Now if $-\beta q+n\geq 0,$ then again from ((b)) this is
$\approx\sup_{R\ll 1}R^{-\alpha},$ which is finite if and only if $\alpha\leq
0.$ At the same time if $-\beta q+n<0,$ then we observe that ((b)) is
$\approx\sup_{R\ll 1}R^{\frac{-\beta q+n}{q}-\alpha},$ which is finite if and
only if $\frac{-\beta q+n}{q}-\alpha\geq 0$ which is further equivalent to
$\alpha+\beta\leq\frac{n}{q}.$
Therefore, after the above conclusion we obtain the following result.

######  Corollary 3.4.

Let $\mathbb{H}^{n}$ be the hyperbolic space, $a\in\mathbb{H}^{n}.$ Let
$|x|_{a}$ denote the hyperbolic distance from $x$ to $a.$ Let
$\alpha,\beta\in\mathbb{R}$ and $1\leq q<\infty.$ Then the following
inequality

(3.21)
$\displaystyle\bigg{(}\int_{\mathbb{H}^{n}}\bigg{(}\int_{\mathbb{B}(0,|x|_{a})}|f(y)|dy\bigg{)}^{q}\left(\sinh{|x|_{a}}\right)^{-\beta
q}dx\bigg{)}^{\frac{1}{q}}\leq
C\bigg{\\{}\int_{\mathbb{H}^{n}}|f(x)|\left(\sinh{|x|_{a}}\right)^{\alpha}dx\bigg{\\}}$

holds for all measurable functions $f:\mathbb{H}^{n}\rightarrow\mathbb{C}$ if
and only if the parameters $\alpha,\beta$ satisfy either of the following
conditions:

    * (a)

for $-\beta q+n\geq 0,$ if and only if $\alpha\leq 0$ and
$\alpha+\beta\geq\frac{n-1}{q},$

    * (b)

for $-\beta q+n<0,$ if and only if $\alpha\leq 0$ and
$\frac{n-1}{q}\leq\alpha+\beta\leq\frac{n}{q}.$

  * (c)

Cartan-Hadamard Manifold: Let $(M,g)$ be a Cartan-Hadamard manifold with
constant sectional curvature $\mathbb{K}_{M}.$ Let $J(t,\omega)$ is a function
of $t$ only. More precisely, if $\mathbb{K}_{M}=-b$ for $b\geq 0,$ then
$J(t,\omega)=1$ if $b=0,$ and
$J(t,\omega)=\left(\frac{\sinh{\sqrt{b}t}}{\sqrt{b}t}\right)^{n-1}$ for $b>0.$
When $b=0,$ then let us assume that $u(x)=|x|_{a}^{-\beta q}$ and
$v(x)=|x|_{a}^{\alpha}$ then the inequality (3.2) holds for $1\leq q<\infty$
if and only if

(3.22) $\displaystyle
A=\sup_{R>0}\bigg{(}\int_{M\backslash\mathbb{B}(0,R)}|z|_{a}^{-\beta
q}dz\bigg{)}^{\frac{1}{q}}\bigg{(}\sup_{z\in\mathbb{B}(a,R)}|z|_{a}^{-\alpha}\bigg{)}<\infty.$

After changing to polar coordinates in (3.22) first term, we get

(3.23) $\displaystyle\sup_{R>0}\bigg{(}\int_{R}^{\infty}\rho^{-\beta
q+n-1}d\rho\bigg{)}^{\frac{1}{q}}\bigg{(}\sup_{z\in\mathbb{B}(a,R)}|z|_{a}^{-\alpha}\bigg{)},$

which is finite if and only if conditions of Corollary 3.3 hold with $Q=n.$
When $b>0$ and if we take weights $u(x)=(\sinh{\sqrt{b}|x|_{a}})^{-\beta q}$
and $v(x)=(\sinh{\sqrt{b}|x|_{a}})^{\alpha},$ then the inequality (3.2) holds
for $1\leq q<\infty$ if and only if

$\displaystyle\sup_{R>0}\bigg{(}\int_{M\backslash\mathbb{B}(a,R)}(\sinh{\sqrt{b}|z|_{a}})^{-\beta
q}dz\bigg{)}^{\frac{1}{q}}\bigg{(}\sup_{z\in\mathbb{B}(0,R)}(\sinh{\sqrt{b}|z|_{a}})^{-\alpha}\bigg{)}<\infty.$

Again changing to polar coordinates,

$\displaystyle\sup_{R>0}\bigg{(}\int_{R}^{\infty}(\sinh{\sqrt{b}\mu})^{-\beta
q}\bigg{(}\frac{\sinh{b}\mu}{\sqrt{b}\mu}\bigg{)}^{n-1}\mu^{n-1}\,d\mu\bigg{)}^{\frac{1}{q}}\bigg{(}\sup_{z\in\mathbb{B}(0,R)}(\sinh{\sqrt{b}|z|_{a}})^{-\alpha}\bigg{)}$
$\displaystyle\simeq\sup_{R>0}\bigg{(}\int_{R}^{\infty}(\sinh{\sqrt{b}\mu})^{-\beta
q+n-1}d\mu\bigg{)}^{\frac{1}{q}}\bigg{(}\sup_{z\in\mathbb{B}(0,R)}(\sinh{\sqrt{b}|z|_{a}})^{-\alpha}\bigg{)},$

which has the same condition as in the case of the hyperbolic space as in
Corollary 3.4.

## acknowledgement

This paper was completed when the second author was visiting Ghent University,
Belgium. She is very grateful to Ghent Analysis & PDE centre, Ghent
University, Belgium for the financial support and warm hospitality during her
research visit. MR is supported by the FWO Odysseus 1 grant G.0H94.18N:
Analysis and Partial Differential Equations, the Methusalem programme of the
Ghent University Special Research Fund (BOF) (Grant number 01M01021) and by
EPSRC grant EP/R003025/2. AS is supported by UGC Non-net fellowship from
Banaras Hindu University, India.

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# Exploring Learned Representations of Neural Networks with Principal
Component Analysis

Amit Harlev
Center for Applied Mathematics
Cornell University
Ithaca, NY 14853
<EMAIL_ADDRESS>
Andrew Engel
Pacific Northwest National Laboratory
Richland, WA 99354
<EMAIL_ADDRESS>Panos Stinis
Pacific Northwest National Laboratory
Richland, WA 99354
<EMAIL_ADDRESS>Tony Chiang
Pacific Northwest National Laboratory
University of Washington
Seattle, WA 98195
<EMAIL_ADDRESS>
###### Abstract

Understanding feature representation for deep neural networks (DNNs) remains
an open question within the general field of explainable AI. We use principal
component analysis (PCA) to study the performance of a k-nearest neighbors
classifier (k-NN), nearest class-centers classifier (NCC), and support vector
machines on the learned layer-wise representations of a ResNet-18 trained on
CIFAR-10. We show that in certain layers, as little as $20\%$ of the
intermediate feature-space variance is necessary for high-accuracy
classification and that across all layers, the first ${\sim}100$ PCs
completely determine the performance of the k-NN and NCC classifiers. We
relate our findings to neural collapse and provide partial evidence for the
related phenomenon of intermediate neural collapse. Our preliminary work
provides three distinct yet interpretible surrogate models for feature
representation with an affine linear model the best performing. We also show
that leveraging several surrogate models affords us a clever method to
estimate where neural collapse may initially occur within the DNN.

##  1 Introduction

In the past several years, DNNs have become a common tool in many scientific
fields and real-world applications. As their use becomes more widespread, it
is more important now than ever to better our understanding of these models.
One way this can be accomplished is by studying their learned representations.
This topic has been explored by many papers in recent years, including methods
such as linear probing ([1, 4, 11, 8]), studying the dimensionality of the
manifold underlying the activations ([2, 13, 14]), and studying the geometry
of the learned representations ([9]).

In this paper, we return to a classical tool for data analysis, principal
component analysis, to help us better understand the learned representations
present in DNNs. While several papers have used PCA to study learned
representations (e.g. [8, 11]), we are the first to study in depth the
performance of multiple surrogate models using varying number of PCs across an
entire CNN. We train a k-nearest neighbors classifier (k-NN), a nearest class-
center classifier (NCC), and a support vector machine (SVM) on each residual
block’s activations after projecting down to the first $d$ principal
components (PCs) and make qualitative observations based on the results.
Studying a pretrained ResNet-18 on the CIFAR10 dataset, we observed that:

  1. 1.

The SVM matches or outperforms the k-NN and NCC across the network.

  2. 2.

The best possible performance of k-NN and NCC models on intermediate layer
activations are completely determined by the first ${\sim}100$ PCs. In fact,
the k-NN model seems to overfit as additional PCs are used.

  3. 3.

The low-variance PCs of intermediate layers contain meaningful information
that improves SVM performance.

  4. 4.

In the latter half of the network, the PCs necessary for $90\%$ of the
classification accuracy account for only $20\%$-$40\%$ of the variance.

##  2 Related work

Figure 1: Diagram showing ResNet-18 architecture with residual blocks labeled.

### Probing intermediate layers.

The idea behind classifier probes is that we can learn more about the behavior
of intermediate layers, and thus neural networks in general, by studying the
suitability of the intermediate representations for the desired task. The term
"probe" was introduced by [1], who observed that the measurements of linear
probes monotonically and gradually increased on trained networks the deeper
they were in the network. [4] observed that k-NN, SVM, and logistic regression
probes all match the performance of a DNN in the last layer and that the k-NN
predictions are almost identical to those of the DNN. [8] projected each
layer’s activations down to the first $d$ (RBF) kernel principal components
before training linear classifiers. They studied changes in performance as
architecture, hyperparameters, and $d$ were varied. While [8] studied early
CNN architectures, we study behaviors of modern residual networks. [11]
introduced SVCCA, a technique combining SVD and canonical correlation
analysis, to study the relationships between representations coming from
different layers, training epochs, and architectures. They show that “trained
networks perform equally well with a number of directions just a fraction of
the number of neurons with no additional training, provided they are carefully
chosen with SVCCA.”

### Intrinsic dimension (ID) of neural network representations.

Another approach to understanding the learned representations of DNNs has been
to study their dimensionality across the network. [14] used tangent plane
approximations to estimate the dimension of feature maps and observed that
they declined quickly across the network. More recently, [2] and [13]
estimated IDs several orders of magnitude smaller than those of [14] using
non-linear methods designed for curved manifolds. They also observed the
layerwise ID profile to have a “hunchback” shape where the ID first increases
and then drastically decreases. [2] compared against “PC-ID”, the number of
PCs required to explain $90\%$ of the variance in the activations. They
observed that (1) layerwise PC-ID profiles were qualitatively the same in
trained and untrained networks and (2) the PC-IDs were one to two orders of
magnitude greater than IDs estimated with non-linear methods. Using this, they
argued that the activations must lie on a highly curved manifold. While this
may be the case, we show that PCA can in fact help find interesting structures
in learned representations. Additionally, we show that while the underlying
manifold may be highly curved, it exists in a low-dimensional subspace that
can be found using PCA.

### Neural collapse.

First defined by [9], neural collapse is a phenomenon observed in the last
layer activations of deep neural networks characterized by several properties,
two of which are: (NC1) within-class variability collapses to zero and the
activations collapse towards their class means and (NC4) the DNN classifies
each activation using the NCC decision rule. Since then, there has been
significant interest in investigating this phenomenon, including several
papers exploring whether this phenomenon manifests in earlier layer’s
activations ([12, 5, 3]). Both [5] and [3] study the performance of the NCC
classifier across the layers of a neural network and observe an increase in
performance the deeper the layer is in the network and the more training
epochs used. [12] shows that the within-class covariance decreases relative to
the between-class covariance as you move deeper into a trained DNN.

(a)

(b)

(c)

Figure 2: Performance of 10-NN (2(a)), NCC (2(b)), and SVM (2(c)) after
projecting activations from each residual block onto first $d$ principal
components.

##  3 Experiment

We used a pre-trained ([10]) ResNet-18 ([6]) with a test accuracy of $92.5\%$
on the CIFAR-10 dataset ([7]). For a given layer, we standardized (mean zero,
std one) the activations from the training data and then used PCA to project
onto the first $d$ PCs. We trained a 10-nearest neighbors model, nearest
class-center model, and soft-margin support vector machine on the resulting
data and then used them to classify the test data after applying the same
standardization and projection learned from the training data. This was done
for each $d=1-20$, $30$, $40$, $50$, $100$, $150$, $200$, $250$, $300$, $400$,
$500$, $750$, $1000$, $1250$, $1500$, $1750$, $2000$ and subsequently at
intervals of $1000$ until reaching the size of the layer. Figure 2 shows the
accuracy by number of PCs for each model. For each model and layer, we also
found the minimum number of PCs needed to attain at least $90\%$ of the best
accuracy attained at that layer and by that model, as well as the variance
explained by those PCs. For example, if model X’s highest attained accuracy on
layer Y was $96\%$, we found the minimum number of PCs for which model X
attained $96\%*0.9=86.4\%$ accuracy. This is shown in Figure 3. We considered
the activations output by the initial max pooling layer and each of the eight
residual blocks present in a ResNet-18— see Figure 1.

##  4 Results

Looking at Figure 2, we see that up until block 4, each of our three models
exhibits different behaviors as we increase the number of PCs, and that from
block 5 onwards, all three models exhibit qualitatively identical behavior. Up
until block 4, the k-NN model’s (2(a)) accuracy increases up to ${\sim}100$
PCs before decreasing significantly, a sign that it may be overfitting. On the
other hand, the NCC model (2(b)) achieves maximum accuracy at around the same
point, but then remains unchanged as more PCs are used. The SVM (2(c))
performs similarly to the k-NN for the first ${\sim}100$ PCs, but continues to
improve in accuracy as the number of PCs increases. It also achieves the best
performance with the original activations (i.e. before projection) across all
layers. All three models see steady increases in accuracy as we move deeper
into the network. On blocks 5 onwards, all three models see a sharp, almost
identical spike up to the true accuracy of the DNN between one and ten PCs,
followed by no change in accuracy beyond that.

In 3(a) we see a “hunchback” profile for the NCC model (and to a lesser
degree, the k-NN model) that matches the “hunchback” ID profile that [2]
observed using a non-linear dimensionality estimator. On the other hand, the
SVM, the only affine-linear method we studied, exhibits a completely different
profile starting very high and then monotonically decreasing. We observe that,
just as in Figure 2, all three models exhibit identical profiles for blocks 5
through 8 and that, excluding block 5, they require only $2$-$3$ PCs to attain
$90\%$ of the accuracy of the DNN. 3(b) shows us that in the latter half of
the network, only $20\%$ to $40\%$ of the variance is needed for accurate
classification, and that this holds true across the entire network for the
non-linear models.

(a)

(b)

Figure 3: For each model: number of PCs (3(a)) and the percentage of variance
explained by those PCs (3(b)) needed to attain $90\%$ of maximum
classification accuracy at each residual block.

##  5 Discussion and conclusion

While the performance of the k-NN and NCC models is determined by the first
${\sim}100$ PCs, the SVM’s performance increases with the number of PCs up to
using the whole space. When considered along with the observations of
intermediate neural collapse of [12], this could perhaps point to there being
a “partially collapsed” subspace in each layer that determines the behavior of
the k-NN and NCC models, while the SVM also accounts for information helpful
to classification in the low variance subspaces. In particular, this means
that the low-variance subspaces contain meaningful information and not just
noise. Additionally, it is interesting to note that the SVM, an affine-linear
model, is the most robust and best performing across all learned
representations of the DNN. While all three models contribute to our intuitive
understanding of how the representation is changing across the network, the
SVM’s accuracy suggests that applications using learned representations might
benefit most from simpler models.

The behavior in blocks 5-8 can also be explained by neural collapse. That is,
the network reaches a “fully collapsed state” at block 5 in which all
activations are approximately equal to their class means, so all three
classifiers perform equally well on very few PCs. Note that had we only
trained one surrogate model, it would not be clear between which layers the
network was “fully collapsing”. However, with three models, Figure 2 and
Figure 3 clearly show that this collapse occurs between the fourth and fifth
residual blocks. Identifying this “collapsing” layer could be a useful tool
for understanding mis-classified training data, as most of the information
used by the DNN for classification is only present prior to that layer. The
notion of intermediate neural collapse is further supported by the fact that
the number of PCs needed for good classification with the SVM decreases
monotonically across the network and that the variance necessary for accurate
classification (by all models) decreases until block 5, which is where we see
“full collapse”.

Since k-NN, NCC, and PCA are all very well understood, the fact that these
non-linear models display the same profile in 3(a) as observed by [2] provides
us a more interpretable way to think about this “hunchbacked” behavior.
Additionally, since the non-linear methods required only ${\sim}100$ PCs or
less throughout the network, this implies that the curved manifold underlying
the activations most likely lives within a relatively low-dimensional
subspace, which can be found using PCA.

Lastly, while it is common to select the number of PCs to keep using metrics
such as accounting for $90\%$ of variance—as seen in [2] and [11]—3(b) shows
that this may not be the best approach for analyzing learned representations,
as the majority of the variance is not necessary for classification.

In this paper, we study learned representations of a ResNet-18 using PCA and
observe multiple interesting behaviors. We hope that our work provides new
intuition and inspires more experiments into the behavior and structure of
learned representations, as well as demonstrates that there may still be more
for us to learn about these complex models using simple techniques.

##  6 Acknowledgements

AH, AE, and TC were partially supported by the Mathematics for Artificial
Reasoning in Science (MARS) initiative via the Laboratory Directed Research
and Development (LDRD) Program at PNNL. PS was partially supported from the
U.S. Department of Energy, Advanced Scientific Computing Research program,
under the Scalable, Efficient and Accelerated Causal Reasoning Operators,
Graphs and Spikes for Earth and Embedded Systems (SEA-CROGS) project (Project
No. 80278). PNNL is a multi-program national laboratory operated for the U.S.
Department of Energy (DOE) by Battelle Memorial Institute under Contract No.
DE-AC05-76RL0-1830

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# A New Limit on Time-Reversal Violation in Beta Decay

H.P. Mumm  National Institute of Standards and Technology, Gaithersburg, MD
20899  CENPA and Physics Department, University of Washington, Seattle, WA
98195  T.E. Chupp  Physics Department, University of Michigan, Ann Arbor, MI,
48104  R.L. Cooper  Physics Department, University of Michigan, Ann Arbor, MI,
48104  K.P. Coulter  Physics Department, University of Michigan, Ann Arbor,
MI, 48104  S.J. Freedman  Physics Department, University of California at
Berkeley and Lawrence Berkeley National Laboratory, Berkeley, CA 94720  B.K.
Fujikawa  Physics Department, University of California at Berkeley and
Lawrence Berkeley National Laboratory, Berkeley, CA 94720  A. García  CENPA
and Physics Department, University of Washington, Seattle, WA 98195
Department of Physics, University of Notre Dame, Notre Dame, IN 46556  G.L.
Jones  Physics Department, Hamilton College, Clinton, NY 13323  J.S. Nico
National Institute of Standards and Technology, Gaithersburg, MD 20899  A.K.
Thompson  National Institute of Standards and Technology, Gaithersburg, MD
20899  C.A. Trull  Physics Department, Tulane University, New Orleans, LA
70118  J.F. Wilkerson  Department of Physics and Astronomy, University of
North Carolina, Chapel Hill, NC 27599  CENPA and Physics Department,
University of Washington, Seattle, WA 98195  F.E. Wietfeldt  Physics
Department, Tulane University, New Orleans, LA 70118

###### Abstract

We report the results of an improved determination of the triple correlation
$D{\bf P}\cdot(\bf{p}_{e}\times\bf{p}_{v})$ that can be used to limit possible
time-reversal invariance in the beta decay of polarized neutrons and constrain
extensions to the Standard Model. Our result is $D=(-0.96\pm 1.89(stat)\pm
1.01(sys))\times 10^{-4}$. The corresponding phase between $g_{A}$ and $g_{V}$
is $\phi_{AV}=180.013^{\circ}\pm 0.028^{\circ}$ (68% confidence level). This
result represents the most sensitive measurement of $D$ in nuclear beta decay.

###### pacs:

24.80.1y, 11.30.Er, 12.15.Ji, 13.30.Ce

††preprint: APS/123-QED

The existence of charge-parity (CP) symmetry violation in nature is
particularly important in that it is necessary to explain the preponderance of
matter over antimatter in the universe SAK91 . Thus far, CP violation has been
observed only in the K and B meson systems CHR64 ; AUB01 ; ABE01  and can be
entirely accounted for by a phase in the Cabbibo-Kobayashi-Maskawa matrix in
the electroweak Lagrangian. This phase is insufficient to account for the
known baryon asymmetry in the context of Big Bang cosmology RIO99 , so there
is good reason to search for CP violation in other systems. As CP and time-
reversal (T) violation can be related to each other through the CPT theorem,
experimental limits on electric dipole moments and T-odd observables in
nuclear beta decay place strict constraints on some, but not all, possible
sources of new CP violation.

Figure 1: A schematic of the emiT detector illustrating the alternating
electron and proton detector segments. The darker shaded proton detectors
indicate the the paired-ring at $z=\pm 10$ cm. The cross section view
illustrates, in an exaggerated manner, the effect of the magnetic field on the
particle trajectories and average opening angle. A P2E3 coincidence event is
shown.

The decay probability distribution for neutron beta decay, $dW$, can be
written in terms of the beam polarization $\bf{P}$ and the momenta (energies)
of the electron ${\bf p}_{e}$ ($E_{e}$) and antineutrino ${\bf p}_{\nu}$
($E_{\nu}$) as JAC57

$\displaystyle dW\propto 1+a\frac{{\bf p}_{e}\cdot{\bf
p}_{\nu}}{E_{e}E_{\nu}}+b\frac{{m}_{e}}{E_{e}}+$ (1) $\displaystyle\hskip
36.135pt{\bf P}\cdot\left(A\frac{{\bf p}_{e}}{E_{e}}+B\frac{{\bf
p}_{\nu}}{E_{\nu}}+D\frac{{\bf p}_{e}\times{\bf
p}_{\nu}}{E_{e}E_{\nu}}\right).$

A contribution of the parity-even triple correlation $D{\bf P}\cdot({\bf
p}_{e}\times{\bf p}_{\nu})$ above the level of calculable final-state
interactions (FSI) implies T-violation. The PDG average of recent measurements
is $D=(-4\pm 6)\times 10^{-4}$ PDG:2010 ; LIS00 ; SOL04 , while the FSI for
the neutron are $\sim 10^{-5}$ CAL67 ; Ando2009 . Complementary limits can be
set on other T-violating correlations, and recently a limit on $R$ has been
published Kozela:2009ni . Various theoretical models that extend the SM, such
as left-right symmetric theories, leptoquarks, and certain exotic fermions
could give rise to observable effects that are as large as the present
experimental limits HER98 . Calculations performed within the Minimal
Supersymmetric Model, however, predict $D\lesssim 10^{-7}$ Drees2003 .

In the neutron rest frame, the triple correlation can be expressed as $D{\bf
P}\cdot({\bf p}_{p}\times{\bf p}_{e})$, where ${\bf p}_{p}$ is the proton
momentum. Thus one can extract $D$ from the spin dependence of proton-electron
coincidences in the decay of cold polarized neutrons. Our measurement was
carried out at the National Institute of Standards and Technology Center for
Neutron Research MUM04 . The detector, shown schematically in Fig. 1,
consisted of an octagonal array of four electron-detection planes and four
proton-detection planes concentric with a longitudinally polarized beam. The
beam, with a neutron capture fluence rate at the detector of $1.7\times
10^{8}$ cm-2 s-1, was defined using a series of 6LiF apertures and polarized
to $>91\%$ (95% C.L.) by a double-sided bender-type supermirror MUM04 . A 560
$\mu$T guide field maintained the polarization direction throughout the
fiducial volume and a current-sheet spin-flipper was used to reverse the
neutron spin direction every 10 s. The symmetric octagonal geometry was chosen
to maximize sensitivity to $D$ while approximately canceling systematic
effects stemming from detector efficiency variations or coupling to the spin
correlations $A$ and $B$ WAS94 ; LIS00 . Each of the four proton segments
consisted of a $2\times 8$ array of silicon surface-barrier detectors (SBDs)
with an active layer 300 mm2$\times$ 300 $\mu$m. Each SBD was contained within
an acceleration and focusing cell consisting of a 94% transmitting grounded
wire-mesh box through which the recoil protons entered. Each SBD, situated
within a field-shaping cylindrical tube, was held at a fixed voltage in the
range $-25$ kV to $-32$ kV. The sensitive regions of the beta detectors were
plastic scintillator measuring 50 cm by 8.4 cm by 0.64 cm thick, with
photomultiplier tube (PMT) readout at both ends. This thickness is sufficient
to stop electrons at the decay endpoint energy of 782 keV. The proton and beta
detectors were periodically calibrated in situ with gamma and beta sources
respectively. Details of the apparatus are presented elsewhere MUM03 ; MUM04 ;
LIS00 .

Data were acquired in a series of runs from October 2002 through December
2003. Typical count rates were 3 s-1 and 100 s-1 for single proton and beta
detectors, respectively, while the coincidence rate for the entire array was
typically 25 s-1. Of the raw events, 12% were eliminated by filtering on
various operational parameters (e.g. coil currents) and by requiring equal
counting time in each spin-flip state. A beta-energy software threshold of 90
keV eliminated detection efficiency drifts due to changes in PMT gain coupled
with the hardware threshold. This was the largest single cut, eliminating 14%
of the raw events. A requirement that a single beta be detected in coincidence
with each proton eliminated 7% of events. All cuts were varied to test for
systematic effects.

Figure 2: Intensity log plot of SBD-scintillator coincidence data showing
proton energy vs delay time. Events near $\Delta t=0$ are prompt coincidences
due primarily to beam-related backgrounds.

The remaining coincidence events were divided into two timing windows: a
preprompt window from -12.3 $\mu$s to -0.75 $\mu$s that was used to determine
the background from random coincidences, and the decay window from -0.5 $\mu$s
to 6.0 $\mu$s as shown in Fig. 2. The recoil proton has an endpoint of 750 eV.
On average it is delayed by $\sim$ 0.5 $\mu$s. The average signal-to-
background ratio was $\sim$ 30/1. The energy-loss spectrum produced by minimum
ionizing particles in 300 $\mu$m of silicon is peaked at approximately 100 keV
and, being well separated from the proton energy spectrum, yielded an
estimated contamination below 0.1%. The final data set consisted of
approximately 300 million accepted coincidence events.

A detailed Monte Carlo simulation was used to estimate a number of systematic
effects. The program Penelope penelope , which has been tested against data in
a variety of circumstances of relevance to neutron decay martin06 , was
embedded within a custom tracking code. All surfaces visible to decay
particles were included. The Monte Carlo was based on the measured beam
distribution upstream and downstream of the fiducial volume MUM04  and
incorporated the magnetic field and electron energy threshold. A separate
Monte Carlo based on the package SIMION SIMION , incorporating the detailed
geometry of the proton cells, was used to model the proton detection response
function.

Achieving the desired sensitivity to $D$ in the presence of the much larger
spin-asymmetries due to $A$ and $B$ depends critically on the measurement
symmetry. To the extent that this symmetry is broken, corrections must be
applied to the measured result. These corrections are listed in Table 1 and
are discussed below. To extract $D$, coincident events were first combined
into approximately efficiency-independent asymmetries

$w^{p_{i}e_{j}}=\frac{N^{p_{i}e_{j}}_{+}-N^{p_{i}e_{j}}_{-}}{N^{p_{i}e_{j}}_{+}+N^{p_{i}e_{j}}_{-}},$
(2)

where $N^{p_{i}e_{j}}_{+}$ is the integrated number of coincident events in
proton detector $i=1...64$, beta detector $j=1...4$, with neutron spin + ($-$)
aligned (anti-aligned) with the guide field. For uniform polarization, ${\bf
P}$, the asymmetries, $w^{p_{i}e_{j}}$, can be written in terms of decay
correlations as

$w^{p_{i}e_{j}}\approx{\bf P}\cdot(A\tilde{\bf K}^{p_{i}e_{j}}_{A}+B\tilde{\bf
K}^{p_{i}e_{j}}_{B}+D\tilde{\bf K}^{p_{i}e_{j}}_{D}),$ (3)

where the $\bf K$’s are obtained from Eqn. 1 by integrating the normalized
kinematic terms over the phase space of the decay, the neutron beam volume,
and the acceptance of the indicated detectors LIS00 . $\tilde{\bf
K}_{A}\propto\langle{\bf p}_{e}/E_{e}\rangle$ and $\tilde{\bf
K}_{B}\propto\langle{\bf p}_{\nu}/E_{\nu}\rangle$ are primarily transverse to
the detector axis but have roughly equal longitudinal components for
coincidence events involving the two beta detectors opposite from the
indicated proton detector (E2 and E3 for P2 as shown in Fig. 1). The
$\tilde{\bf K}_{D}$’s, however, are primarily along the detector axis and are
opposite in sign for the two beta detectors. Thus for each proton detector we
can choose an appropriate combination of detector pairs that is sensitive to
the $D$-correlation but that largely cancels the parity-violating $A$ and $B$
correlations. One such combination is

$v^{p_{i}}=\frac{1}{2}(w^{p_{i}e_{R}}-w^{p_{i}e_{L}}),$ (4)

where $e_{R}$ and $e_{L}$ label the electron-detector at approximately 135∘
giving a positive and negative cross-product ${\bf p}_{p}\times{\bf p}_{e}$
respectively (P2E3 vs P2E2 as shown in Fig. 1). Proton cells at the detector
ends accept decays with larger longitudinal components of $\tilde{\bf K}_{A}$
and are more sensitive to a range of effects that break the detector symmetry.
We therefore define $\bar{v}$ as the average of the values of $v$ from the
sixteen proton-cells at the same $|z|$, i.e. $z=\pm$2, $\pm$6, $\pm$10, and
$\pm$14 cm. Each set of detectors corresponds to a paired-ring with the same
symmetry as the full detector, e.g. the shaded detectors in Fig. 1. We then
define

$\tilde{D}=\frac{\bar{v}}{P\bar{K}_{D}},$ (5)

where $\bar{K}_{D}=0.378$ is the average of ${\hat{z}}\cdot(\tilde{\bf
K}^{p_{i}e_{R}}_{D}-\tilde{\bf K}^{p_{i}e_{L}}_{D})$ determined by Monte
Carlo. The experiment provides four independent measurements corresponding to
each of the four paired-rings. The systematic corrections to $\tilde{D}$
presented in Table I yield our final value for $D$. Eqn. 5 is based on the
following: 1) accurate background corrections, 2) uniform proton and electron
detection efficiencies, 3) cylindrical symmetry of the neutron beam and
polarization, and 4) accurate determination of $\bar{K}_{D}$, $P$, and spin
state.

Table 1: Systematic corrections and combined standard uncertainties (68%
confidence level). Values should be multiplied by $10^{-4}$.

Source | Correction | Uncertainty
---|---|---
Background asymmetry |  $0$a | $0.30$
Background subtraction | $0.03$ | $0.003$
Electron backscattering | $0.11$ | $0.03$
Proton backscattering |  $0$a | $0.03$
Beta threshold | $0.04$ | $0.10$
Proton threshold | $-0.29$ | $0.41$
Beam expansion, magnetic field | $-1.50$ | $0.40$
Polarization non-uniformity |  $0$a | $0.10$
ATP - misalignment | $-0.07$ | $0.72$
ATP - Twist |  $0$a | $0.24$
Spin-correlated fluxb |  $0$a | $3\times 10^{-6}$
Spin-correlated pol. |  $0$a | $5\times 10^{-4}$
Polarizationc |  |  $0.04$d
$\bar{K}_{D}$c |  | $0.05$
Total systematic corrections | $-1.68$ | $1.01$

a Zero indicates no correction applied. b Includes spin-flip time, cycle
asymmetry, and flux variation. c Included in the definition of $\tilde{D}$. d
Assumed polarization uncertainty of 0.05.

Backgrounds not properly accounted for contribute two systematic errors: 1)
multiplicative errors due to dilution of the asymmetries, and 2) spin-
dependent backgrounds that can lead to a false $D$. Errors in background
subtraction, as well as possible spin-dependent asymmetries in this
background, have a small effect. The multiplicative correction to the value of
$w^{p_{i}e_{j}}$ due to backscattered electrons was determined using the Monte
Carlo. The uncertainty given in Table 1 reflects the 20% uncertainty assigned
to the backscattering fractions due to limitations of the detector and beam
model and due to limited knowledge of backscattering at energies below a few
hundred keV. Proton backscattering, though observable, produces a negligible
effect on $\tilde{D}$.

In principle, the values of $w^{p_{i}e_{j}}$ are independent of the absolute
efficiencies of the proton and electron detectors; however, they do depend on
any energy dependence of the efficiencies through the factors $\langle{\bf
p}_{e}/E_{e}\rangle$. Spatial variation of the efficiencies breaks the
symmetry assumed in combining proton-cell data into paired-rings. Beta energy
thresholds were observed to vary less than 20 keV across the detector,
implying the almost negligible correction given in Table 1. Proton detector
efficiency variations, however, were more significant. Lower energy proton
thresholds varied across the detector and over the course of the experiment.
These thresholds combined with the spin-dependence of the accelerated proton
energy spectra can result in significant deviations in the values of
$w^{p_{i}e_{j}}$, though the effect on the the value of $v^{p_{i}}$ is largely
mitigated because the low-energy portion of the proton energy spectrum is
roughly the same for the $e_{R}$ and $e_{L}$ coincidence pairs. To estimate
the proton-threshold-nonuniformity effect on $\tilde{D}$, spin-dependent
proton energy spectra were generated by Monte Carlo for all proton-detector-
electron-detector pairings and convoluted with model detector response
functions based on fits to the average proton-SBD spectra. The average fit
parameters were varied over a range characteristic of the observed variations
during the run. Representative thresholds were then applied to determine the
effect on $\tilde{D}$. An alternative and consistent estimate was derived by
correcting the values of $w^{p_{i}e_{j}}$ on a day-by-day basis using the
spectrum centroid shift and empirically determined functional form of the
spectrum at the threshold.

Beam expansion from a radius of 2.5 cm to 2.75 cm combined with the magnetic
field breaks the symmetry of the detector because the average proton-electron
opening angle for each proton-electron detector pair is modified. Monte Carlo
calculations using measured upstream and downstream density profile maps were
used to calculate the correction given in Table 1. Possible inaccuracies in
the determination of the beam density were estimated and their implications
explored with the Monte Carlo code. The effect on the value of $v^{p_{i}}$ as
a function of position is illustrated in Fig. 3.

Figure 3: Solid (open) squares show the values of $v$ averaged over the four
planes for proton cells on the even (odd) side of the proton detection plane
(the side with P2 vs P1 in Fig. 1). Monte Carlo results are indicated by
lines. The broken symmetry, due to the combination of the magnetic field and
beam expansion, is evident in the shift in the crossing point from the
detector center.

For a symmetric beam, contributions to $v^{p_{i}}$ due to transverse
polarization cancel for opposing proton planes; however small azimuthal beam
asymmetries can affect this cancellation. This asymmetric-beam transverse-
polarization (ATP) effect is proportional to both $\sin\theta_{P}$, the angle
of the average neutron-spin orientation with respect to the detector axis, and
$\sin(\phi_{P}-\phi^{p_{i}})$, where $\phi^{p_{i}}$ is the effective azimuthal
position of the proton cell, and $\phi_{P}$ is the azimuthal direction of the
neutron polarization. To study this effect transverse-polarization calibration
runs with $\theta_{P}=90^{\circ}$ and several values of $\phi_{P}$ were taken
over the course of the experiment. In these runs the ATP effect was amplified
by $\approx 200$. The values of $\sin\theta_{P}$ and $\phi_{P}$ for the
experiment were determined using the calibration runs and Monte Carlo
corrections for the beam density variations. To estimate the effect, the
extreme value of sin$\theta_{P}=12.8\times 10^{-3}$ and the range of
$-31.5^{\circ}<\phi_{P}<112.2^{\circ}$ were used. The uncertainty is due to
uncertainties in the angles $\theta_{P}$ and $\phi_{P}$. The effect of
nonuniform beam polarization is also given in Table 1. Time-dependent
variations in flux, polarization, and the spin-flipper, as well as the
uncertainty in the instrumental constant $\bar{K}_{D}$, can be shown to
produce asymmetries proportional to $\tilde{D}$. These effects are listed in
Table 1.

Correlations of $\tilde{D}$ with a variety of experimental parameters were
studied by varying the cuts and by breaking the data up into subsets taken
under different conditions of proton acceleration voltage and number of live
SBDs as shown in Fig. 4. A linear correlation of $\tilde{D}$ with high-
voltage, revealed by the cuts study, yields $\chi^{2}=5.6$ with 11 DOF
compared to 10.4 for 12 DOF for no correlation. The acceleration-voltage
dependence of the focusing properties was extensively studied by Monte Carlo
with no expected effect, and we intrepret the 2.1 sigma slope as a statistical
fluctuation.

To improve the symmetry of the the detector, we combine counts for the entire
run to determine the values of $v^{p_{i}}$ and to extract $\tilde{D}$ for each
paired-ring. A blind analysis was performed by adding a constant hidden factor
to Eqn. 2. The blind was only removed once all analyses of systematic effects
were complete and had been combined into a final result. The weighted average
of $\tilde{D}$ for the four paired-rings is $0.72\pm 1.89$ with
$\chi^{2}=0.73$ for 3 DOF.

Figure 4: Results for $\tilde{D}$ by run subset. Uncertainties are
statistical. The fully functioning paired-rings used for each subset are
indicated: 1-4 indicates all paired-rings were used.

The result including all corrections from Table 1 is

$D=(-0.96\pm 1.89(stat)\pm 1.01(sys))\times 10^{-4}.$

Our result represents the most sensitive measurement of the $D$ coefficient in
nuclear beta decay. Assuming purely vector and axial-vector currents,
$\phi_{AV}=180.013^{\circ}\pm 0.028^{\circ}$ which is the best direct
determination of a possible CP-violating phase between the axial and vector
currents in nuclear beta decay. Previously the most sensitive measurement was
in 19Ne, with $D=(1\pm 6)\times 10^{-4}$ CAL85 .

The authors acknowledge the support of the National Institute of Standards and
Technology, U.S. Department of Commerce, in providing the neutron facilities
used in this work. This research was made possible in part by grants from the
U.S. Department of Energy Office of Nuclear Physics (DE-FG02-97ER41020, DE-
AC02-05CH11231, and DE-FG02-97ER41041) and the National Science Foundation
(PHY-0555432, PHY-0855694, PHY-0555474, and PHY-0855310).

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