Like a bird on the wire,
Like a drunk in a midnight choir
I have tried in my way to be free111Leonard Cohen: Bird on the wire..

To Stephen Smale, at his $80$ \- th birthday

# Turning Washington’s heuristics in favor of Vandiver’s conjecture

Preda Mihăilescu  Mathematisches Institut der Universität Göttingen
<EMAIL_ADDRESS>
(Date: Version 2.0 )

###### Abstract.

A famous conjecture bearing the name of Vandiver states that $h_{p}^{+}=1$ in
the $p$ \- cyclotomic extension of $\mathbb{Q}$. Heuristics arguments of
Washington, which have been briefly exposed in [La], p. 261 and [Wa], p. 158
suggest that the Vandiver conjecture should be false, if certain conditions of
statistical independence are fulfilled. In this note we assume that
Greenberg’s conjecture is true for the $p{\rm-th}$ cyclotomic extensions and
prove an elementary consequence of the assumption that Vandiver’s conjecture
fails for a certain value of $p$: the result indicates that there are deep
correlations between this fact and the defect $\lambda^{-}i(p)$, where $i(p)$
is like usual the irregularity index of $p$, i.e. the number of Bernoulli
numbers $B_{2k}\equiv 0\bmod p,1<k<(p-1)/2$. As a consequence, if one combines
the various assumptions in Washington’s heuristics, these turn, on base of the
present result, into an argument in favor of the Vandiver’s conjecture.

##  1\. Introduction

Let $p$ be an odd prime and $\mathbb{K}=\mathbb{Q}[\zeta]$ be the $p{\rm-th}$
cyclotomic field and $G=\mbox{ Gal }(\mathbb{K}/\mathbb{Q})$. If $X$ is a
finite abelian group, we denote by $X_{p}$ its $p$ \- Sylow group; let
$A=\mathcal{C}(\mathbb{K})_{p}$, the $p$ \- Sylow subgroup of the class group
$\mathcal{C}(\mathbb{K})$ and $h^{+},h^{-}$ the sizes of $A^{+}$ respectively
$A^{-}$. In a letter to Kronecker from 1857, Kummer refers to $p\nmid h^{+}$
as a noch zu beweisender Satz, a theorem yet to prove (see also [Wa], p. 158).
The fact was stated later as a conjecture by Vandiver.

In [La], p. 261 Washington gives an heuristic argument which suggests that
there might be an asymptotic amount of $O(\log\log(N))$ of primes $p\leq N$
for which $\lambda(A_{\infty}^{-})=i(p)+1$, where $i(p)$ is the irregularity
index of $p$, i.e. the number of Bernoulli numbers $B_{2k},1<k<(p-1)/2$ that
vanish modulo $p$. In [Wa], p.158, Washington starts with a naive argument, on
base of which the cyclotomic unit $\eta_{2k}:=e_{2k}(1-\zeta)^{\sigma-1}$ (see
below for the definition of the idempotents $e_{j}\in\mathbb{F}_{p}[G]$) may
be a $p{\rm-th}$ power with probability $1/p$: this yields a probability of
more than one half, for the failure of Vandiver’s conjecture, so the argument
is obviously too crude. Washington considers then that the probabilities that
a Bernoulli number vanishes modulo $p$ and the one that the corresponding
cyclotomic unit $\eta_{2k}$ is a $p$-power are independent: this heuristic
leads to a frequence of $O(\log\log(n))$ primes $p<n$ for which the conjecture
fails. As a consequence, various specialists in the field expect that the
conjecture should not always hold. Our result in this note, shows that if
Vandiver’s conjecture fails, then one has the additional condition
$\lambda^{-}>i(p)$. If one considers this condition also as statistically
independent (!) from the two conditions in Washington’s heuristiscs, then the
same argument suggest that there may be $O(1)$ primes $p<n$ for which
Vandiver’s conjecture fails, thus possibly none. Therefore the elementary
result in this note may be understood as one that turns Washington’s
heuristics into an argument in favor of Vandiver’s conjecture.

In this paper we prove an elementary fact, which implies that the failure of
Vandiver’s conjecture has an impact on the value of $\lambda$, and thus two
events which were supposed to be uncorrelated in the heuristic approach:
namely $h_{p}^{+}\neq 0$ and $\lambda^{-}>i(p)$ are not independent. No direct
consequence can be drawn as to the truth of the Kummer - Vandiver conjecture;
however we have an explicite theorem which indicates an unknown dependence,
and also a method of investigation which may be extended for the purpose of
investigating more possible consequences of the assumption that the Kummer-
Vandiver conjecture is false.

The result of this paper is the following:

######  Theorem 1.

Let $p$ be an odd prime with irregularity index $i(p)=1$. If $h_{p}^{+}>1$,
then $\lambda^{-}\geq 2$.

Since $\lambda^{-}>1$ is an implication of $h_{p}^{+}>1$, the two events
cannot be considered as independent events, each one with probability $1/p$.
But this implication can also suggest that the probability that $\eta_{2k}$ is
a $p{\rm-th}$ power has rather the probability $1/p^{2}$ than $1/p$, since it
implies the vanishing of a higher order Bernoulli number. Either way, we
consider that our elementary result should suggest that it is worthwhile to
consider that Vandiver’s conjecture might be true, and pursue the
investigation for the reasons why this may be the case. For this purpose, the
central idea of our proof can be extended, with additional detail, to the
general case, and this shall be done in a subsequent paper.

Note that we restrict our analysis, for simplicity, to the case of
irregularity index $1$. However, this is the critical case in Washington’s
heuristics, and if the assumption of “statistical independence” is close to
reality222Washington mentions explicitly that this is the crucial and critical
in the various heuristics of this kind., then the probability of failure of
the conjecture for higher values of $i(p)$ can only be smaller, so the
argument stays valid.

##  2\. Proof of the Theorem

We let $\mathbb{K}=\mathbb{Q}[\zeta]$ be the $p{\rm-th}$ cyclotomic extension
and $\mathbb{K}_{n}=\mathbb{K}[\zeta^{1/p^{n}}],n\geq 1$, the $p^{n}{\rm-th}$
extension. The galois groups are

$\displaystyle G=\mbox{ Gal }(\mathbb{K}/\mathbb{Q})$ $\displaystyle=$
$\displaystyle\\{\sigma_{a}\ :\ a=1,2,\ldots p-1,\
\zeta\mapsto\zeta^{a}\\}\cong(\mathbb{Z}/p\cdot\mathbb{Z})^{*},$
$\displaystyle G_{n}=\mbox{ Gal }(\mathbb{K}_{n}/\mathbb{Q})$ $\displaystyle=$
$\displaystyle
G\times\langle\tau\rangle,\quad\tau(\zeta_{p^{n}})=\zeta_{p^{n}}^{1+p},$

so $\tau$ generates $\mbox{ Gal }(\mathbb{K}_{n}/\mathbb{K})$, in particular.
If $g\in\mathbb{F}_{p}$ is a generator of $(\mathbb{Z}/p\cdot\mathbb{Z})^{*}$,
then $\sigma=\sigma_{g}$ generates $G$ multiplicatively. We write $\jmath\in
G$ for complex multiplication. For $\sigma\in G$ and
$R\in\\{\mathbb{F}_{p},\mathbb{Z}_{p},\mathbb{Z}/(p^{m}\cdot\mathbb{Z})\\}$ we
let $\varpi(\sigma)\in R$ be the value of the Theichmüller character on
$\sigma$; for $R=\mathbb{F}_{p}$ we may also write $\hat{\sigma}$ for this
values. The orthogonal idempotents $e_{k}\in R[G]$ are

$e_{k}=\frac{1}{p-1}\sum_{a=1}^{p-1}\varpi^{k}(\sigma_{a})\cdot\sigma_{a}^{-1}.$

If $X$ is a finite abelian $p$ \- group on which $G$ acts, then
$e_{k}(\mathbb{Z}_{p})$ acts via its approximants to the $p^{m}{\rm-th}$
order; we shall not introduce additional notations for these approximants. A
fortiori, complex conjugation acts on $X$ splitting it in the canonical plus
and minus parts: $X=X^{+}\oplus X^{-}$, with
$X^{+}=X^{1+\jmath},X^{-}=X^{1-\jmath}$. The units of $\mathbb{K}$ and
$\mathbb{K}_{n}$ are denoted by $E,E_{n}$ and the cyclotomic units by
$C,C_{n}$. The Iwasawa invariants $\lambda,\lambda^{-}$ are related to the
cyclotomic $\mathbb{Z}_{p}$-extension
$\mathbb{K}_{\infty}=\cup_{n}\mathbb{K}_{n}$ and
$A_{n}=(\mathcal{C}(\mathbb{K}_{n}))_{p}$ are the $p$-parts of the ideal class
groups of $\mathbb{K}_{n}$. They form a projective sequence with respect to
the relative norms $N_{m,n}=\mbox{\bf
N}_{\mathbb{K}_{m}/\mathbb{K}_{n}},m>n\geq 1$ and $\mbox{\bf
A}=\varprojlim_{n}A_{n}$. We also write $A$ for $A_{1}$. We shall write for
simplicity $A(\mathbb{L})=(\mathcal{C}(\mathbb{L}))_{p}$ for the $p$-part of
the class group of an arbitrary number field $\mathbb{L}$, so
$A=A(\mathbb{K})$, etc.

We fix now an odd prime $p$ such that

  * 1\.

Greenberg’s conjecture holds for $p$, so $A^{+}$ is finite and
$\lambda^{+}=0$.

  * 2\.

Vandiver’s conjecture fails for $p$.

  * 3\.

There is a unique irregular index $2k$ such that
$A_{p-2k}=\varepsilon_{p-2k}A\neq\\{1\\}$. Additionally $A_{2k}\neq\\{1\\}$,
as a consequence of 2.

Under these premises, we show that
$\mathbb{Z}_{p}\hbox{-rk}(\varepsilon_{p-2k}\mbox{\bf A}>1$, which is the
statement of the theorem. We prove the statement by contraposition, so we
assume that $\mathbb{Z}_{p}\hbox{-rk}(\varepsilon_{p-2k}\mbox{\bf A})=1$.
Since there is a unique irregular index, the minimal polynomial of A is
linear. Let $\mathbb{H}_{n}/\mathbb{K}_{n}$ be the maximal $p$-abelian
unramified extensions. They split in plus and minus parts according to
$A_{n}=A_{n}^{+}\oplus A_{n}^{-}$ and our assumption implies that
$\mathbb{H}_{n}^{+}/\mathbb{K}_{n}$ are cyclic extensions of degree

$d_{n}:=[\mathbb{H}_{n}^{+}:\mathbb{K}_{n}]=|A_{n}^{+}|.$

We may also consider $\mathbb{H}_{n}^{+}$ as the compositum of
$\mathbb{K}_{n}$ with the full $p$-part of the Hilbert class field of
$\mathbb{K}_{n}^{+}\subset\mathbb{K}_{n}$, the maximal reals subextension of
$\mathbb{K}_{n}$: thus $\mathbb{H}_{n}^{+}$ is a canonical subfield,
corresponding by the Artin map to $A_{n}^{+}$. It follows that
$\mathbb{H}_{n}^{+}/\mathbb{K}_{n}^{+}$ is an abelian extension, and thus
$\mathbb{H}_{n}^{+}$ is a CM field (see also [Wa], Lemma 9.2 for a detailed
proof). There is a canonic construction of radicals from $A_{n}^{-}$, such
that
$\mathbb{H}_{n}\cdot\mathbb{K}_{m}\subset\mathbb{K}_{m}[(A_{m}^{-})^{1/p^{m}}]$
for sufficiently large $m$. As a consequence of Greenberg’s conjecture holding
for $\mathbb{K}$, there is an $n_{0}\geq 1$ such that
$|A_{n}^{+}|=|A_{n_{0}}^{+}|$ for all $n\geq n_{0}$ and for such $n$, let
$a_{n}\in A_{n}^{-}$ generated this cyclic group. Let $\mathfrak{Q}\in a_{n}$
and $\alpha_{0}\in\mathbb{K}_{n}^{\times}$ with
$(\alpha_{0})=\mathfrak{Q}^{{\rm ord}\ (a_{n})}$; there is an
$\alpha=\eta\cdot\alpha_{0}^{1-\jmath},\eta\in\mu_{p^{n}}$ which is well
defined up to roots of unity, such that
$\mathbb{H}_{n}^{+}\subset\mathbb{K}_{n}[\alpha^{1/p^{n}}]$. The radical
$B_{n}$ of $\mathbb{H}_{n}^{+}$ is then the multiplicative group generated by
$\alpha$ and $(\mathbb{K}_{n}^{\times})^{d_{n}}$.

Since $\mathbb{H}_{n}^{+}/K_{n}$ is cyclic, a folklore result, which we prove
for completeness in Lemma 3 of the Appendix below, implies that

(1) $\displaystyle
A(\mathbb{H}_{n}^{+})=(\mathcal{C}(\mathbb{H}_{n}^{+}))_{p}=\\{1\\}.$

A classical result, proved by Iwasawa [Iw] in a general cohomological
language, states that for an arbitrary galois extension
$\mathbb{L}/\mathbb{F}$ of finite number fields, there is a canonical
isomorphism

(2) $\displaystyle H^{1}(\mbox{ Gal
}(\mathbb{L}/\mathbb{F}),E(\mathbb{L}))\cong\mathcal{A}(\mathbb{L}),$

where $\mathcal{A}(\mathbb{F})$ are Hilbert’s ambig ideals, i.e. the ideals of
$\mathbb{L}$ which are invariant under $\mbox{ Gal }(\mathbb{L}/\mathbb{F})$,
factored by the principal ideals of $\mathbb{F}$. These can be either totally
ramified ideals or ideals from $\mathbb{F}$ that capitulate completely (become
principal) in $\mathbb{L}$. We shall in the sequel often consider the homology
groups $H^{0},H^{1}$ for the unit groups. We can then write, for simplicity

$H^{i}(\mathbb{L}/\mathbb{F}):=H^{i}(\mbox{ Gal
}(\mathbb{L}/\mathbb{F}),E(\mathbb{L}))\quad\hbox{for}\quad i=0,1.$

The isomorphism above restricts also to one of $p$-parts of the respective
groups; furthermore, complex conjugation also induces canonical isomorphisms
of the plus and minus parts of $H^{i}$. The extensions
$\mathbb{H}_{n}^{+}/\mathbb{K}_{n}$ being cyclic of degree $d_{n}$, the
Herbrand quotient is $d_{n}$ and thus

$H^{1}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})=d_{n}\cdot
H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n}).$

We claim that
$\left(H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})\right)^{+}=\\{1\\}$. Indeed,
the ambig ideals in an unramified extension are capitulated ideals. In our
case, since $d_{n}=|A_{n}^{+}|$ by definition, we have exactly
$|\mathcal{A}(\mathbb{H}_{n}^{+})|=d_{n}^{2}$. This follows from the fact that
the plus part capitulates completely, while the minus part is cyclic too and
generates the radical of the extension. Consequently $\mathfrak{Q}^{{\rm ord}\
(a_{n})/d_{n}}$ becomes principal in $\mathbb{H}_{n}^{+}$, which confirms that

$|\mathcal{A}(\mathbb{H}_{n}^{+})|=d_{n}^{2}\quad\hbox{ and
}|\mathcal{A}(\mathbb{H}_{n}^{+})|^{-}=|\mathcal{A}(\mathbb{H}_{n}^{+})|^{+}=d_{n}.$

Therefore, $|H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})|=d_{n}$. The roots of
unity $\zeta_{p^{n}}\not\in\mbox{\bf
N}_{\mathbb{H}_{n}^{+}/\mathbb{K}_{n}}(E(\mathbb{H}_{n}^{+}))$: indeed, if
$\zeta_{p^{m}}=N(\delta)$ for $\delta\in E(\mathbb{H}_{n}^{+})$ and $m\leq n$,
then $\varepsilon=\delta/\overline{\delta}$ is well defined in the CM field
$\mathbb{H}_{n}^{+}$ and it is a root of unity, by Dedekind’s unit Theorem –
so $\varepsilon\in\mathbb{K}_{n}$. Moreover, we have $\mbox{\bf
N}_{\mathbb{H}_{n}^{+}/\mathbb{K}_{n}}(\varepsilon)=\varepsilon^{d_{n}}=\zeta_{p^{m}}^{2}$.
Since $p$ is odd, it follows that $\mu_{p^{n}}/\mu_{p^{n}/d_{n}}\subset
H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})$; by comparing orders of the groups,
we conclude that

$H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})=\mu_{p^{n}}/\mu_{(p^{n}/d_{n})}=\left(H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})\right)^{-}.$

We have proved:

######  Lemma 1.

Notations being like above,

$\left(H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})\right)^{+}=\\{1\\}.$

In particular

(3) $\displaystyle\mbox{\bf
N}_{\mathbb{H}_{n}^{+}/\mathbb{K}_{n}}(E^{+}(\mathbb{H}_{n}^{+}))=E^{+}(\mathbb{K}_{n}),$

where for a CM field $\mathbb{F}$ we write
$E^{+}(\mathbb{F})=\\{e\cdot\overline{e}:e\in E(\mathbb{F})\\}$.

In our case, $E^{+}$ are the real units and the units of $\mathbb{K}_{n}^{+}$,
resp. $\mathbb{H}(\mathbb{K}_{n}^{+})\subset\mathbb{H}_{n}^{+}$; the prime $p$
is odd and we are interested in $p$-parts, so the implicit exponent $2$ in the
above definition has no further consequences: the norm is surjective on the
real units in our class field. Then
$\mathbb{H}_{n+1}^{+}=\mathbb{K}_{n+1}\cdot\mathbb{H}_{n}^{+}$ and we have a
commutative diagram of fields. By computing
$H^{0}(\mathbb{H}_{n+1}^{+}/\mathbb{K}_{n})$ in two ways, over the
intermediate field $\mathbb{K}_{n+1}$ and respectively over
$\mathbb{H}_{n}^{+}$, we obtain from (3) that

(4)
$\displaystyle\left(H^{0}(\mathbb{K}_{n+1}/\mathbb{K}_{n})\right)^{+}=\left(H^{0}(\mathbb{H}^{+}_{n+1}/\mathbb{H}^{+}_{n})\right)^{+}.$

The core observation of this proof is

######  Proposition 1.

Let $\lambda_{n}=1-\zeta_{p^{n}}$. Then the ramified prime
$\wp_{n}=(\lambda_{n})\subset\mathbb{K}_{n}$ above $p$ splits totally in
$\mathbb{H}^{+}_{n}$ in $p$-principal ideals. Moreover, if $\nu\in\mbox{ Gal
}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})$ is a generator of this cyclic group,
then there is a prime $\pi_{n}\in\mathbb{H}_{n}^{+}$ with $\mbox{\bf
N}_{\mathbb{H}_{n}^{+}/\mathbb{K}_{n}}(\pi_{n})=\lambda_{n}$.

###### Proof.

Since $\wp_{n}=(\lambda_{n})$ is principal, the Principal Ideal Theorem
implies that it splits completely in the unramified extension
$\mathbb{H}_{n}^{+}/\mathbb{K}_{n}$ and since $A(\mathbb{H}_{n}^{+})=\\{1\\}$
by (3), the primes above $\wp_{n}$ are $p$-principal. Let
$E^{\prime}(\mathbb{F})$ denote the $p$-units of the number field
$\mathbb{F}$, i.e. the units of the smallest ring containing $E(\mathbb{K})$
and in which all the primes above $p$ are invertible. In particular
$E_{n}^{\prime}=E_{n}[1/\lambda_{n}]$; it is customary to denote by
$A^{\prime}_{n}$ the $p$-part of the ideal class group of the $p$-integers.
Since $\mathbb{H}_{n}^{+}/\mathbb{K}_{n}$ splits the prime above $p$ and
$A_{n}^{+}=(A^{\prime}_{n})^{+}$ and $H^{0}(\mbox{ Gal
}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n}),E^{\prime}(\mathbb{H}_{n}^{+}))=H^{0}(\mathbb{H}_{n}^{+}/\mathbb{K}_{n})$.
In particular, the norm $\mbox{\bf
N}_{\mathbb{H}_{n}^{+}/\mathbb{K}_{n}}:E^{\prime}(\mathbb{H}_{n}^{+})\rightarrow
E^{\prime}(\mathbb{K}_{n})$ is surjective, so there is a prime
$\pi_{n}\in\mathbb{H}_{n}^{+}$ mapping on $\lambda_{n}$.

The proof of this proposition is made particularly simple by the use of (3).
However, a more involved proof shows that the facts hold in more generality
and the primes above $\lambda$ are principal in any subfield of the Hilbert
class field $\mathbb{H}_{n}$. ∎

As a consequence of the proposition, we see that
$\mathcal{A}(\mathbb{H}_{n}^{+})=\\{1\\}$. Indeed, by Lemma 3, the class group
$A(\mathbb{H}_{n}^{+})=\\{1\\}$, and the only primes that ramify in
$\mathbb{H}_{n+1}^{+}/\mathbb{H}_{n}^{+}$ lay above $p$, so they are principal
by Lemma 1. There are consequently no real ambig ideals in
$\mathbb{H}_{n}^{+}$. On the other hand, for $n$ such that
$|A^{+}_{n}|=|A_{n+1}^{+}|$, the capitulation kernel $P_{n}:=\mbox{ Ker
}(\iota_{n,n+1}:A_{n}^{+}\rightarrow A_{n+1}^{+})$ is an
$\mathbb{F}_{p}$-space of dimension $d=p-{\rm rank}\ (\mbox{\bf A}^{+})=p-{\rm
rank}\ (A_{n})=1$. We obtain a contradiction with (4), which shows that if
$i(p)=1$, then either $\mathbb{Z}_{p}\hbox{-rk}(A^{-})>1$ or $h_{p}^{+}=1$.
This completes the proof of the Theorem.

##  3\. Three detailed variants of the proof

Let $m$ be the first index for which
$|A_{m}^{(}\mathbb{K})|=|A_{m+1}^{+}(\mathbb{K})|=|A^{+}(\mathbb{K})|$.
Consider now $\mathbb{H}_{n}^{+}=\mathbb{H}_{m}\cdot\mathbb{K}_{n}^{+}$ for
$n\geq m$. Since
$[\mathbb{H}_{n}^{+}:\mathbb{K}_{n}^{+}]=q=|A(\mathbb{K})^{+}|$ for all $n>m$,
we see that $\mbox{ Ker }(A^{+}(\mathbb{H}_{m})\rightarrow
A^{+}(\mathbb{H}_{\infty})=1$.

Iwasawa’s Theorem 12 in [Iw1] implies that $H^{1}(\mbox{ Gal
}(\mathbb{H}_{n}/\mathbb{H}_{m}),E^{\prime}(\mathbb{H}_{n}))=1$, and the fact
that the primes above $p$ are principal leads by a computation of Herbrand
quotients to $E(\mathbb{H}_{m})=N_{n,m}(E(\mathbb{H}_{n}))$ for all $n>m$. We
have

$\displaystyle N_{\mathbb{H}_{n},\mathbb{K}^{+}_{m}}(E(\mathbb{H}_{n}))$
$\displaystyle=$ $\displaystyle
N_{\mathbb{H}_{m}/\mathbb{K}^{+}_{m}}(E(\mathbb{H}_{m}))=E(\mathbb{K}_{m}^{+})$
$\displaystyle=$ $\displaystyle
N_{n,m}(N_{\mathbb{H}_{n}/\mathbb{K}_{n}^{+}}(E(\mathbb{H}_{n}))=N_{n,m}(E(\mathbb{K}_{n}^{+})).$

It follows thus that $N_{n,m}(E(\mathbb{K}_{n}^{+}))=E(\mathbb{K}_{m}^{+})$
for all $n>m$. However, for sufficiently large $n$ we have
$N_{n,m}(E(\mathbb{K}_{n}^{+}))=C(\mathbb{K}_{m}^{+})$ and this would require
that $E(\mathbb{K}_{m}^{+})=C(\mathbb{K}_{m}^{+})$, which is the contradiction
above.

We give below333I thank Hiroki Takahashi for some critical questions and an
interesting dialogue, which suggested that the details provided in this
section may be useful for readers interested to understand better the specific
conditions of the $p{\rm-th}$ cyclotomic extensions, compared for instance
with some real quadratic extensions in which Greenberg’s conjecture is known
to hold, yet $A^{+}=p$ and $A^{-}$ is cyclic. some additional detail about the
units in $\mathbb{H}_{n}$ which lead to two variants of this proof and reveal
interesting details.

###  3.1. Metacyclotomic units

Let $E(\mathbb{H}_{n})$ be the units of $\mathbb{H}_{n}^{+}$ and
$E(\mathbb{K}_{n}),C(\mathbb{K}_{n})$ the units, resp. cyclotomic units of
$\mathbb{K}_{n}$. We shall construct some norm coherent sequences of units in
$E(\mathbb{H}_{n})$ and relate them to the cyclotomic. For this, fix some
large integer $N$. We assume for simplicity that the primes above $p$ in
$\mathbb{H}_{n}$ are principal and derive some specific units on base of this
assumption. Since we are only interested in $p$-parts, raising to a power
coprime to $p$ can be neglected.

We shall denote the norms $\mbox{\bf
N}_{\mathbb{H}_{n}/\mathbb{H}_{m}}=\mbox{\bf
N}_{\mathbb{K}_{n}/\mathbb{K}_{m}}$ by $N_{n,m}$ and let
$\mathcal{N}=\mbox{\bf N}_{\mathbb{H}_{n}/\mathbb{K}_{n}}$ for all $n$
sufficiently large. We choose $\pi_{2N}\in\mathbb{H}_{2N}$ a prime with
$\mathcal{N}(\pi_{2N})=1-\zeta_{p^{2N}}$ and let $\pi_{n}=N_{2N,n}(\pi_{2N})$
for all $n<2N$. The inertia group $I(\pi_{2N})\cong\mbox{ Gal
}(\mathbb{K}_{2N}/\mathbb{Q})$. We let $\sigma^{\prime},\tau^{\prime}\in
I(\pi_{2N})$ be lifts of $\sigma,\tau\in\mbox{ Gal
}(\mathbb{K}_{2N}/\mathbb{Q})$, the generators of the subgroups with $p-1$ and
$p^{2N-1}$ elements, respectively. We identify $T=\tau^{\prime}-1$ a lift of
$T$ to $\mbox{ Gal }(\mathbb{H}_{2N}/\mathbb{H}_{m})$, where $m$ is an integer
such that $|A^{+}(\mathbb{K}_{m})|=|A^{+}(\mathbb{K}_{m+1})|=q$, say.

We fix an other large integer $M>>N$ and define the lifted idempotents

$\varepsilon^{\prime}_{j}=\frac{1}{p-1}\sum_{\varsigma\in<\sigma^{\prime}>}\omega^{j}(\varsigma)\varsigma^{-1}\in\mathbb{Z}[\mbox{
Gal }(\mathbb{H}_{n}/\mathbb{Q})],$

where $\omega:<\sigma^{\prime}>\rightarrow\mathbb{Z}$ approximates the
Teichmüller character to the power $p^{M}$. We also let

$\tilde{\varepsilon}_{j}=\frac{1}{p-1}\sum_{\varsigma\in<\sigma^{\prime}>}\varpi^{j}(\varsigma)\varsigma^{-1}\in\mathbb{Z}_{p}[\mbox{
Gal }(\mathbb{H}_{n}/\mathbb{Q})],$

with $\varpi$ the $p$-adic Teichmüller character.

Then we define
$E^{(2j)}(\mathbb{K}_{n}^{+})=\varepsilon_{2j}E(\mathbb{K}_{n}^{+})$, so
$\tilde{\varepsilon}_{2}j(E(\mathbb{K}_{n}^{+})/E^{p^{M}}(\mathbb{K}_{n}^{+}))=E^{(2j)}(\mathbb{K}_{n}^{+})/(E^{(2j)}(\mathbb{K}_{n}^{+}))^{p^{M}}$.
Moreover, $m$ verifies for all $n>m$

(5) $\displaystyle E(\mathbb{K}_{n})=C(\mathbb{K}_{n})\cdot
E(\mathbb{K}_{m}),\quad\forall n>m.$

Indeed, this condition is equivalent to
$[E(\mathbb{K}_{n}):C(\mathbb{K}_{n})]=|A_{n}^{+}|=[E(\mathbb{K}_{m}):C(\mathbb{K}_{m})]=|A_{m}^{+}|$
for all $n>m$, a condition which is implied by our assumption (e.g. Fukuda).
It follows also that
$[E(\mathbb{K}_{l}):C(\mathbb{K}_{l})]=|A^{+}_{l}|=p|A_{l-1}^{+}|$ for
$1<l\leq m$, and if $\varepsilon_{l}\in e_{2k}E(\mathbb{K}_{l})$ generates
this component up to $p^{M}{\rm-th}$ powers, then

$E(\mathbb{K}_{n})=\langle\varepsilon_{m},\eta_{n}\rangle_{\mathbb{Z}[T]}.$

We show

######  Lemma 2.

Notations being like above, there is a module
$\overline{C}(\mathbb{H}_{n})\subset E(\mathbb{H}_{n})$, of finite index and
norm coherent for all $n\leq 2N$. Moreover, if $m$ is the smallest index such
that $|A_{m}^{+}|=|A^{+}_{m+1}|$, then

(6) $\displaystyle
E(\mathbb{H}_{n})=E(\mathbb{H}_{m})\cdot\overline{C}(\mathbb{H}_{n})\quad\forall
n\geq m.$

###### Proof.

Let $\eta_{n}=(\pi_{n}\cdot\overline{\pi_{n}})^{\sigma^{\prime}-1}$ with
$\pi_{n}$ defined above, let $\delta_{n}=(\pi_{n}\cdot\overline{\pi_{n}})^{T}$
and $C(\mathbb{H}_{n})=\eta_{n}^{\mathbb{Z}[\mbox{\small Gal
}(\mathbb{H}_{n}/\mathbb{Q})]}$ be the $\mathbb{Z}$-module generated by
$\eta_{n}$, for all444We define the metacyclotomic units only for $n<N$, in
order to make sure that they are norms from a large extension above
$\mathbb{H}_{n}$. By avoiding the limit process
$\cap_{N}N_{N,n}(E^{\prime}(\mathbb{H}_{n}))$ we obtain in addition explicit
uniformizors $\pi_{n}$, which are norm coherent. $n<N$. By definition,
$\mathcal{N}(C(\mathbb{H}_{n}))=C(\mathbb{K}_{n})$ and the modules
$C(\mathbb{H}_{n})$ are norm coherent. In particular, if $e\in
C(\mathbb{H}_{l})$ then there is a unit $e^{\prime}\in C(\mathbb{H}_{2N})$
with $e=N_{2N,l}(e^{\prime})$.

Let further $(a_{n})_{n\in\mathbb{N}}\in A^{+}_{n}\setminus(A_{n}^{+})^{p}$ be
a norm coherent sequence of classes; thus $a_{n}^{q}=1$ for all $n\geq m$ and
$q=\exp(A^{+})$. Let $\mathfrak{Q}\in a_{2N}$ be a totally split prime and
$\gamma\in\mathbb{H}_{2N}$ be such that
$\mathfrak{Q}\mathcal{O}(\mathbb{H}_{2N})=(\gamma)$; if
$\mathfrak{Q}_{n}=N_{2N,n}(\mathfrak{Q})$ and $\gamma_{n}=N_{2N,n}(\gamma)$,
then $\mathfrak{Q}_{n}\mathcal{O}(\mathbb{H}_{n})=(\gamma_{n})$ for all
$n<2N$. We let $c_{n}=\gamma_{n}^{s}\in E(\mathbb{H}_{n})$, a norm coherent
sequence of units for $n\leq 2N$. Let
$\mathcal{C}(\mathbb{H}_{n})=c_{n}^{\mathbb{Z}[s]}\cdot\delta_{n}^{\mathbb{Z}[T,s]}$.
Then $\mathcal{C}(\mathbb{H}_{n})$ also form a norm coherent sequence and we
define
$\overline{C}(\mathbb{H}_{n})=C(\mathbb{H}_{n})\cdot\mathcal{C}(\mathbb{H}_{n})$,
which is a further norm coherent sequence of units in $E(\mathbb{H}_{n})$.

We note that $\mbox{ Ker }(\mathcal{N}:E(\mathbb{H}_{n})\rightarrow
E(\mathbb{K}_{n}^{+}))=E^{s}(\mathbb{H}_{n})\cdot\mathcal{C}(\mathbb{H}_{n})$.
Letting
$\mathcal{E}(\mathbb{H}_{n})=E(\mathbb{H}_{n})/(E^{s}(\mathbb{H}_{n})\cdot\mathcal{C}(\mathbb{H}_{n}))$,
it follows from (3) that

$\mathcal{E}(\mathbb{H}_{n})\cong E(\mathbb{K}_{n}),$

and the norm $\mathcal{N}$ is an isomorphism between these two modules. As a
consequence, if $c_{0}\in\mathcal{C}(\mathbb{H}_{n})$ generates this module
and $e_{2j}\in E(\mathbb{H}_{n}),j=0,1,\ldots,\frac{p-3}{2}$ have non trivial
image in $\mathcal{E}$ and map to a set of units $d_{j};j=0,1,\ldots,(p-3)/2$
which generate $E(K_{n}^{+})$ as a $\mathbb{Z}[T]$-module and such that
$d_{j}$ generates
$\varepsilon_{2j}E(\mathbb{K}_{n}^{+})/E(\mathbb{K}_{n}^{+})^{p^{M}}$ for
fixed, arbitrarily large $M>0$, then

$\langle
c_{0};e_{0},\ldots,e_{(p-3)/2}\rangle_{\mathbb{Z}[T,s]}=E(\mathbb{H}_{n}).$

We let $e_{n}\in E(\mathbb{H}_{n})$ be such that $\mathcal{N}(e_{n})$
generates $\varepsilon_{2j}E(\mathbb{K}_{n})/E^{p^{M}}(\mathbb{K}_{n})$ for
some fixed, large $M$. We then consider the systems

$\displaystyle\mathcal{H}_{n}$ $\displaystyle=$
$\displaystyle\left\\{c_{n}^{s^{j}}\ :\
j=0,1,\ldots,p-2\right\\}\bigcup\left\\{\delta_{n}^{s^{j}T^{l}}\ :\
j=0,1,\ldots,p-1;l=0,\ldots,p^{n-1}-1\right\\}$
$\displaystyle\bigcup\left\\{\varepsilon^{\prime}_{2j}\eta_{n}^{s^{i}\cdot
T^{l}}\ :\
i=0,1,2,\ldots,p-1;j=1,2,\ldots,\frac{p-3}{2};l=0,\ldots,p^{n-1}-1\right\\},$
$\displaystyle\overline{\mathcal{H}}_{n}$ $\displaystyle=$
$\displaystyle\left\\{c_{n}^{s^{j}}\ :\
j=0,1,\ldots,p-2\right\\}\bigcup\left\\{\delta_{n}^{s^{j}T^{l}}\ :\
j=0,1,\ldots,p-1;l=0,\ldots,p^{n-1}-1\right\\}$
$\displaystyle\bigcup\left\\{(e_{n}^{(2j)})^{s^{i}\cdot T^{l}}\ :\
i=0,1,2,\ldots,p-1;j=1,2,\ldots,\frac{p-3}{2};l=0,\ldots,p^{n-1}-1\right\\}.$

One verifies that $|\mathcal{H}_{n}|=\mathbb{Z}\hbox{-rk}(E(\mathbb{H}_{n}))$;
an application of the Nakayama lemma to
$\overline{C}(\mathbb{H}_{n})/\overline{C}(\mathbb{H}_{n})^{p^{M}}$ implies
that the system is also a generating system for
$\overline{C}(\mathbb{H}_{n})$. Moreover $\overline{\mathcal{H}}_{n}$
generates $E(\mathbb{H}_{n})/E^{p^{M}}(\mathbb{H}_{n})$ for $n\leq m$. These
systems are reminiscent of Hilbert’s relative units in his Theorem 91. It
follows that $\overline{C}(\mathbb{H}_{n})$ has finite index in
$E(\mathbb{H}_{n})$ for all $n<2N$.

Assume now that (6) is false, and there is some $m^{\prime}>m$ and
$e_{m^{\prime}}\in E(\mathbb{H}_{m^{\prime}})\setminus
E(\mathbb{H}_{m})\cdot\overline{C}(\mathbb{H}_{m^{\prime}})$; we can assume
without restriction of generality that $e_{m^{\prime}}\in
E^{(2j)}(\mathbb{H}_{m^{\prime}})$ (see below for the definition of the
components $E^{(2j)}(\mathbb{H}_{m^{\prime}})$), since at least one component
verifies the condition. Then, by (5)

$\mathcal{N}(e_{m^{\prime}})\in
E(\mathbb{K}_{m})\cdot\overline{C}(\mathbb{K}_{m^{\prime}})=\mathcal{N}(E(\mathbb{H}_{m})\cdot\overline{C}(\mathbb{H}_{m^{\prime}})).$

There is thus a unit $e\in
E(\mathbb{H}_{m})\cdot\overline{C}(\mathbb{H}_{m^{\prime}})$ such that
$e_{m^{\prime}}=e\cdot w,w\in\mbox{ Ker
}(\mathcal{N}:E(\mathbb{H}_{m^{\prime}})\rightarrow
E(\mathbb{K}_{m^{\prime}}))$. Since $\mathcal{C}_{n}$ are norm coherent and
$(1-\varepsilon^{\prime}_{0})\mbox{ Ker
}(\mathcal{N}:E(\mathbb{H}_{m^{\prime}})\rightarrow
E(\mathbb{K}_{m^{\prime}}))=(1-\varepsilon^{\prime}_{0})E(\mathbb{H}_{m^{\prime}})^{s}$,
we can assume that
$w\in(1-\varepsilon^{\prime}_{0})E(\mathbb{H}_{m^{\prime}})$, so
$w=(e_{m^{\prime}}^{\prime})^{s}$ with
$e_{m^{\prime}}^{\prime}\in(1-\varepsilon^{\prime}_{0})E(\mathbb{H}_{m^{\prime}})$.
By repeating the same argument, we find that
$e_{m^{\prime}}^{\prime}=e^{\prime\prime}\cdot w^{\prime}$, where
$w^{\prime}\in\mbox{ Ker }(\mathcal{N}:E(\mathbb{H}_{m^{\prime}})\rightarrow
E(\mathbb{K}_{m^{\prime}}))$. It follows by induction that $e_{m^{\prime}}\in
E(\mathbb{H}_{m})\cdot\overline{C}(\mathbb{H}_{m^{\prime}})\cdot
E(\mathbb{H}_{m^{\prime}})^{s^{u}}$ for any $u>0$. Since
$s^{p}=pv,v\in(\mathbb{Z}_{p}[s])^{\times}$, it follows that
$e_{m^{\prime}}\in
E(\mathbb{H}_{m})\cdot\overline{C}(\mathbb{H}_{m^{\prime}})\cdot
E(\mathbb{H}_{m^{\prime}})^{p^{M^{\prime}}}$ for arbitrary $M^{\prime}>0$. But
the index $[E(\mathbb{H}_{m^{\prime}}):\overline{C}(\mathbb{H}_{m^{\prime}})]$
is finite, so the claim must be true. Therefore $m$ is defined as the smallest
index after which $|A_{n}^{+}|$ stabilizes ∎

We obtain the following explicite variant of the first proof given above.
Since $E(\mathbb{H}_{m})\neq\overline{C}(\mathbb{H}_{m})$ while the quotient
of these modules is finite, there is a unit $e\in
E(\mathbb{H}_{m})\setminus\overline{C}(\mathbb{H}_{m})$, such that
$e^{p}\in\overline{C}(\mathbb{H}_{m})$. Consequently, there is a
metacyclotomic unit $c_{m+1}\in\overline{C}(\mathbb{H}_{m+1})$ such that
$N_{m+1,m}(c_{m+1})=e^{p}$. Let
$d=c_{m+1}/e\not\in\overline{C}(\mathbb{H}_{m+1})$, by choice of $e$. By
definition, $N_{m+1,m}(d)=1$ and thus Hilbert 90 implies that
$d=\gamma^{\omega_{m}}$ for some $\gamma\in\mathbb{H}_{m+1}$. The ideal
$(\gamma)$ is ambig, and since $A^{+}(\mathbb{H}_{m+1})=1$, it follows that
$\gamma$ must be a product of principal ramified primes. Thus
$\gamma=\pi_{m+1}^{x}\cdot\delta$, for some $x\in\mathbb{Z}[\mbox{ Gal
}(\mathbb{H}_{m+1}/\mathbb{Q})]$ and $\delta\in E(\mathbb{H}_{m+1}$. But then
$\pi_{m+1}^{x\omega_{m+1}}\in\overline{C}(\mathbb{H}_{m+1})$ by definition of
$\overline{C}$. Moreover, the structure identity (6) implies that
$\delta^{\omega_{m}}\in\overline{C}(\mathbb{H}_{m+1})^{\omega_{m}}\subset\overline{C}(\mathbb{H}_{m+1})$.
Altogether, we find that $d\in\overline{C}(\mathbb{H}_{m+1})$ and thus
$e\in\overline{C}(\mathbb{H}_{m+1})$ too. This contradicts the choice
$e\not\in\overline{C}(\mathbb{H}_{m})$, given the (6): units which are
metacyclotomic in $\mathbb{H}_{m+1}$ must be so already in $\mathbb{H}_{m}$.
Note that this contradiction makes explicite the argument of Iwasawa used in
the first proof.

###  3.2. Local units and components

We consider the structure of the local units
$U(\mathbb{K}_{n}),U(\mathbb{H}_{n})$. For the first, it is known that
$U(\mathbb{K}_{n}^{+})=\bigoplus_{k}\varepsilon_{2k}x_{n}^{\Lambda}$ are
$\Lambda$-cyclic modules, generated by some $x_{n}\in U(\mathbb{K}^{+}_{n})$
(e.g. [Wa], §8). Since $p$ is totally split, we also have
$U(\mathbb{H}_{n})\cong(U(\mathbb{K}_{n}^{+}))^{p}$, as cartesian product. Let
$\xi_{n}=(x_{n},1,1,\ldots,1)$ under the Chinese Remainder Theorem, with the
first component being the projection to $\mathbb{H}_{n,\wp_{n}}$ and $\wp_{n}$
a ramified prime above the initially chosen $\wp=(\pi)$. Then
$\tilde{\varepsilon}_{2j}\xi_{n}$ defines the $2j$-component of
$U(\mathbb{H}_{n})$ as a $\Lambda[s]$-module. Since $\nu$ acts transitively on
the projections in $U(\mathbb{H}_{n})$, it follows that these modules are also
canonically given by

(7) $\displaystyle U_{2j}(\mathbb{H}_{n})=(U(\mathbb{K}_{n}^{+}))^{p}$

where the right hand side is a cartesian of copies corresponding to the
completions at the various primes above $p$. The action of $\mbox{ Gal
}(\mathbb{K}_{n}^{+}/\mathbb{Q})$ by conjugation on $\nu$ implies

$\displaystyle\nu^{\sigma}=\nu^{\varpi(\sigma)^{2k}},\quad\nu^{\tau}=\nu^{p+1},$

with $\varpi$ being the Teichmüller character. From this, one verifies without
using the construction above, that $(U(\mathbb{K}_{n}^{+}))^{p}$ is a canonic
$\mathbb{Z}_{p}[\mbox{ Gal }(\mathbb{H}/\mathbb{Q})]$-module which well
defined by (7). We say that $U_{2j}(\mathbb{H}_{n})$ is the $2j{\rm-th}$
component of $U(\mathbb{H}_{n})$ and have

$U(\mathbb{H}_{n})=\bigoplus_{j=0}^{(p-3)/2}U_{2j}(\mathbb{H}_{n}).$

Let $\mbox{\bf N}_{l}=\cap_{n}N_{n,l}(U(\mathbb{H}_{n}))$ for $l\geq 1$. Then
class field theory implies that $U(\mathbb{H}_{l})/\mbox{\bf N}_{l}\cong\prod
U^{(1)}(\mathbb{Z}_{p})$ is the product of the $p$ copies of
$U^{(1)}(\mathbb{Z}_{p})$ in $U(\mathbb{H}_{l})$. Since $\mathbb{Z}_{p}\subset
U_{0}(\mathbb{K}_{n}^{+})$ it follows from the definition (7) that
$U(\mathbb{H}_{l})/\mbox{\bf N}_{l}\hookrightarrow U_{0}(\mathbb{H}_{n})$, and
therefore

(8) $\displaystyle U_{2j}(\mathbb{H}_{m})\subset\mbox{\bf N}_{m}\quad\hbox{for
all}\quad j\neq 0.$

We identify the units $E(\mathbb{H}_{n})$ with their diagonal embedding
$E(\mathbb{H}_{n})\hookrightarrow U(\mathbb{H}_{n})$. Since
$E(\mathbb{H}_{n})/E(\mathbb{H}_{n})^{p^{M}}$ is a $\mathbb{Z}_{p}$-module, we
can use the components of the local units for defining the components of
global units by

$\displaystyle E^{(2j)}(\mathbb{H}_{n})$ $\displaystyle=$ $\displaystyle
E(\mathbb{H}_{n})\cap\left(U_{2j}(\mathbb{H}_{n})\cdot
E(\mathbb{H}_{n})^{p^{M}}\right),$ $\displaystyle C^{(2j)}(\mathbb{H}_{n})$
$\displaystyle=$ $\displaystyle
E^{(2j)}(\mathbb{H}_{n})\cap\overline{C}(\mathbb{H}_{n})$

By definition of $U_{2j}$, we have $E^{(2l)}(\mathbb{H}_{n})\cap
E^{(2j)}(\mathbb{H}_{n})=E(\mathbb{H}_{n})^{p^{M}}$ for $l\neq j$.

Let now $e\in E^{(2k)}(\mathbb{H}_{m})\setminus
N_{m+1,m}(E^{(2k)}(\mathbb{H}_{m+1}))$. Note that by (6), the existence of
such a unit is granted. But then, by applying $\mathcal{N}$ we deduce that
$\varepsilon_{2k}E(\mathbb{K}_{m})=\varepsilon_{2k}C(\mathbb{K}_{m})$.
However, our initial assumption implies that
$\varepsilon_{2k}A^{+}_{m}\neq\\{1\\}$ and the main conjecture thus leads to
$[\varepsilon_{2k}E(\mathbb{K}_{m}):\varepsilon_{2k}C(\mathbb{K}_{m})]\neq 1$.
Therefore, we conclude that there is unit $e$ with the claimed properties. The
Hasse norm principle implies by the above that $e\in
N_{m+1,m}(\mathbb{H}_{m+1}^{\times})$. Let thus
$x\in\mathbb{H}_{m+1}^{\times}$ with $N_{m+1,m}(x)=e$. Then
$(x)=\mathfrak{X}^{\omega_{m}}$, since
$H^{1}(\langle\nu\rangle,I(\mathbb{H}_{n}))$ vanishes for the ideals in the
cyclic extension $\mathbb{H}_{n}/\mathbb{K}_{n}$. Moreover, $\mathfrak{X}$ is
not $p$-principal: otherwise, if $(t,p)=1$ and $\mathfrak{X}^{t}=(\xi)$ is
principal, then $x^{t}=\xi^{\omega_{m}}\cdot e_{1}$ for some unit $e_{1}\in
E(\mathbb{H}_{m+1})$. If the $ut\textasciiacute+vp=1$ we have

$e=N_{m+1,m}(e^{v}\cdot\xi^{\omega_{m}}e_{1})=N_{m+1,m}(e^{v}\cdot e_{1})\in
N_{m+1,m}E(\mathbb{H}_{m+1}),$

and we had assumed $e\not\in N_{m+1,m}E(\mathbb{H}_{m+1})$. Consequently
$A(\mathbb{H}_{m+1})\neq\\{1\\}$, which is a contradiction to the Lemma 3 that
completes the proof.

Note that universal norms behave differently in the positive components and in
the zero component, the latter being by (8) the one which localizes the norm
residue defects along the cyclotomic $\mathbb{Z}_{p}$-extension of any base
field. This fact may indicate the major distinction between zero components
and positive indexed components.

##  4\. Appendix

For the sake of completeness, we give a proof of the following

######  Lemma 3.

Let $\mathbb{K}$ be a number field and $A$ be the $p$ \- part of its class
group, while $\mathbb{H}$ is the $p$ \- part of its Hilbert class group. If
$A$ is cyclic, then $A(\mathbb{H}):=\mathcal{C}(\mathbb{H})_{p}=\\{1\\}$.

###### Proof.

Since $A$ is cyclic, $\mbox{ Gal }(\mathbb{H}/\mathbb{K})\cong A$ is a cyclic
group and the ideals of a generating class $a\in A$ are inert and become
principal in $\mathbb{H}$. Let $\sigma=\varphi(a)\in\mbox{ Gal
}(\mathbb{H}/\mathbb{K})$ be a generator and $s=\sigma-1$. Suppose that $b\in
A(\mathbb{H})\setminus A(\mathbb{H})^{(s,p)}$ is a non trivial class and let
$\mathfrak{Q}\in b$ be an ideal above a rational prime
$\mathfrak{q}\subset\mathbb{K}$, which splits completely in
$\mathbb{H}/\mathbb{K}$: such a prime must exist, by Tchebotarew’s Theorem. If
$b^{\prime}=[\mathfrak{q}]$, then the order of $b^{\prime}$ must be a power of
$p$, since this holds for $\mathfrak{Q}$; thus $b^{\prime}\in
A(\mathbb{K})=\langle a\rangle$. But we have seen that the ideals from $a$
capitulate in $\mathbb{H}$, so $b=1$, in contradiction with our choice.
Therefore $b^{\prime}=1$ and thus $\mbox{\bf N}_{\mathbb{H}/\mathbb{K}}(b)=1$.
Furtwängler’s Hilbert 90 Theorem for ideal class groups in cyclic extensions
[Fu] says that $\mbox{ Ker }(\mbox{\bf N}:A(\mathbb{H})\rightarrow
A(\mathbb{K}))\subset A(\mathbb{H})^{s}$, and this implies that $b\in
A(\mathbb{H})^{s}$, which contradicts the choice of $b$ and completes the
proof. ∎

## References

  * [Fu] P. Furtwängler: Über die Reziprozitätsgesetze zwischen $l$-ten Potenzresten in algebraischen Zahlkörpern, wenn $l$ eine ungerade Primzahl bedeutet, (German) Math. Ann. 58, 1-50 (1904).
  * [Hi] David Hilbert: The Theory of Algebraic Numbers (Zahlbericht), Translated by Iain Adamson, Edited by Iain Adamson, Franz Lemmermayer and Norbert Schappacher; Springer (1998).
  * [La] S. Lang: Cyclotomic fields I and II, First Edition, Springer (1978,80)
  * [Iw] K. Iwasawa: A note on the group of units of an algebraic number fields, Journal de Math. Pures et Appl. 35/121 (1956), pp. 189-192.
  * [Iw1] K. Iwasawa: On $\mathbb{Z}_{\ell}$ \- extensions of number fields, Ann. Math. Second Series, (1973), 98, 247 – 326.
  * [Wa] Lawrence Washington: Introduction to cyclotomic fields, Springer, Graduate Texts in Mathematics 83, 2-nd edition (1996).


# DeepWSD: Projecting Degradations in Perceptual Space to Wasserstein Distance
in Deep Feature Space

Xingran Liao  City University of Hong KongHong KongChina
<EMAIL_ADDRESS>,  Baoliang Chen  City University of Hong
KongHong KongChina<EMAIL_ADDRESS>,  Hanwei Zhu  City University of
Hong KongHong KongChina<EMAIL_ADDRESS>,  Shiqi Wang  City
University of Hong KongHong KongChina<EMAIL_ADDRESS>,  Mingliang Zhou
Chongqing University174 shazheng street, Shapingba DistrictChongqingChina
<EMAIL_ADDRESS>and  Sam KWONG  City University of Hong Kong83 Tat
Chee Ave, Kowloon TongHong KongChina999077<EMAIL_ADDRESS>
(2022)

###### Abstract.

Existing deep learning-based full-reference IQA (FR-IQA) models usually
predict the image quality in a deterministic way by explicitly comparing the
features, gauging how severely distorted an image is by how far the
corresponding feature lies from the space of the reference images. Herein, we
look at this problem from a different viewpoint and propose to model the
quality degradation in perceptual space from a statistical distribution
perspective. As such, the quality is measured based upon the Wasserstein
distance in the deep feature domain. More specifically, the 1D Wasserstein
distance at each stage of the pre-trained VGG network is measured, based on
which the final quality score is performed. The deep Wasserstein distance
(DeepWSD) performed on features from neural networks enjoys better
interpretability of the quality contamination caused by various types of
distortions and presents an advanced quality prediction capability. Extensive
experiments and theoretical analysis show the superiority of the proposed
DeepWSD in terms of both quality prediction and optimization. The
implementation of our method is publicly available at https://github.com/Buka-
Xing/DeepWSD.

Full-reference IQA, Wasserstein distance, Statistical model for image
representation

††journalyear: 2022††copyright: acmcopyright††conference: Proceedings of the
30th ACM International Conference on Multimedia ; October 10–14, 2022; Lisbon,
Portugal.††booktitle: Proceedings of the 30th ACM International Conference on
Multimedia (MM ’22), October 10–14, 2022, Lisbon, Portugal††price:
15.00††isbn: 978-1-4503-9203-7/22/10††doi: 10.1145/3503161.3548193††ccs:
Computing methodologies Image representations††ccs: Networks††ccs: General and
reference Metrics

##  1\. Introduction

Full-reference image quality assessment (FR-IQA), which establishes the
quantitative projection between perceptual quality and computational models
(Wang and Bovik, 2006), plays an essential role in monitoring the quality of
multimedia processing systems. In addition, the FR-IQA models have been shown
to excel in quality optimization of image restoration tasks, such as
denoising, deblurring and superresolution (Ding et al., 2020a, 2021a).

The design philosophies of the FR-IQA models are supported by the statistics
of natural scenes, the prior knowledge of the Human Visual System (HVS), as
well as the understanding of distortion (Duanmu et al., 2021). In particular,
it is widely accepted that extraction of the features that account for
naturalness from reference and distortion images is important. The well-known
structural similarity (SSIM) index (Wang et al., 2004a) focuses on the
structural information from a scene. The deep image structure and texture
similarity (DISTS) method (Ding et al., 2020b) adopts the pretrained deep
convolutional network (VGG16) learned from abundant images to extract
features. The FR-IQA models also attempt to mimic the visual perception in
HVS. The feature similarity index (FSIM) (Zhang et al., 2011) adopts the log-
Gabor filters to mimic such mechanism, supported by the psychophysical
evidence (Fischer et al., 2007; Thillou and Gosselin, 2006) that log-Gabor
filters are highly correlated with the shallow visual system. The complex
wavelet structural similarity (CW-SSIM) index (Sampat et al., 2009) performs
the wavelet transformation to mimic the visual system, which is also supported
by neurological experiments (Morrone and Burr, 1988). Moreover, the FR-IQA
methods also consider the distortion types and levels between the reference
and distortion images. In image information and visual quality (VIF) (Sheikh
and Bovik, 2006) and the information fidelity criterion (IFC) (Sheikh et al.,
2005), distortions are treated as information loss and images are constructed
as Gaussian scale mixture models. Then, mutual information is used to quantify
the information loss. In the most apparent distortion metric (MAD) (Larson and
Chandler, 2010), distortions are classified as apparent and invisible
distortions. As such, both the pyramid structure and log-Gabor filters are
used to identify the distortion type and strength.

The FR-IQA shall be regarded as the fidelity measure in a wider sense, due to
the existence of the reference image. Typically, the common pipeline is
comparing the prominent visual information that governs the visual quality
from pristine and distorted images, in spatial, frequency, or deep feature
domains. These mathematical models, which are expected to model the perceptual
functionalities in assessing the visual quality, typically involve making
comparisons. As such, a typical deterministic way is explicitly performing
feature level comparison, where the features are defined in a wider sense.
However, the nonlinearities of the complicated perceptual system, as well as
the capability of quality assessment without the pristine reference for HVS,
persuade us to re-examine the quality assessment in a statistical way. In
particular, we propose the deep feature domain Wasserstein distance (DeepWSD)
for FR-IQA. The proposed method attempts to relate the FR-IQA to the minimal
efforts that account for the conversion between the distributions of the
pristine and distorted images in the deep feature domain. The amount of
minimal effort is quantified based upon the concept of the Wasserstein
distance (WSD) (Peyre and Cuturi, 2019), a widely used statistical measure
between two distributions.

The proposed DeepWSD is intrinsically aligned with the efficient coding
hypothesis that the efficient coding neural system is ought to match the
statistics of the signals that they attempt to represent (Ming and Holt,
2009), from the perspective of natural scene statistics. Regarding the human
perception, the free-energy principle (Friston et al., 2006; Friston, 2010; Gu
et al., 2014; Zhai et al., 2011) also reveals that the internal generative
model with which the human vision tries to explain the perceived scene, is
able to produce the intrinsic reference for statistical discrepancy
evaluation. The visual quality, which is governed by this discrepancy, is
characterized by the Wasserstein distance here. Moreover, our method does not
rely on training with quality labels or any controllable parameters for
tuning, such that it is highly robust to diverse types of distortions.
Experimental results show that the proposed DeepWSD delivers accurate
predictions over a series of datasets, and enjoys the advantage of strong
generalization capability and high interpretability.

##  2\. Related Works

###  2.1. FR-IQA with Feature Comparisons

One widely accepted philosophy in FR-IQA design is comparing perceptually
meaningful visual features. From the statistical perspective, image features
can be treated as invariants of the distribution (Qi et al., 2021) which
contain important information of images. Distortions typically occur (Hu,
1962) along with the contamination of these invariants. Traditional methods
extract image features in the pixel or the frequency domain, based upon the
similarity measure or the Euclidean norm. Regarding SSIM (Wang et al., 2004a),
the luminance, contrast, and structure are treated as features in a wider
sense, and they are compared by the similarity measure.

In FSIM (Zhang et al., 2011), the high phase congruent features in the
frequency domain are extracted by the log-Gabor filters and features are also
compared by the similarity index. FSIM (Zhang et al., 2011) presents
outstanding performance with increased computation cost. In Gradient Magnitude
Similarity Deviation (GMSD) (Xue et al., 2014), image gradients are extracted
by Prewitt operators and compared by simply using the Euclidean norm, due to
the observation that gradient map of images is more sensitive to various types
of distortions (Xue et al., 2014).

Due to the excellent performance of deep neural networks in feature
representation, deep features can also be compared to indicate the visual
quality. In Learned Perceptual Image Patch Similarity (LPIPS) (Zhang et al.,
2018), different deep network encoders have been used to extract different
deep features. Subsequently, LPIPS adopts weighted MSE for comparisons,
presenting better performance compared to most traditional FR-IQA methods
(Zhang et al., 2018). In DISTS (Ding et al., 2020b), motivated by the design
of the modified SSIM (Brunet et al., 2012), deep features are compared in a
new way, delivering more accurate subjective ratings. Based upon DISTS (Ding
et al., 2020b), the A-DISTS (Ding et al., 2021a) which separates the structure
and texture across space and scale, further improves the performance of DISTS
(Ding et al., 2020b). However, the explicit comparisons of the features in
spatial, frequency, or deep learning domains do not specifically consider the
image quality from the statistical perspective, while a large literature has
been dedicated to the importance of naturalness from the statistical point of
view in biological vision (Simoncelli and Olshausen, 2001).

###  2.2. Wasserstein Distance

The Wasserstein distance originates from optimal transformation (Peyre and
Cuturi, 2019) and has been widely used in different vision tasks. In
particular, it characterizes the differences between two distributions by
computing how difficult a specific distribution can be transformed into
another. In image retrieval (Rubner et al., 2000), the Wasserstein distance
was used to compare the similarity between the histogram of color, texture,
and outlines of images. It delivers higher retrieval accuracy than traditional
Euclidean norm and Chi-square distance (Yang et al., 2015). In image
reconstruction and generation tasks (Chen et al., 2020; Arjovsky et al.,
2017), the Wasserstein distance also exhibits promising capability on texture
generation compared to the Euclidean norm and the Kullback-Leibler divergence
(Sanchez et al., 2018). Moreover, using the Wasserstein distance in
Generative-Adversarial networks can better prevent mode collapse than using
the Kullback-Leibler divergence (Arjovsky et al., 2017). In image style
transfer (Chen et al., 2021a), the distillation method based on Wasserstein
distance also exhibits that such distance can better maximize the lower bound
of mutual information between source and target images compared to the
Kullback-Leibler divergence, leading to the reduction of artifacts. However,
to the best of our knowledge, the Wasserstein distance has not been exploited
in the FR-IQA.

##  3\. Quality Assessment Method

###  3.1. Methodology

In DeepWSD, we evaluate the quality of distorted images by using 1D
Wasserstein distance (Snow and Lent, 2016). In particular, we first adopt the
VGG16 network (Simonyan and Zisserman, 2015) which projects images into deep
feature representations, and subsequently, the 1D WSD is used to compare the
distributions of the input images as well as all the deep features. In the
following paragraphs, we first provide a brief introduction of WSD, then the
proposed method is detailed.

####  3.1.1. Preliminary of WSD

Given two multidimensional random variables $M$ and $N$ with their
distributions denoted as $\mathcal{X}$ and $\mathcal{Y}$, respectively, the
Wasserstein distance is defined as,

(1)
$W_{l}(M,N)={{\left(\underset{J\in\mathcal{J}(\mathcal{X},\mathcal{Y})}{\mathop{\inf}}\,\int{||m-n|{{|}_{l}}\;dJ(m,n)}\right)}^{{1}/{l}\;}},$

where $m$ and $n$ are the masses of $M$ and $N$. J $\in$
$\mathcal{J}(\mathcal{X},\mathcal{Y})$ is the joint distribution of $(M,N)$.
$l$ is the order of the $l$-norm. Typically, $l$ can be set as 1 or 2, and in
our experiments we set $l=2$. Supposing $F(m)=P\\{M\leqslant m\\}$ and
$G(n)=P\\{N\leqslant n\\}$, as indicated in (Peyre and Cuturi, 2019), when $M$
and $N$ are one-dimension, Eqn. (1) has a closed-form solution. It computes
the integral of the inverse function of the probability distribution function
of $\mathcal{X}$ and $\mathcal{Y}$,

(2)
$W_{l}(M,N)={{\left(\int_{0}^{1}{||{{F}^{-1}}(z)-{{G}^{-1}}(z)||_{l}\;dz}\right)}^{{1}/{l}\;}},$

where $z$ is the implicit variable which is used to integral $F^{-1}(\cdot)$
and $G^{-1}(\cdot)$ from 0 to 1. $F^{-1}(\cdot)$ and $G^{-1}(\cdot)$ represent
the inverse functions of $F(\cdot)$ and $G(\cdot)$, respectively.

####  3.1.2. The Proposed DeepWSD

In DeepWSD, we adopt the one-dimension form of WSD for the distribution
divergence measuring. More specifically, as shown in Fig. 1, the input
reference image and distorted image are denoted as $P$ and $Q$, respectively.
Then we utilize the pre-trained VGG16 network (Simonyan and Zisserman, 2015)
to extract their deep features at five stages. In particular, we denote the
extracted feature of $P$ and $Q$ at the $i-th$ stage as $\tilde{P}_{i}$ and
$\tilde{Q}_{i}$, $i\in\\{1,5\\}$. Herein, the VGG16 network was originally
used to deal with the image classification task and has been widely used in
different CV tasks including super-resolution (Wang et al., 2019) and
denoising (Sil et al., 2019). In LPIPS (Zhang et al., 2018) and DISTS (Ding et
al., 2020b), features of the VGG16 network were also proved to be useful for
quality assessment. Subsequently, we reshape the $P$, $Q$, $\tilde{P}_{i}$,
and $\tilde{Q}_{i}$ to 1D vectors thus the WSD can be performed as follows,

(3)
$\mathcal{D}_{wsd}\left(P,Q\right)=W_{l}\left(P,Q\right)+\sum_{i=1}^{5}{W_{l}\left(\tilde{P}_{i},\tilde{Q}_{i}\right)}.$

In addition to WSD, we further employ the adaptive Euclidean distance (denoted
as $\mathcal{D}_{eul}$) to pursue high fidelity, which can be formulated as
follows,

(4) $\displaystyle\mathcal{D}_{eul}\left(P,Q\right)=$ $\displaystyle
g(W_{l}\left(P,Q\right))\times||P-Q||_{2}$
$\displaystyle+\sum_{i=1}^{5}{\left(g\left(W_{l}\left(\tilde{P}_{i},\tilde{Q}_{i}\right)\right)\times||\tilde{P}_{i}-\tilde{Q}_{i}||_{2}\right)},$

where the definition of $g(s)$ is given by,

(5) $g(s)=\frac{1}{{{(s+10)}^{2}}\sqrt{\exp(-{1}/{(s+10)}\;)}}.$

The combination of Euclidean norm and adaptive weight $g(s)$ mimics a special
mechanism when HVS quantifies the visual quality. In particular, Larson et al.
(Larson and Chandler, 2010) pointed out that the most apparent distortions are
first perceived in a high priority. Then, scrupulous viewers may find certain
parts of the images contain subtle distortions. By considering the
$\mathcal{D}_{wsd}$ and $\mathcal{D}_{eul}$ simultaneously, the distance
scores of each layer are finally averaged and further processed by a
logarithmic function (denoted as $\log(\cdot)$), deriving the proposed DeepWSD
measure as follows,

(6)
$DeepWSD\left(P,Q\right)=\log\left(\frac{1}{6}\mathcal{D}_{wsd}\left(P,Q\right)+\frac{1}{6}\mathcal{D}_{eul}\left(P,Q\right)\right).$

As a deep learning based quality measure, the DeepWSD avoids the training with
quality labels, enjoying the great advantages in terms of generalization
capability. Moreover, the proposed DeepWSD does not rely on any hyperparameter
that is determined empirically regarding the generation of the final score. We
believe that these are the directions in which the greatest benefit eventually
lies.

Figure 1. Overall structure of DeepWSD. It contains the raw image and five feature extraction stages in VGG16. ‘Avg’ means averaging scores in different layers and ‘Log’ indicates the logarithm function. Figure 2. (a) original image; (b) structure map of (a) extracted by SSIM; (c) feature map sampled in the 4-th layer of VGG16. Figure 3. Quality measure results in feature domain by different quality models. $I_{a}^{R}$ and $I_{b}^{R}$ are two reference images. $I_{a}^{D1}$ ($I_{b}^{D1}$ ) and $I_{a}^{D2}$ ($I_{b}^{D2}$) are two distorted versions of $I_{a}^{R}$ ($I_{b}^{R}$ ) with different quality ratings. The scores under each distorted image are the final outputs of corresponding IQA models and the scores of the feature maps are calculated by the feature comparison method adopted in each IQA model.  Table 1. Comparison between our method and state-of-the-art methods on four classical datasets. Bold and uderline numbers correspond to the first and second best methods in each column, respectively. Method | TID2013 (Ponomarenko et al., 2015) | LIVE (Sheikh et al., 2010) | CSIQ (Larson and Chandler, 2010) | IVC (Ninassi et al., 2006)
---|---|---|---|---
PLCC | SRCC | KRCC | PLCC | SRCC | KRCC | PLCC | SRCC | KRCC | PLCC | SRCC | KRCC
PSNR | 0.6773 | 0.6876 | 0.4963 | 0.8648 | 0.8726 | 0.6773 | 0.8195 | 0.8101 | 0.6014 | 0.6804 | 0.6499 | 0.4781
SSIM (Wang et al., 2004a) | 0.7767 | 0.7271 | 0.5454 | 0.9341 | 0.9479 | 0.7963 | 0.8523 | 0.8656 | 0.6807 | 0.7708 | 0.9018 | 0.7223
MS-SSIM (Wang et al., 2004b) | 0.8292 | 0.7887 | 0.6049 | 0.9399 | 0.9513 | 0.8048 | 0.8895 | 0.9133 | 0.7393 | 0.7913 | 0.8980 | 0.7203
NLPD (Laparra et al., 2016) | 0.8398 | 0.8491 | 0.6785 | 0.9316 | 0.9372 | 0.7781 | 0.9234 | 0.9343 | 0.7695 | 0.7815 | 0.8202 | 0.6093
FSIM (Zhang et al., 2011) | 0.8233 | 0.8549 | 0.6549 | 0.9608 | 0.9672 | 0.8814 | 0.9187 | 0.9379 | 0.7683 | 0.8161 | 0.9263 | 0.7537
CW-SSIM (Sampat et al., 2009) | 0.5846 | 0.7560 | 0.5580 | 0.5346 | 0.9082 | 0.7739 | 0.6808 | 0.9441 | 0.7594 | 0.5630 | 0.5834 | 0.4072
VIF (Sheikh and Bovik, 2006) | 0.8672 | 0.8429 | 0.6526 | 0.9343 | 0.9603 | 0.8284 | 0.9132 | 0.9121 | 0.7432 | 0.7391 | 0.7273 | 0.5590
MAD (Larson and Chandler, 2010) | 0.8267 | 0.7680 | 0.6154 | 0.9682 | 0.9669 | 0.8425 | 0.9505 | 0.9468 | 0.7975 | 0.8704 | 0.8698 | 0.6671
PieAPP (Prashnani et al., 2018) | 0.8501 | 0.8479 | 0.6828 | 0.9079 | 0.9279 | 0.8266 | 0.8823 | 0.8833 | 0.7305 | 0.6594 | 0.7187 | 0.5249
DISTS (Ding et al., 2020b) | 0.8624 | 0.8483 | 0.6574 | 0.9360 | 0.9542 | 0.8112 | 0.9284 | 0.9289 | 0.7675 | 0.8993 | 0.9138 | 0.7267
LPIPS (Zhang et al., 2018) | 0.7324 | 0.6696 | 0.4970 | 0.9343 | 0.9324 | 0.7782 | 0.8936 | 0.8758 | 0.6893 | 0.8715 | 0.9044 | 0.7386
DeepWSD | 0.8697 | 0.8741 | 0.6932 | 0.9609 | 0.9624 | 0.8378 | 0.9503 | 0.9652 | 0.8297 | 0.9119 | 0.9231 | 0.7570

###  3.2. Connections with Existing Models

The new design philosophy of DeepWSD can be better understood by exploring the
connections with existing methods. First, the traditional quality measures,
based upon the point-wise signal difference, structural similarity
quantification, or feature-domain comparisons, suggest that exploiting the
strong dependencies in visual representation is important in modeling the
visual quality. One intuitive example is shown in Fig. 2, which suggests that
both structures and deep features reflect such dependencies. The proposed
DeepWSD naturally considers such correlations in the deep feature domain due
to its high relevance in visual perception.

Second, it is widely acknowledged that the features from the pre-trained VGG
network correlate well with the visual perception (Ding et al., 2020b; Zhang
et al., 2018). As such, the straightforward point-wise feature map
comparisons, or conducting the structure and texture similarity modeling,
could better quantify the visual quality. The proposed method adopts the deep
learning features as well, while the principle philosophy underlying the
companions has been advanced to a new level with the probability modeling on
the distribution. As shown in Fig. 3, the distortions introduced cause the
significant quality degradation in the feature domain, demonstrating that deep
features possess a strong capability for sensing distortions. Moreover, we can
observe that even the two distorted images, $I_{a}^{D1}$ and $I_{a}^{D2}$
present totally different quality (1.53 v.s. 3.60), the feature comparison
results obtained by DISTS are very close at each stage. On the contrary, the
quality measured by our DeepWSD could better reflect the distortions and such
variations. The quality unawareness is also presented by LPIPS in Fig. 3 when
compared with DeepWSD, further revealing the effectiveness of WSD performed on
deep feature space. In (Delbracio et al., 2021), the optimization of the WSD
of deep features has been shown to be effective in low-level visual
processing, providing further evidence that the exploration of the WSD for
quality assessment is meaningful.

Third, the assumption that HVS is highly adapted to the natural scene has been
a golden hypothesis in no-reference IQA. Due to the unavailability of the
pristine images, the visual quality is quantified from the perspectives that
how those regularities are modified along with the distortion (Chen et al.,
2021b). The FR-IQA methods, which shall be understood as the fidelity
measurement, typically do not rely on distributions. The proposed approach,
although belonging to FR-IQA, fundamentally lies between the evaluation of
fidelity and quality, which are sometimes driven by competing factors (Yeganeh
and Wang, 2013). The visual quality is highly relevant to naturalness which
has been long recognized as the subject quantity relying on the statistics of
natural scene distribution. As such, the visual quality evaluation relying on
the distribution of features could reliably capture low-level statistics by
measuring the destruction against the feature distribution of pristine images.

##  4\. Experimental Results

We evaluate the performance of DeepWSD on six IQA datasets, including CSIQ
(Larson and Chandler, 2010), TID2013 (Ponomarenko et al., 2015), LIVE (Sheikh
et al., 2010), IVC (Ninassi et al., 2006), KADID-10k (Lin et al., 2019) and
LIVE-MultiDist (Jayaraman et al., 2012). Those databases contain various
distortion types and distortion levels. In particular, the KADID-10k (Lin et
al., 2019) dataset contains 81 original reference images and 25 distortion
types, resulting in a total number of 10,125 images. In LIVE-MultiDist
(Jayaraman et al., 2012), two mixed distortion types are involved: blur with
Gaussian white noise and blur with JPEG compression, which target at the
application of image acquisition and transmission. We compare DeepWSD with 11
state-of-the-art FR-IQAs, including PSNR, SSIM (Wang et al., 2004a), MS-SSIM
(Wang et al., 2004b), NLPD (Laparra et al., 2016), FSIM (Zhang et al., 2011),
CW-SSIM (Sampat et al., 2009), VIF (Sheikh and Bovik, 2006), MAD (Larson and
Chandler, 2010), perceptual image-error assessment through pairwise preference
(PieAPP) (Prashnani et al., 2018), DISTS (Ding et al., 2020b) and LPIPS (with
VGG16 utilized as the feature extractor) (Zhang et al., 2018). Among them,
PSNR, SSIM (Wang et al., 2004a), MS-SSIM (Wang et al., 2004b) and NLPD
(Laparra et al., 2016) are FR-IQAs computed in the pixel domain. FSIM (Zhang
et al., 2011), CW-SSIM (Sampat et al., 2009), VIF (Sheikh and Bovik, 2006) and
MAD (Larson and Chandler, 2010) are computed in the frequency domain. PieAPP
(Prashnani et al., 2018), DISTS (Ding et al., 2020b), LPIPS (Zhang et al.,
2018) and DeepWSD are FR-IQAs computed in the deep feature domain. We use
three measures to evaluate the quality assessment results, including Pearson
linear correlation coefficient (PLCC), Spearman rank correlation coefficient
(SRCC), and Kendall rank correlation coefficient (KRCC). In addition, a four-
parameter regression model is used to visualize the score predicting results,
and the definition is given by

(7) $\bar{D}=(a_{1}-a_{2})/(1+\exp(-(D-a_{3})/|a_{4}|)+a_{2}),$

where $\bar{D}$ is the fitted score and $D$ is the raw score.
$\\{a_{i}\\}_{i=1}^{4}$ are the parameters. For DeepWSD, we set $l=2$, making
the quality measure more sensitive to outliers. For the one-dimensional case
of the Wasserstein distance, we split the images into patches and reshape them
as vectors. The patch size is set to $4\times 4$. A larger patch size could
persuade DeepWSD to pay more attention to global differences.

Table 2. Quality prediction results on the KADID-10k and LIVE-MultiDist datasets. Bold and underline numbers correspond to the first and second best methods in each column. Herein, it is worth mentioning DISTS was trained on the KADID-10k dataset.  Method | KADID-10k (Lin et al., 2019) | LIVE-MultiDist (Jayaraman et al., 2012)
---|---|---
PLCC | SRCC | KRCC | PLCC | SRCC | KRCC
PSNR | 0.6747 | 0.6665 | 0.4836 | 0.7622 | 0.7664 | 0.5830
SSIM (Wang et al., 2004a) | 0.7217 | 0.7325 | 0.5572 | 0.8652 | 0.8255 | 0.6157
MS-SSIM (Wang et al., 2004b) | 0.8012 | 0.8029 | 0.6088 | 0.8826 | 0.8795 | 0.6431
NLPD (Laparra et al., 2016) | 0.8131 | 0.8444 | 0.6461 | 0.8697 | 0.8536 | 0.6837
FSIM (Zhang et al., 2011) | 0.8511 | 0.8542 | 0.6648 | 0.8386 | 0.8932 | 0.6961
CW-SSIM (Sampat et al., 2009) | 0.5493 | 0.7444 | 0.5594 | 0.3936 | 0.4636 | 0.3166
VIF (Sheikh and Bovik, 2006) | 0.7927 | 0.7906 | 0.6014 | 0.8947 | 0.8874 | 0.5428
MAD (Larson and Chandler, 2010) | 0.8232 | 0.7965 | 0.6210 | 0.8542 | 0.8712 | 0.6943
PieAPP (Prashnani et al., 2018) | 0.7889 | 0.7857 | 0.6493 | 0.8763 | 0.8881 | 0.6897
DISTS (Ding et al., 2020b) | 0.8589 | 0.8893 | 0.7100 | 0.8951 | 0.9330 | 0.7478
LPIPS (Zhang et al., 2018) | 0.7003 | 0.7200 | 0.5313 | 0.8350 | 0.8078 | 0.6083
DeepWSD | 0.8434 | 0.8883 | 0.7076 | 0.8952 | 0.9073 | 0.7558

###  4.1. Performance Validations

Figure 4. Visualization of regression results. The $x$-axis is the fitted
scores by Eqn. (7), and the $y$-axis is the MOS. $R=\sqrt{1-RSS/TSS}$ is the
goodness of fit. Herein, $RSS$ and $TSS$ are sum of squares of deviations and
total sum of squares. A larger $R$ implies the stronger relationship between
the predicted scores and MOS values.  Figure 5. Optimization results by the
DISTS, LPIPS, and DeepWSD. The noise-to-land and the white-to-Lena experiments
are conducted. ‘Q’ means the quality score predicted by each IQA measure.

Quality Prediction. The experimental results on the six datasets are shown in
Table 1 and Table 2, from which we can observe our method achieves the best
performance on CSIQ, TID2013, LIVE, IVC, and LIVE-MultiDist datasets in terms
of PLCC, SRCC and KRCC. For KADID-10k, although the pre-trained model DISTS
achieves the best results, our method is still comparable without any training
procedure introduced. Moreover, the linear regression results in Fig. 4 show
that our method presents strong linearity with MOSs, further demonstrating the
superior performance of our method.

There are two reasons behind this phenomenon. First, applying WSD as the
distance measure for features from VGG16 networks could better align with the
visual perception, rooted in the view that the quality is not perceived by
direct comparisons. Second, the proposed DeepWSD can deal with different
distortion types with satisfied generalization ability and does not rely on a
set of hyper-parameters. This is reflected in the challenging KADID-10k and
LIVE-MultiDist datasets, on which the promising performance can be also
achieved.

Quality Optimization. As indicated in (Ding et al., 2021b), a promising way
for FR-IQA models evaluation is to use them as objectives for quality
optimization. Following this vein, in Fig. 5, we attempt to reconstruct the
reference image from the noise image and the white image using different
quality measures. This procedure can be formulated as
$arg\min_{P}\;D\left(P,Q\right)$, where $D(P,Q)$ is the FR-IQA model.
Moreover, $Q$ is the reference image and $P$ is the reconstructed image
initialized by random noise and a pure white image. We compare our
optimization results with the two latest deep IQA models, LPIPS and DISTS.
From Fig. 5, the unpleasant artifacts are usually introduced in the
optimization results of DISTS, revealing the fidelity may not be well
maintained. The LPIPS also presents low-quality results, especially in the
rich-texture regions, and cannot fully reconstruct the Lena image when the
initial image is the white image. Compared with both DISTS and LPIPS, our
method reconstructs the reference image with both high quality and high
fidelity, demonstrating the WSD is able to measure the feature distance in a
more perceptual way.

Table 3. Comparison results with different patch sizes and different network structures. The DeepWSD w/P4, DeepWSD w/P8 and DeepWSD w/P16 represent the methods with patch sizes set to 4,8,16, respectively. The DeepWSD w/VGG16, DeepWSD w/AlexNet and DeepWSD w/SqueezeNet represent the methods using corresponding network structures. Bold and underline denote the first and second highest scores, respectively. Method | TID2013 (Ponomarenko et al., 2015) | LIVE (Sheikh et al., 2010) | CSIQ (Larson and Chandler, 2010) | KADID-10k (Lin et al., 2019)
---|---|---|---|---
PLCC | SRCC | KRCC | PLCC | SRCC | KRCC | PLCC | SRCC | KRCC | PLCC | SRCC | KRCC
DeepWSD w/P4 | 0.8697 | 0.8741 | 0.6932 | 0.9609 | 0.9624 | 0.8378 | 0.9503 | 0.9652 | 0.8297 | 0.8434 | 0.8883 | 0.7076
DeepWSD w/P8 | 0.8512 | 0.8662 | 0.6749 | 0.9586 | 0.9546 | 0.8237 | 0.9379 | 0.9521 | 0.8232 | 0.8154 | 0.8678 | 0.6887
DeepWSD w/P16 | 0.8323 | 0.8489 | 0.6577 | 0.9471 | 0.9448 | 0.8135 | 0.9246 | 0.9503 | 0.8189 | 0.7931 | 0.8423 | 0.6703
DeepWSD w/VGG16 | 0.8697 | 0.8741 | 0.6932 | 0.9609 | 0.9624 | 0.8378 | 0.9503 | 0.9652 | 0.8297 | 0.8434 | 0.8883 | 0.7076
DeepWSD w/AlexNet | 0.8471 | 0.8266 | 0.6479 | 0.9237 | 0.9518 | 0.8175 | 0.9478 | 0.9515 | 0.7952 | 0.7825 | 0.8094 | 0.6076
DeepWSD w/SqueezeNet | 0.8354 | 0.8673 | 0.6855 | 0.9536 | 0.9567 | 0.8321 | 0.9381 | 0.9597 | 0.8127 | 0.8393 | 0.8801 | 0.7027

###  4.2. Ablation Studies

Figure 6. Ablation study results of DeepWSD, DeepWSD w/o $g(s)$, WSD and
DeepWSD w/o Pixel on TID2013 and KADID-10k datasets.

Influence of Different Patch Sizes. To explore the best patch size for WSD
performing on VGG layers, we set various patch sizes in our proposed method
and present the comparison results on TID2013 and KADID-10k datasets in Table
3. We denote the model with patch size set to 4, 8, 16 as DeepWSD w/P4,
DeepWSD w/P8 and DeepWSD w/P16, respectively. From the table, we can observe
that the best performance is achieved by setting the patch size as 4. The
reason lies in that the smaller patch size could benefit the capture of tiny
local distortion.

Ablation of Adaptive Weight $g(x)$. In Eqn. (4), the adaptive wight $g(x)$ is
adopted for the Euclidean distance measuring term in the pixel domain. To
verify the effectiveness of $g(x)$, we ablate it from Eqn. (4), leading to the
model denoted as DeepWSD w/o $g(x)$. The comparison results on TID2013 and
KADID-10k datasets are shown in Fig. 6, in which we observe the performance
drop significantly in terms of both SRCC and PLCC. This phenomenon
demonstrates that the adaptive wight involved in fidelity measurement plays an
important role in pursuing perceptual consistency.

WSD Performed in Pixel Domain. As we discussed before, the WSD performed in
deep feature space can capture the pixel-correlation in a quality-aware
manner. To verify such an assumption, we incorporate the WSD in the pixel
domain i.e., we measure the WSD of the reference image and distorted image
patch-by-patch in the RGB space for image quality estimation. The comparison
results on TID2013 and KADID-10k datasets are shown in Fig. 6. We could
observe the superior performance our DeepWSD can achieve, demonstrating that
deep features provide more appropriate feasible space for quality measuring.
Although conducting the WSD directly on the pixel domain is not optimal, we
find it can be complementary to the feature domain WSD. As shown in Fig. 6,
when we ablate the pixel information from the DeepWSD, the results (denoted as
DeepWSD w/o Pixel) show the performance drop in terms of PLCC (0.865 v.s
0.875) on the TID2013 dataset and KRCC (0.708 v.s 0.706) on the KADID-10k
dataset.

Different Network Structures. To explore the performance of WSD on different
feature encoders, we use the same philosophy to build WSD on the structure of
AlexNet (Krizhevsky et al., 2017) and SqueezeNet (Iandola et al., 2017). The
results are shown in TABLE 3. We observe that the VGG16 backbone can achieve
the best performance on different datasets, revealing its effectiveness and
high generalization capability. Our observation is also in line with the
results of LPIPS (Zhang et al., 2018).

##  5\. Conclusions

In this paper, we have developed a new quality measure DeepWSD and shown that
simplifying comparing the deep features from the perspective of the
probability distribution is effective for quality assessment. The proposed
quality measure does not rely on plentiful hyper-parameters and is constructed
with the pre-trained network without tuning with quality labels. Experimental
results show that the proposed method is well correlated with the subjective
evaluations of image quality, based upon the conventional distortions as well
as the distortions that typically occur in real-application scenarios.

The message we are trying to send here is not that the traditional FR-IQA
measure with direct feature comparisons should be abandoned. Rather, it is
highly expected that the design of FR-IQA should also assimilate the idea of
NR-IQA methods, the majority of which are based upon statistical modeling. In
particular, the adoption of the WSD that serves as the metric on probability
distributions enables the use of a rich groundwork of distribution based
methods in FR-IQA.

##  6\. Acknowledgments

This work was supported in part by the Key Project of Science and Technology
Innovation 2030 supported by the Ministry of Science and Technology of China
(Grant No. 2018AAA0101301), the National Natural Science Foundation of China
Grant 61672443, 62022002 and 62176027, in part by Hong Kong GRF - RGC General
Research Fund 9042816 (CityU 11209819) and 9042958 (CityU 11203820). Also,
this work is supported by the Hong Kong Innovation and Technology Commission
(InnoHK Project CIMDA).

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