\section{Introduction}

Two sets $A, B$  of positive integers are called \emph{additive complements} if their sumset $A+B$
contains all but finitely many positive integers. The counting functions of additive complements clearly satisfy
\begin{equation}\label{trivi}
A(x) B(x) \geq x-r, \end{equation}
where $r$ is the number of positive integers not represented as a sum. It is easy to construct sets, separating odd and even
places in a digital representation, for which equality holds for infinitely many values of $x$. These sets have the property that
 \[ \limsup A(x) B(x)/x > 1 . \]
Hanani asked whether this is always the case for infinite additive complements. This was answered by Danzer\cite{danzer64},
who first constructed infinite additive complements such that
\begin{equation}\label{hyp} A(x)B(x)/x \to1. \end{equation}
We shall call such additive complements \emph{exact}. This property is less exotic than it seems; powers of a fixed integer
do have an exact complement, as do all sufficiently thin sets \cite{r96a, r01a}.

Narkiewicz\cite{narkiewicz59} proved an important property of exact complements. He considered a wider class.

\begin{Th}[Narkiewicz's dichotomy]\label{nark}
Let $A,B$ be infitite sets of positive integers such that the number $r(x)$ of integers up to $x$ not contained in
their sumset $A+B$ satisfies $r(x)=o(x)$. Under condition \eq{hyp} we have
\begin{equation}\label{smallbig}
A(2x)/A(x) \to 1, \ B(2x)/B(x)\to 2, \end{equation}
or this holds with the roles of $A,B$ exchanged. If \eq{smallbig} holds, then for $\varepsilon>0$ and $x>x_0(\varepsilon)$ we have
\begin{equation}\label{smallbig2}
  A(x) < x^\varepsilon, \ B(x) > x^{1-\varepsilon} . \end{equation}
\end{Th}

 This shows that
polynomial sequences do not have an exact complement. The set of primes does not have either, for less obvious reasons \cite{r98f}.

For the sequel we will assume that \eq{smallbig} holds, that is, $A$ is small and
$B$ is large.

For exact complements S\'ark\"ozy and Szemer\'edi\cite{ssz94} proved that if \eq{hyp} holds,
then $A(x)B(x)-x \to \infty$. (While this paper actually appeared in 1994, the result was already announced in the 1966
edition of Halberstam and Roth's book Sequences\cite{sequences}.) They remark that their proof shows that
 \[ A(x)B(x)-x = o\big(A(x)\big) \]
cannot hold, and they conjecture that
 \[ A(x)B(x)-x = O\big(A(x)\big) \]
may be possible.

Chen and Fang\cite{chenfang} disproved this conjecture and considerably improved S\'ark\"ozy and Szemer\'edi's bound.
Their result shows that even
\begin{equation}\label{hatvany}
A(x)B(x)-x = O\big(A(x)^c\big) \end{equation}
cannot hold for any constant $c$.

The aim of this paper is to improve Chen and Fang's result and to show by means of an example that there
is precious little room for further improvement.

Write
 \[ a^*(x) = \max \{ a\in A, a \leq x .\}\]

\begin{Th}\label{estimate}
  Let $A, B$ be infinite  sets of positive integers such that the number $r(x)$ of integers up to $x$ not contained in
their sumset $A+B$ satisfies $r(x)=o(x)$. Suppose they satisfy \eq{hyp} and the notation corresponds to
\eq{smallbig}.  If $r(x)= o\bigl( a^*(x) \bigr)  $, then             we have
\begin{equation}\label{also}
    A(x) B(x) -x > \bigl(1-o(1)\bigr) \frac{a^*(x)}{A(x)} . \end{equation}
\end{Th}

The reason that this excludes \eq{hatvany} is that Narkiewicz's dichotomy \eq{smallbig2}   implies that
 \[ A(x) = A\bigl(a^*(x)\bigr) < a^*(x)^\varepsilon \]
hence $a^*(x)$ is larger than any power of $A(x)$. Chen and Fang's result, though stated in quite different terms,
is equivalent to the lower bound
 \[ \frac{2}{3} \sqrt{a^*(x)} .\]
The proof of Theorem \ref{estimate} is based on their argument, with some parts improved.

Clearly the bound in \eq{also} cannot be improved to $a^*(x)$, since for  $x\in A$  we have  $a^*(x)=x$, and this would contradict
\eq{hyp}. However, it is possible that such an improvement holds whenever  $a^*(x)$ is small compared to $x$.
It is also a natural question, also formulated by Chen and Fang, whether one can give an \emph{absolute} lower bound,
say $ A(x) B(x) -x> \log x$. We show this is not the case.

\begin{Th}\label{pelda}
  Let $\omega$ be a function tending to infinity arbitrarily slowly. There are additive complements satisfying \eq{hyp} such that
for infinitely many values of $x$ we have
\begin{equation}\label{felso}
 A(x) B(x) - x < \min \bigl(\omega(x), c a^*(x) \bigr) \end{equation}
with some constant $c$.
\end{Th}

\section{The lower estimate}

\begin{Lemma} \label{deltaszigma}
  Let $U,V$ be finite sets of integers. Put
   \[ \sigma(n) = \#\{ (u, v): u\in U, v\in V, u+v=n \}, \ \delta(n) = \#\{ (u, v): u\in U, v\in V, v-u=n \} . \]
   We have
    \[ \sum_{\sigma(n)>1} (\sigma(n)-1) \geq \frac{1}{|U|}  \sum_{\delta(n)>1} (\delta(n)-1)  . \]
\end{Lemma}

\begin{proof}
  We have
   \[ \sum \sigma(n) = \sum \delta(n) = |U| |V|,\]
\[ \sum \sigma(n)^2 = \sum \delta(n)^2 \]
by double-counting the quadruples satisfying $u+v=u'+v'$, which can be rearranged as $v-u'=v'-u$, and $\sigma(n)\leq |U|$ for all $n$.
Hence
 \[ \sum_{\delta(n)>1} (\delta(n)-1) \leq   \sum \bigl(\delta(n)^2 -\delta(n) \bigr) =  \sum \bigl(\sigma(n)^2 -\sigma(n) \bigr) \leq |U|  \sum_{\sigma(n)>1} (\sigma(n)-1).  \]
\end{proof}
This estimate can be doubled, as $ \delta(n)-1 \leq \bigl(\delta(n)^2 -\delta(n) \bigr)/2$ whenever ${\delta(n)>1}$, but we cannot utilize this improvement.

There are sets $U,V$ for which this estimate is correct up to a constant factor. It is likely that the sets for which we shall apply
this lemma are not of this kind, but I do not see any way to show this.

\begin{Lemma} \label{aatlag}
  Assume that the sets $A,B$ satisfy \eq{hyp} and \eq{smallbig}. Then
\begin{equation}\label{smallbiga}
A(cx)/A(x) \to 1 \end{equation}
uniformly in any range $c_1<c<c_2$ with $0<c_1<c_2$;
\begin{equation}\label{smallbigb}
\ B(cx)/B(x)\to c \end{equation}
uniformly in any range $c<c_2$ with $0<c_2$.
 Furthermore
   \begin{equation}\label{aa}
  \sum_{a\in A, a\leq x} a = o(xA(x)) .  \end{equation}
\end{Lemma}
\begin{proof}
  For $c=2^k$ with a (positive or negative) integer $k$ the claim \eq{smallbiga} follows from an iterated application of
\eq{smallbig}. For general $c$ the claim for $A$ follows from the monotonicity of $A(x)$. For $B$ from \eq{hyp} we get
\eq{smallbigb} for the same range; the range can be extended down to 0 by the monotonicity of $B(x)$.

  To see \eq{aa} note that the sum with $a\leq \varepsilon x$ contributes at most $\varepsilon xA(x)$, and the sum with $a>\varepsilon x$ contributes at most
   \[ x \bigl(A(x)-A(\varepsilon x)\bigr) =o(xA(x))  \]
by \eq{smallbiga}.
\end{proof}

\begin{proof}[Proof of the Theorem.]
  Fix an integer $x$ and put $U=A\cap[1,x]$,  $V=B\cap[1,x]$. We use the notations $\sigma, \delta$ as in Lemma \ref{deltaszigma}.
We have
   \[ A(x)B(x)-x = |U| |V| -x = y + z - r,\]
   where
    \[ y =  \sum_{\sigma(n)>1} (\sigma(n)-1) \]
    counts the excess multiplicities,
     \[ z = \#\{n: n>x, n\in U+V \} \]
     counts the unnecessarily large sums, and $r=r(x)$ is the number of integers not in $A+B$.

     Let $t=a^*(x)$. Adding $t$ to any $b\in B, b>x-t$ we get a sum $>x$, so
      \[ z \geq B(x)-B(x-t) .\]
If $t \geq x/2$, we use only this and \eq{smallbig} with $c=(x-t)/x$ to conclude
 \[ z \geq \left(1- \frac{x-t}{x}  -o(1) \right)  B(x) \sim \frac{t}{x} B(x) \sim  \frac{t}{A(x)}.\]
(This argument works for $t>cx$ with any fixed $c>0$, but fails for very small $t$, which is the typical situation.)

Assume now $t<x/2$. We are going to estimate $y$. Put $V'=B\cap[1, x-t]$. We will consider the sets $V'+U$, $V'-U$, and
use $\sigma', \delta'$ to denote the corresponding representation functions.

We have
 \[ \sum \delta'(n) = |U| |V'| = A(x) B(x-t) .\]
 As $U \subset [1,t]$ and $V'\subset[1, x-t]$, we have $V'-U \subset [1-t, x-t-1]$. We show that few sums lie in $[1-t, t]$.
Indeed, if $b-a \leq t$ with $a\in U, b\in V'$, then $b \leq a+t$, so for an $a\in A$ there are at most $B(a+t)$ possible choices of $b$,
This gives altogether
  \[ \sum_{a\in U} B(a+t) < (1+o(1))  \sum_{a\in U} \frac{a+t}{A(a+t)} \]
  by \eq{hyp}. As $A(a+t)=A(t)=A(x)= |U|$ in this range, the sum is equal to
   \[ t + \frac{1}{|U|}  \sum_{a\in U} a = (1+o(1)) t \]
by Lemma \ref{aatlag}. Hence
  \[ \sum_{a\in U} B(a+t) < (1+\varepsilon) t . \]

This means that at least $ A(x) B(x-t) - (1+\varepsilon) t$ pairs give a difference in the interval $[t+1, x-t-1]$, which contains
less than $x-2t$ integers. Consequently
 \[   \sum_{\delta'n)>1} (\delta'(n)-1)  > \bigl(A(x) B(x-t) - (1+\varepsilon) t \bigr) -(x-2t) =A(x) B(x-t) -x + (1-\varepsilon) t. \]
 We now apply Lemma \ref{deltaszigma} to the sets  $U, V'$ to conclude
  \[   \sum_{\sigma'(n)>1} (\sigma'(n)-1) \geq \frac{1}{|U|} \bigl( A(x) B(x-t) -x + (1-\varepsilon) t \bigr) = B(x-t)- \frac{x-(1-\varepsilon)t}{A(x)} . \]
Clearly $\sigma(n) \geq \sigma'(n)$ for all $n$, so
   \[ y =  \sum_{\sigma(n)>1} (\sigma(n)-1) \geq   \sum_{\sigma'(n)>1} (\sigma'(n)-1). \]
   Adding the estimates we obtain
    \[ A(x)B(x)-x+r=y+z \geq B(x)- \frac{x-(1-\varepsilon)t}{A(x)} = \frac{A(x)B(x)-x}{A(x)} +\frac{(1-\varepsilon)t}{A(x)},\]
which can be rearranged as
\[ A(x)B(x)-x\geq  \frac{(1-\varepsilon)t}{A(x)-1} -  \frac{rA(x)}{A(x)-1}  .  \]

\end{proof}

\section{The construction}

We prove Theorem \ref{pelda}.

Take an increasing sequence $p_1, p_2, \ldots$ of primes such that $k^3 < p_k < (k+1)^3$, possibly with finitely many exceptions.
We shall construct a sequence of integers $u_k$ such that $u_k > ku_{k-1}$, $p_k|u_k$ and finite sets $A_i$ of integers such that
 \[ A_1 = \{ 1, 2, \ldots, p_1\}, \ A_k \subset (u_k, 2u_k)   \text{ for } k \geq 2,   \]
 \[ |A_1| = p_1, \ |A_k| = p_k - p_{k-1} \text{ for } k \geq 2, \]
 hence
  \[ |A_1 \cup A_2 \cup \ldots \cup A_k | = p_k , \]
  and the set $A_1 \cup A_2 \cup \ldots \cup A_k$ is a complete set of residues modulo $p_k$. One of the complements will be
   \[ A = \bigcup_{k=1}^\infty A_k . \]

   To specify the other set we put
    \[ B_k = \{ n: p_k | n, \ ku_k < n < (k+3) u_{k+1}  \} \]
and
 \[ B = \bigcup_{k=1}^\infty B_k . \]

First we prove that such sets  $A_k$
exist, provided the sequence $u_k$ increases sufficiently fast.

\begin{Lemma}
  There are integers $v_k$, depending only on the primes $p_j$, such that sets $A_k$ with the above described properties can
be found whenever $u_k > v_k $ for all $k$.
\end{Lemma}

\begin{proof}
Write
 \[ \delta = \prod_{i=k}^\infty \left(1- \frac{p_k-1}{p_j} \right) \]
 and choose $r$ so that
  \[ \sum_{i=r+1}^\infty \frac{1}{p_i} < \frac{\delta}{4p_k} . \]
The positivity of $\delta$ and the existence of $r$ follows from the convergence of the series $\sum 1/p_i$.
Write $q=p_k p_{k+1} \ldots p_r$. We show that  suitable sets can be found if $u_k > v_k = 2q/\delta$.

We will construct the sets $A_k$ recursively. Given $A_1, \ldots, A_{k-1}$, a necessary condition for the existence of $A_k$
is that the elements of  $A_1 \cup A_2 \cup \ldots \cup A_{k-1}$ be all incongruent modulo $p_k$. Hence the property which we shall preserve
during the induction is:

``the elements of  $A_1 \cup A_2 \cup \ldots \cup A_k$ are all incongruent modulo $p_j$ for every $j \geq k$.''
We assume this holds for $k-1$ and we build $A_k = \{ a_1, a_2, \ldots, a_{ p_k - p_{k-1}}  \} $.

 Suppose
$a_1, \ldots, a_{t-1}$ are already found. We want to find $a_t$ so that $m=p_k - p_{k-1} +t-1$ residue classes are
forbidden for each  $p_j$, $j \geq k$. In each interval of length $q$ there are
 \[ q  \prod_{i=k}^r \left(1- \frac{m}{p_j} \right)  > \delta q \]
integers which avoid the $m$ forbidden residue classes modulo all  $p_j$, $k \leq j \leq r$. In the interval $(u_k, 2u_k)$
this means at least $\delta u_k-q$ candidates.

Next we count the numbers in forbidden residue classes modulo  $p_j$, $j>r$.
The number of integers in a residue class $ a \pmod p$ in the interval $(u_k, 2u_k)$ is exactly
 \[ \left[ \frac{2u_k-a-1}{p}\right] - \left[ \frac{u_k-a}{p}\right] \leq \frac{2u_k}{p}, \]
 assuming that $p<2u_k$. We use this estimate for $p_j<2u_k$. This excludes less than
  \[ p_k  \sum_{i=r+1}^\infty \frac{2u_k}{p_i} < (\delta/2) u_k \]
integers.

 Finally, if  $p_j>2u_k$, then there are no new excluded integers. Indeed, the only integer satisfying
$ n \kong{a}{p_j}$ with some $a\in A_1 \cup A_2 \cup \ldots \cup A_{k-1} \cup \{ a_1, \ldots, a_{t-1} \} $ is $a$ itself, which was already excluded
(even several times) by previous congruences.

This leaves us  at least $(\delta/2) u_k-q$ integers to choose from, which is positive if $u_k > 2q/\delta$.
\end{proof}

Now we show that $A, B$ are additive complements, then estimate $A(x)B(x)-x$.

To prove the first claim, take an arbitrary $n>3u_1$. It satisfies
 \[ (k+2)u_k < n \leq (k+3) u_{k+1} \]
 with some $k$. Select $a\in A$ so that
  \[ a\in A_1 \cup A_2 \cup \ldots \cup A_k, \ a \kong{n}{p_k} .\]
As $1 \leq a < 2u_k$, the integer $b=n-a$ satisfies $ku_k < b < (k+3) u_{k+1}$ and $p_k|b$, so $b\in B_k$.

Now we estimate $B(x)$ for a typical $x$. This number satisfies $ku_k<x \leq (k+1) u_{k+1}$ for some $k$. All blocks
$B_j, j>k$ lie above $x$. An initial segment of $B_k$ gives
 \[ B_k(x) \leq \frac{x-ku_k}{p_k} \]
 elements. To estimate the contribution of smaller blocks note that
  \[ |B_j| \leq \frac{(j+3) u_{j+1} - ju_j} {p_j},   \]
  hence
  \[ B(x) \leq  B_k(x) + |B_{k-1}| + |B_{k-2}| + \ldots + |B_1| \]
   \[ \leq \frac{x}{p_k} + \sum_{j=2}^k \left( \frac{j+2}{p_{j-1}} - \frac{j}{p_j} \right) u_j .\]
This estimate is not quite exact, since possibly only a segment of $B_{k-1}$ is contained in our interval, and the sets
$B_j$ are not disjoint; takint these into account would not substantially improve our result.

By our assumption about the rate of growth of the sequence $p_j$ the coefficient of $u_j$ in the above formula is $O(j^{-3})$,
that is,
 \[ B(x)  \leq \frac{x}{p_k} + c_1 \sum_{j=2}^k \frac{u_j}{j^3} < \frac{x}{p_k} + c_2 \frac{u_k}{k^3} \]
by our assumption about the rate of growth of the sequence $u_j$.

Since $A_{k+2}$ consists  already of elements $>u_{k+2} > (k+1) u_{k+1}$, we have $A(x) \leq p_{k+1}$, consequently
 \[ A(x) B(x) - x < \frac{p_{k+1}-p_k}{p_k} x +  c_2 \frac{u_kp_{k+1}}{k^3} = O(x/k) = o(x), \]
which shows that these sets are indeed exact complements.

For $x=u_{k+1}$ we have $A(x)=p_k$ and $u_k<a^*(x) < 2u_k$, so
\[ A(x) B(x) - x < c_2 \frac{u_kp_{k}}{k^3} < c_3 u_k < c_3a^*(x) , \]
and also
 \[  c_3 u_k < \omega(x) , \]
provided the sequence $u_j$ grows so fast that $\omega(u_{k+1}) > u_k$.
These estimates show the bound \eq{felso}.

\section{Concluding remark}

All known constructions of exact complements use a variant of this approach, namely combining a complete set of residues
modulo some integers $p_k$ (primes here, other sorts of integers in other papers, depending on the situation) and
multiples of these $p_k$ in an interval. The difficulty is that multiples of $p_k$ are needed for a time after the
appearance of the firts few multiples of $p_{k+1}$, which creates multiply represented sums. I see no way to eliminate or reduce
this effect, nor a way to improve the lower estimate which would then vindicate this overkill.

   \bibliographystyle{amsplain}


\section{\label{sec:intro}Introduction}
The need for a cosmological constant, or an equivalent effect, was revived when stronger than expected fainting in type Ia supernovae, used as standard candles, was detected \cite{Perlmutter:1997zf,Riess:1998cb,Perlmutter:1998np}. This was interpreted as cosmic acceleration and confirmed later from cosmic microwave background radiation, clusters and baryon acoustic oscillation measurements \cite{Hinshaw:2012aka,Bennett:2012zja,Aghanim:2012vda,Ade:2013sjv}.

The first common interpretation of such constant portraits it as quantum vacuum energy. However this raises more problems, such as that from naive evaluations of the full quantum vacuum energy, i.e. of order the Planck scale \cite{Martin}, compared with the $\Lambda$ value extracted from observed acceleration, offers one of the largest discrepancy in physics, coined the fine tuning problem \citep{Weinberg:1988cp,Weinberg:2008zzc}.

The fine tuning problem proceeds from the dominance at early time of the Planck scale on the quantum vacuum scale. It should have therefore given initial values for all running constants, including the cosmological constant. Since $\Lambda$ is constant, its value is naively expected to be set from the energy conditions of the Planck era, instead of the hundred order of magnitude smaller observed value from cosmic dynamics. Moreover, as the cosmological constant is non-varying, it also entails the coincidence problem \citep{Amendola:1999er,TocchiniValentini:2001ty,Zimdahl:2001ar,Zimdahl:2002zb}.

That extra problem stems from the apparently unnatural proximity between the present epoch and the moment of emergence of the cosmological constant as dominant in the cosmic energy density balance \cite{Frieman:2008sn,Li:2012dt}. Such surprising coincidence between values of $\Lambda$, expected to come from the Universe evolution's initial conditions, and today's Universe matter energy density content  would require at least an explanation.

The standard understanding of the cosmological constant is review, e.g. in \cite{Martin}.

General relativity's (GR) curved spacetime can be generalised, to include torsion on the manifold, in Einstein-Cartan theories \cite{Dona,Giulini,nakahara,hatcher}. Their two limiting cases for equivalent degrees of freedom allow to recover
\begin{enumerate}
    \item the torsion-less GR limit and
    \item the curvature-less Teleparallel Equivalent to GR case \cite{Deandrade,Baez,Arcos}.
\end{enumerate}
This work focusses on the GR limit of Einstein-Cartan theories.

The Gravitational action contains further freedom in the form of boundary terms \cite{Baekler,Dyer} and manifold topology \cite{donaldson,Sengupta,Kaul}. We restrain here the latter with an action, built on the Einstein-Hilbert (EH) and Vielbein-Einstein-Palatini (VEP) equivalence \cite{Dadhich}, that adds to GR's well known features, the spacetime topology-consistent Chern-type topological invariants, as well as the cobordism.

Such approach remarkably gives rise to a topologically related term which, under certain conditions, acts as an \textit{effective cosmological constant}.

This very general action allows this paper to argue that the cosmological constant can have a topological origin rather than being generated at the universe's formation from quantum vacuum considerations. The restriction of such action, naturally featuring torsion and an
\textit{effective cosmological constant} related to topology, to the usual GR condition can be effected by dynamical systems stability considerations. Since the resulting \textit{effective cosmological constant} connects to the topology of the Universe via a value we argue to be proportional to the finite inverse average black hole volume, and no longer to the large Planck density, it renders moot the fine tuning problem. A preliminary discussion of these results can be found in \cite{LeDelliou:2019mus}. This paper presents a detailed technical account of this work.

Topology as a source of cosmology has been considered in the works of \cite{Alexander1} and \cite{Alexander2}.In those papers, the topological terms are coupled trivially over closed manifolds, contrary to our approach. This allows for the classical field equations to remain unchanged, only affecting the quantum equations. This is obtained using the Ashtekar variables formalism, while we keep our work at the classical level and use Einstein-Cartan formalism.

We begin by introducing the topological invariants articulation with the Vielbein-Einstein-Palatini action in Sec.~\ref{sec:model}. We introduce the supplementary dynamics induced by these invariants in the action, which result, in Sec. \ref{sec:cosmtop}, in the effective cosmological constant and, as a byproduct and on a practical ground, in the recovery of GR. Section \ref{sec:TCCeval} presents how this topological constant value can be estimated for our Universe. Finally, we discuss and conclude our findings in Sec.~\ref{sec:Conclusion}.

\section{\label{sec:model}The model}

In this section we briefly outline the mechanism in which the torsion-less classical gravitational action inherits torsional degrees of freedom when topological terms are added. By topological terms we mean the so called characteristic classes well defined for any $4n$-manifold, in this $n=1$. This procedure will suggest the natural upgrade of the cosmological constant term in the \textit{Einsteinian} tradition of gravitational theories to a functional with degrees of freedom related to topological information of the space-time manifold. Appendix \ref{sec:cartan} provides a brief account of the formalism and overall notation.

In \ref{subsec:comact} is demonstrated that, in the case in which the space-time manifold $\mathcal{M}$ has a finite volume $\text{Vol} \left( \mathcal{M} \right)$ and a boundary $\partial \mathcal{M}$, the cosmological constant $\Lambda$ is functionally related to the Dirichlet to Neumann operator $\Upsilon \left[ \chi \right]$ (DN operator). Recall that this operator maps the Dirichlet boundary conditions to those of Neumann for any well posed boundary value problem\footnote{Informally, a physical observable $\mathcal{O}$, associated to differential form $\hat{\mathcal{O}} \in \Omega \left( \mathcal{M} \right)$, can be uniquely defined as a Dirichlet problem, i.e. by knowing its value over space-time
boundary $\hat{\mathcal{O}} |_{ \partial \mathcal{M}}$, or as a Neumann problem, i.e. by knowing the value of its normal derivatives $L_n \hat{\mathcal{O}} |_{ \partial \mathcal{M}}$ over the space-time boundary, with $n$ the outward unit vector over the boundary $\partial \mathcal{M}$. The Dirichlet to Neumann operator allows to move freely between the two types of problems.}. The exact expression is simply:
\begin{align*}
\Lambda = \kappa \Upsilon \left[ \chi \right] = \frac{1}{ \text{Vol} \left( \mathcal{M} \right)} \int\limits_{\partial \mathcal{M}} \chi \;\;,
\end{align*}
which is valid for $\chi$, a $3$-form satisfying the well posed boundary value problem appearing in (\ref{DNop}) and Proposition \ref{DNoperator}, with $\kappa = 8 \pi G c^{-4}$ being the gravitation coupling constant. This result strikes as being of a mostly topological nature. Moreover, if the kernel of the DN operator $\Upsilon \left[ \chi \right]$ is known, lower bounds for the so called Betti numbers (see Eq.~\ref{betti}) can be obtained, which we can intuitively understand as counting the $n$-dimensional punctures of a manifold. Hence, the topology of the manifold $\mathcal{M}$ is restricted.

This observation enlighten us is two ways:

\begin{enumerate}[i)]
\item The cosmological constant, for the conditions presented, is related to topological rather than vacuum energy considerations. Furthermore, it is inversely proportional to $\text{Vol} \left( \mathcal{M} \right)$ which naturally favors small values for $\Lambda$, opening the possibility to alleviate or even solve the fine tuning problem.

\item  The cosmological constant can be understood as some type of functional of certain fields rather than as a fixed parameter of the theory.
\end{enumerate}

Hence, inspired by the above observations we upgrade the cosmological constant $\Lambda$ to a functional $\tilde{\lambda}$. We expect this new quantity to also be characterized by topological degrees of freedom such that this new functional defines a well posed variational problem when the field equations are tackled.

The model is completely characterized in Appendix \ref{sec:The model} but here are discussed its most important features. Let the index set $J = \left\{ N, P , E \right\} $ with $j \in J$ be the collection of subscripts related to the Nieh-Yan, Pontryagin and Euler characteristic classes, respectively. We focus on the family of actions having the following three distinctive sectors:
\begin{align}\label{Bodyaction}
S & := S_{G} \left[e^c, \omega^a_{\;\;b} , \tilde{\lambda} \right] + S_{T} \left[e^c, \omega^a_{\;\;b}, \varphi_j \right] +  S_M \left[ \Psi \right] \; ,
\end{align}
where the first term is the pure gravitational action (see Eq.~\ref{EHactionmod}), $e^c$ are the tetrad, $\omega^a_{\;\;b}$ is the total connection and $\tilde{\lambda}$ is the newly upgraded \textit{cosmological functional}. The second term is the \textit{topological action}, named after the fact that it explicitly includes the three characteristic classes $C_j$ $(j \in J)$ for the characteristic classes:
\begin{align}
\label{Pontryagin1} \small C_{P}  & = \frac{1}{8 \pi^2} R^a_{\;\;b} \wedge R^b_{\;\;a} \;\;,\\
\label{Euler1} \small C_{E} & = \frac{1}{8 \pi^2} R^{ab} \wedge R^{\left( * \right)}_{ba} \;\;, \\
\label{NiehYan1}\small C_N & = T^a \wedge T_a - R_{ab} \wedge \Sigma^{ab} \;\;,
\end{align}
(See also \ref{TopManifold}) which are consistent with the space-time manifold $\mathcal{M}$. Each characteristic class is coupled to its corresponding $\varphi_j$ zero-forms (see Eq.~ \ref{Stop}) (functions) having dimensions of $ \left[ \texttt{length}^{2} \right]$ for $j=E,P$ and being dimension-less for $j=N$.

The last term in (\ref{Bodyaction}) stands for an explicit matter action (see Eq.~\ref{SMat}) of the Dirac type, where $\Psi:=\left\{ \bar{\psi}, \psi \right\}$ is a collected term for the spinor fields. As usual, a matrix representation $\gamma^a$ of the Clifford algebra $\left\{ \gamma^a , \gamma^b \right\} = 2 \eta^{ab} $ ( with $\eta^{ab}$ the Minkowskian metric) is considered, such that $\bar{\psi }:= \psi ^{\dagger} \gamma^{0}$ and $\psi^{\dagger}$ is the conjugate transpose of $\psi$. By specifying the cosmological functional to behave as the Yukawa-type of interaction
\begin{align}\label{Yukawa}
\tilde{\lambda} := \bar{\psi} \lambda \left[ \varphi_j \right] \psi \;\; ,
\end{align}
we are in position to obtain the variation of Eq.~(\ref{Bodyaction}). Notice that $\lambda$ now concentrates all the information and becomes the only source of interaction between the spinor fields and the coupling zero-forms. The field equations (for $\delta e^a$, $\delta \omega_{ab}$, $\delta \varphi_j$, $\delta \bar{\psi}$ and $\delta \psi$ , respectively)\footnote{Originally the field equations suppose the presence of a fermion potential $V \left( \Psi \right)$. However, since this potential is not involved in the final calculation, we prefer to omit it out of clarity.} read:
\begin{align}
\label{fe1} & 0 = \left[ R^{\left( *\right)}_{ab} - \frac{2 \lambda}{3} \left| \psi \right|^2_{\psi} \hat{\Sigma}^{(*)}_{ab} \right] \wedge e^{b} + i d \varphi_{N} \wedge T_{a} + \Xi_a  \;\; , \\
\nonumber & 0 = d_{\omega} \hat{\Sigma}^{(*)}_{ab} + i 2 d \hat{\varphi}_P \wedge R_{ab} - i 2 d \hat{\varphi}_E \wedge R^{\left( * \right)}_{ab}  + \\
\label{fe2} & \quad \quad \quad \quad \quad - i d \varphi_{N} \wedge \hat{\Sigma}^{\flat}_{ab} - \tau_{ab} \;, \\
\label{fe3} & 0  = - i \frac{\delta \lambda}{ \delta \varphi_j } \left| \psi \right|^2_{\psi} d \mu - C_j \;\;\;\;\;\; , \;\;\;\;\;\; j = \left\{ E, P, N \right\} \;\; , \\
\label{fe4} & \; 0 = \gamma^a D_{\omega} \psi - \frac{\lambda}{4} \psi \, e^a \;\;\;\;\; , \;\;\;\; 0 = D_{\omega} \bar{\psi} \gamma^a + \frac{\lambda}{4} \bar{\psi} \, e^a \;\;\; ,
\end{align}
where we have defined $\left| \psi \right|^2_{\psi} := \bar{\psi} \psi$ and the normalized quantity $\hat{\varphi}_j := \frac{\varphi_j}{\left( 4 \pi \right)^2}$ for brevity, $d_{\omega}$ stands for the exterior covariant derivative, $D_{\omega}$ stands for the gauge covariant derivative (see definition in Eq.~\ref{gaugecovder}) and we have defined the quantities:
\begin{align}
\label{fevarious} & \Xi_{a} := \frac{ \delta \mathfrak{L}_{M}}{\delta e^a} = 2 \mathfrak{Re} \left\{ \bar{\psi} \gamma^{b} D_{\omega} \psi \right\}\wedge \hat{\Sigma}^{(*)}_{ab} \;\;
; \;\; i := \sqrt{-1} \;\;\; ; \;\;\;  \\
\nonumber & \tau_{ab} := \frac{\delta \mathfrak{L}_M}{\delta \omega^{ab} } = \frac{1}{4} \bar{\psi} \sigma_{abc} \psi \star e^c \; ; \; \sigma^{a_1 \cdots a_k}:= \gamma^{\left[ a_1 \right.} \cdots \gamma^{\left. a_k \right]} \, .
\end{align}

Notice the non-GR terms in the field Eqs.~(\ref{fe1}) and (\ref{fe2}), i.e. the ones that are multiplied by the coupling zero-forms and/or its exterior derivatives. These are the terms that ultimately differentiate the current theory with that of the strict GR.

Being so, we can interpret the cosmological functional $\tilde{\lambda}$ as the source of the topology of the space-time manifold $\mathcal{M}$. As discussed in Appendix \ref{TopManifold}, a direct consequence of this modelling is that it ensures the space-time manifold $\mathcal{M}$ to be oriented, simply connected and smooth.

In this way, we show that the couplings zero forms $\varphi_j$ ($j \in J$) can actually be naturally interpreted as scalar matter fields. This becomes clear once the total $2$-form curvature $R_{ab}$ (see Eq.~\ref{Riemanntensorgral}) is obtained, its torsion related part will have terms directly dependant on $\varphi_j$ with a clear dynamical interpretation.

\section{\label{sec:cosmtop}Obtaining a cosmological constant from topology}

A thorough exposition of the detailed calculations is presented in Appendix \ref{sec:sol}. We outline here the general procedure used to solve the field equations. We stress that, given the strong non-linearities of the field equations, we opted for an approach mixing formal and heuristic methodologies in which some physical restrictions have been made in order to obtain closed analytical solutions.

\subsection{\label{subsec:nonlinear} Managing the non-linearities of the canonical field equations}

Basically, due to the form of the topological action, we are allowing for the presence of torsion in the field equations, so the VEP first order formalism becomes ideal to treat this feature. We recall that the total connection $1$-form $\omega^{a}_{\;\;b}$ can always be written as:
\begin{align*}
\omega^{a}_{\;\;b} & = \bar \omega^{a}_{\;\;b} + K^{a}_{\;\;b}  & \in \Omega^1 \left( \mathcal{M} \right) \; ,
\end{align*}
where $\bar \omega^{a}_{\;\;b}$ is the torsion-less Levi-Civita connection, while $ K^{a}_{\;\;b}$ is the torsion related contortion $1$-form. The latter becomes the only responsible for the torsion $2$-form as:
\begin{align*}
T^a & = K^{a}_{\;\;b} \wedge e^b & \in \Omega^2 \left( \mathcal{M} \right) \; .
\end{align*}

By means of the same connection decomposition, the total curvature $2$-form can then be written as (see Appendix \ref{subsec:VEPformalism}, Eq.~(\ref{torcurv}) for details):
\begin{align}
R^a_{\;\;b} & =  \bar{R}^a_{\;\;b} +  \Theta^a_{\;\;b}  & \in \Omega^2 \left( \mathcal{M} \right) \;\;,\label{RiemanntensorgralB}
\end{align}
where $\bar{R}^a_{\;\;b}$ is the torsion-less part of the curvature, formally equivalent to that of a GR curvature $2$-form, while the term $\Theta^a_{\;\;b}$ concentrates all the contributions from the torsion related quantities by means of the contortion $K^{a}_{\;\;b}$ as:
\begin{align}\label{torcurvB}
\Theta^a_{\;\;b} := d_{\omega} K^a_{\;\;b} - K^a_{\;\;c} \wedge K^{c}_{\;\;b} \quad \in \Omega^2 \left( \mathcal{M} \right) \;\;.
\end{align}

Focusing back on the first two field Eqs.~(\ref{fe1}) and (\ref{fe2}), it can be seen that a GR-like behavior is obtained back if the coupling zero-forms $\varphi_j$ ($j \in J$) become slowly varying functions within the torsion-less regions of space-time. The possibility of this behavior for the coupling zero-forms is related to the existence of stability regions (see Appendix \ref{sec:borel} and the references therein) for a turn around in the dynamical system defined by the field Eqs.~(\ref{fe1} - \ref{fe4}). We can therefore expect to relate the vanishing of the torsion with the derivatives of such a slow varying function. This can be equivalently thought of as a contortion which is at least proportional to the derivatives of the coupling zero-forms $\varphi_j$ ($j \in J$). Consequently, this observation plus the natural decomposition of the curvature $2$-form $R_{ab}$ (\ref{RiemanntensorgralB}), suggests a contortion $1$-form:
\begin{align}\label{contortionbody}
K_{ab} = 4i \star \left( d \varphi_N \wedge \hat{\Sigma}^{\flat}_{ab} \right) \;\; \in \;\; \Omega^1 \left( \mathcal{M} \right) \;\;\;.
\end{align}
This contortion cancels the undesired non-linear terms appearing in the field Eqs.~(\ref{fe1}) and (\ref{fe2}) via the torsion related part of the curvature $\Theta_{ab}$ (See Eq.~(\ref{torcurvB})).

It is through $\Theta_{ab}$ also that a term proportional to the inner product $\left| d \varphi_N \right|^2$ appears in the field equations. When interpreted as a kinetic term, the coupling zero-form $\varphi_N$ inherits a matter field status. It is then just a matter of rearranging the action  (\ref{Bodyaction}), assuming the correctness of the contortion (\ref{contortionbody}), so that the coupling zero-forms can be gathered in a canonical scalar matter field action, justifying our claim on the interpretation of them as matter fields. Moreover, after some algebra and with the help of Eqs.~(\ref{fe4}) to (\ref{fe1}), we obtain the following expression for the torsion-less curvature $2$-form:
\begin{align*}
& \bar{R}_{ab} = \frac{ 2 \left( \alpha_{\psi} \lambda \left[ \varphi_j \right] + \left| d \varphi_N \right|^2 \right)}{3} \hat{\Sigma}^{\flat}_{ab} \;\; ; \;\; \alpha_{\psi} := \frac{\left| \psi \right|^2_{\psi}}{4} \in \mathbb{R} \;\; .
\end{align*}
which is formally equivalent to the analogous field equation obtained in a GR-like theory. The last expression has the right behavior in the limiting case where torsion is turned off, as expected.

Tackling the second field equation, we see that (\ref{fe1}) only contains one of the coupling zero form, i.e. $\varphi_N$, while in comparison (\ref{fe2}) had all three $\varphi_j$ ($j \in J$). Thus, in order for the expression $\Theta_{ab}$, depending only on the coupling zero-form $\varphi_N$, to be sufficient to correct the non-GR terms in (\ref{fe2}) we should expect some dependency between the coupling zero-forms. Moreover, when inserting the calculated total curvature $R_{ab}$ and its Lie dual $R^{(*)}_{ab}$ (See expression \ref{curvaturetotal}) into the second field equation (\ref{fe2}), we obtain:
\begin{align*}
& \tau_{ab} = d_{\omega} \hat{\Sigma}^{(*)}_{ab} - \frac{4i}{3} \left[ \alpha_{\psi} \lambda d \hat{\varphi}_E + \left( i 2 \Delta \varphi_N \right) d \hat{\varphi}_P \right] \wedge \hat{\Sigma}^{(*)}_{ab} + \\
& + \frac{4 i}{3} \left[ \alpha_{\psi} \lambda d \hat{\varphi}_P - \left( i 2 \Delta \varphi_N \right) d \hat{\varphi}_E - \frac{3}{4} d \varphi_{N} \right] \wedge \hat{\Sigma}^{\flat}_{ab} + \\
& + 2 d \varphi_N  \wedge \left[ \left( d \hat{\varphi}_E \wedge K_{ab} \right) + \left( d \hat{\varphi}_P \wedge K_{ab} \right)^{(*)} \right] \;\; ,
\end{align*}
for which we naturally define the quadrature expressions:
\begin{align*}
d \hat{\varphi}_E & = \frac{3}{4} \frac{ i 2 \Delta \varphi_N }{ \left( \alpha_{\psi} \lambda \right)^2 + \left( 2 \Delta \varphi_N \right)^2} d \varphi_N \;\; , \\
d \hat{\varphi}_P & = \frac{3}{4} \frac{ \alpha_{\psi} \lambda }{ \left( \alpha_{\psi} \lambda \right)^2 + \left( 2 \Delta \varphi_N \right)^2 } d \varphi_N \;\; .
\end{align*}
We are effectively writing $\varphi_E$ and $\varphi_P$ in terms of $\varphi_N$. These relations restore the original Eq.~(\ref{fe2}) into the usual generalized spin equation for the analogous GR-like theories in the VEP formalism. The entire process then is summed up in the transformation $\lambda \left[ \varphi_j \right] \rightarrow \lambda \left[ \varphi_N \right]$.

\subsection{\label{subsec:topocalc} Tackling the topological field equations}

At this point, we are in position of calculating the characteristic classes either directly from (\ref{fe3}) or from the set (\ref{Pontryagin} - \ref{NiehYan}). We must ensure that these two approaches are compatible and, at the same time, we will shed some light on the characterization of the functional $\lambda\left[ \varphi_N \right]$.

For the Nieh-Yan characteristic class $C_N$, those two approaches are immediately compatible if the expression
\begin{align}\label{lambda1}
\lambda \left[ \varphi_N \right] = - \frac{\left| d \varphi_N \right|^2}{\left| \psi \right|^2_{\psi}} + \frac{4 \Lambda}{\left| \psi \right|^2_{\psi}}
\end{align}
is satisfied, where the second term is consistent with our assumption that the functional is constant when torsion is turned off. Hence, we can similarly calculate directly the Euler characteristic class $C_E$, also completely consistent but not necessarily illustrative for our purposes, as well as the Pontryagin characteristic class $C_P$, which needs the following restriction for compatibility:
\begin{align}\label{Dalembertian}
3 \left( \alpha_{\psi} \lambda \right)^2 = \left( 2 \Delta \varphi_N \right)^2 \; \Rightarrow \; \Delta \varphi_N = \pm \frac{\sqrt{3}}{2} \alpha_{\psi} \lambda \;\;.
\end{align}
When combined with Eq.~(\ref{lambda1}) characteristic class, Eq.~(\ref{Dalembertian}) yields the second characterization for the functional $\lambda \left[ \varphi_N \right]$ as:
\begin{align}\label{lambda2}
\frac{\delta \ln \left| \lambda \right| }{\delta \varphi_N} = \frac{\sqrt{3}}{4} \;\; \Rightarrow \;\; \lambda = \frac{4 \Lambda}{\left| \psi \right|^2_{\psi}} \exp \left( \pm \frac{\sqrt{3}}{4} \varphi_N \right) \; ,
\end{align}
where the amplitude of the exponential function has been normalised coherently with (\ref{lambda1}). If we postulate dynamical stability, then the kinetic term associated to the zero-form $\varphi_N$ should behave as a Lyapunov function and we obtain the following expression:
\begin{align*}
\left| \frac{d \varphi_N}{4}  \right|^2 & = \Lambda \left\{ 1 - \exp \left( - \frac{\sqrt{3} }{4} \left| \varphi_N  \right|  \right) \right\} \;\; ,
\end{align*}
which can be used to characterize the GR-like regions of the space-time manifold $\mathcal{M}$, via the condition $d \varphi_N = 0$, as equivalent to $\varphi_N = 0$. We have identified this region as the submanifold $\mathcal{N} \subseteq \mathcal{M}$.

The exact form for the coupling zero-form $\varphi_N$ must be a result of solving the boundary value problem stated in Proposition (\ref{boundaryvalue}), which is out of the scope of this paper.

\subsection{\label{subsec:estimation} The cosmological constant.}

Calculating now the topological numbers, we obtain:

\begin{enumerate}[i)]
\item the Nieh-Yan number $ n_N = 0 $, from Eq.~(\ref{NYdirect}). This might be considered as the model's partial answer to the absence of
global manifestation of a physical quantity that accounts for presence of torsion in the observation. In other words, the model gives no hint of torsion, neither locally, in the GR-like regions, nor globally, since $n_Y$ is exactly null.

\item the Pontryagin number $n_P = 0$, from Eq.~(\ref{CP12}). Physically, this prevents the space-time manifold $\mathcal{M}$ to have an orientation reversing diffeomorphism \cite{donaldson}. However this also prevents the CPT type of symmetry expected from the quantum substructure. At the same time, having a non-zero Pontryagin number means that $\mathcal{M}$ cannot be the boundary of an oriented compact higher dimensional manifold \cite{donaldson}.

\item the Euler number:
\begin{align}\label{nEbody}
 n_E = - \frac{16 \Lambda^2}{3 \left( 4 \pi \right)^2} \left\| u^2_{\hat{\varphi}_N} \right\|^2_{L^2} \; ; \; u_{\hat{\varphi}_N} = \exp \left( - \frac{\sqrt{3}}{4} \left| \varphi_N \right| \right) \; ,
\end{align}
from Eq.~(\ref{CE12}), which is finite by topology since by construction $n_E \in \mathbb{Z}$ (See Theorem \ref{topnumber}). Considering that $n_E$ can be written as (see Sec.~\ref{subsec:characteristicclasses}):
\begin{align}\label{nEbody1}
n_E = - 2 k^2_E b_3 \;\;\; ,
\end{align}
where we have defined
\begin{align}\label{kEB}
k^2_E := 1 - \frac{1 + b}{ b_3}  \lesssim 1 \quad ;
\end{align}
here $2b$ is the number of $2$-\textit{punctures} and $b_3$ is the number of $3$-\textit{punctures} in $\mathcal{M}$, that is the number of $2$- and $3$- dimensional holes in  $\mathcal{M}$. Since we have not observed $2$-dimensional topological defects but black holes, which induce surface holes in space slices, i.e. $3$-dimensional holes in space-time, are pervasive in observation, we claim $b_3$ should dominate for any sensible space-time manifold $\mathcal{M}$, and interpret it to be essentially the number of black holes in the Universe.
\end{enumerate}

We thus obtain from  Eq.~(\ref{nEbody}) the exact result:
\begin{align}\label{lambdaexactbody}
\Lambda^2 = \frac{ 3 \left( 4 \pi \right)^2 k^2_E b_3 }{8 \left\| u^2_{\hat{\varphi}_N} \right\|^2_{L^2}} \quad ,
\end{align}
for which we can give the estimate (See also the discussion in Sec. \ref{subsec:characteristicclasses}):
\begin{align}\label{uestimatebody}
0 \leq \left\| u^2_{\hat{\varphi}_N} \right\|^2_{L^2} \simeq \frac{\text{Vol} \left( \partial \mathcal{N} \right)}{\mathcal{C}^2} < \infty
\end{align}
where $\mathcal{C}^2$ is the Cheeger or isoperimetric constant \cite{Benson} with dimensions of $\left[ \texttt{length} \right]$. It has been calculated in certain specific cases. Finally, Eq.~(\ref{uestimate}), combined with (\ref{lambdaexactbody}), (\ref{manifoldregions}) and (\ref{eq:Topolambda1}) we have the topological cosmological constant estimate:
\begin{align}\label{eq:Topolambdabody}
 \Lambda \approx  \frac{ 2 \pi \, k_E \, \mathcal{C} }{ \left\langle \frac{2}{3} \text{Vol} \left( \partial \mathcal{M} \right) \right\rangle^{\frac{1}{2}} }  \;\; .
\end{align}

\section{\label{sec:TCCeval} Topological cosmological constant evaluation}

The resulting expression (\ref{eq:Topolambdabody}) for the topological value of the cosmological constant can be then confronted with its observed value.
To evaluate it, we need to consider the three key components:
\begin{enumerate}
    \item From (\ref{nEbody1}), we have that $k_E=\sqrt{-\frac{n_E}{2b_3}}$, the ratio of the Euler number to what we have interpreted as the number of black holes (BHs) in the manifold ($b_3=\mathcal{N}_{BH}$). Since we argue that for any sensible space-time, $b_3$ should dominate the other terms, we take $k_E\approx 1$,
    \item $\mathcal{C}$, the isoperimetric or Cheeger constant,  measures the ratio between the $3$-surface of boundary and the $4$-volume of the bulk of the spacetime considered, which is generally unknown but has been evaluated in some cases \cite{ASENS, Hoffman} to be of order $\sim 10^1$,
    \item $\langle \text{Vol} \left( \partial \mathcal{M} \right) \rangle$, the average volume of the boundary of the space-time.
\end{enumerate}
We shall evaluate it in what follows.This detailed evaluation is presented in Appendix \ref{sec:BHsurf}.

\subsection{Average boundary volume}

From the assumption on $b_3$, the evaluation of the space-time boundary follows from assuming
\begin{enumerate}
    \item the boundary is made of BHs,
    \item the hypersurface of the boundary is the sum of all their outer horizons. These are approximated by their Schwarzschild horizons, neglecting Kerr horizons deformations and merger history variations,
    \item the distribution of BH in the Universe is well approximated by the observed distribution (on our past lightcone),
    \item the resulting BH average volume can be approximated by the volume of the average BH. Given that the available distribution concerns BH masses, we used the volume of the average BH mass rather than an average volume which distribution we have no access at all.
\end{enumerate}
We evaluate the average BH volume by first getting the volume for a given mass, then using knowledge of the Universe's BH distribution to evaluate the average BH mass. The resulting volume of the average BH is then taken as BH average volume.

\subsubsection{Volume of a given BH}

The total volume of the boundary of a BH of given mass $M$ until its evaporation is evaluated considering it appears at creation with initial mass $M$ and evaporates through Hawking radiation \cite[e.g.][]{Wald:1984rg,Carroll:2004st}. The formation phase is neglected since its dynamical time is expected to be considerably much less than the formation time, and a simple isolated Schwarzschild BH is considered. The resulting volume read as
\begin{align}
     \text{Vol}_{BH} \left( M \right)=&1.96\times 10^{87}\left(\frac{M}{M_\odot}\right)^5 m^3.\label{eq:BHBounVol}
\end{align}

\subsubsection{Evaluation of the average BH mass}

We are limited by our present knowledge of the BH mass distribution. We chose to evaluate it using Refs~\cite{Kovetz:2016kpi,Mutlu-Pakdil2016,Garcia-Bellido:2017fdg,Christian:2018mjv}. We used those references to evaluate averages and variances for the masses of Stellar made, Primordial, Intermediate mass and Super Massive BHs (SMBHs). Despite being aware of the limits on knowledge about the BH population, we resolved to treat available information as informed views and indications on it and present the information we were able to extract as signs of the potential of the model. From the theoretical figure of \cite{Garcia-Bellido:2017fdg} (Fig 7, indication of shape and proportions), we expect the BH distribution to present 4 peaks: Stellar BHs (SBHs), Primordial BHs (PBHs), Intermediate mass BHs (IMBHs) and Super Massive BHs (SMBHs).

Fig 10 of \cite{Mutlu-Pakdil2016} evaluates the peak of SMBHs, while
figs 2 and 6 of \cite{Kovetz:2016kpi} assesses astrophysical BHs from gravitational waves (GW),
and can be combined with Fig.~1 from the theoretical study of \cite{Christian:2018mjv} to evaluate the peak of SBHs.

The SBHs part of the distribution can proceed from fitting both Fig.~1 from \cite{Christian:2018mjv} and fig. 6 of \cite{Kovetz:2016kpi}, with a power law distribution. Then, taking their geometrical mean while variance is evaluated from their maximum variations for both standard deviations, yields an evaluation with standard deviation as $M_S=10^{1.09^{+0.27}_{-0.48}}$.

The SMBHs distribution can be assessed with Fig. 10 of \cite{Mutlu-Pakdil2016} as it appears as a double power law. An evaluation of that distribution mean and variance for SMBHs provides $M_{SM}=10^{11.69^{+0.20}_{-0.39}}$.

Since knowledge for the two remaining peaks does not exceed that presented in Fig 7 of \cite{Garcia-Bellido:2017fdg}, we used it as indication of the relative peak heights and standard deviations of their Gau\ss ian shapes, since the mass axis is providing numerical values, to obtain estimates for the remaining peaks for Primordial and Intermediate mass BH, $M_P=10^{1.96^{+0.14}_{-0.21}}M_\odot$ and $M_I=10^{4.19^{+0.14}_{-0.21}}M_\odot$. We are well aware that these estimates are not based on strict modeling nor observations but, since no possibility was given to evaluate the errors on this method, we considered the variances obtained as lower estimates of errors and continued to treat them as mere variances.

As those peaks span several orders of magnitude in mass, we estimated the mean mass from these evaluations by combining them together in a weighted geometric mean. Putting together those evaluations with a geometrical weighted average and conservative treatment of variance yields $\langle M \rangle= \left(M_S^{1.63} M_P^{2.58} M_I M_{SM}^{1.32}\right)^\frac{1}{6.53}$, an estimate of the average BH mass
\begin{align}
    \left\langle M_{BH}\right\rangle \sim & 10^{4.04^{+0.49}_{-0.61}}M_\odot. \label{eq:AvgBHmassU}
\end{align}

\subsubsection{boundary volume}

Putting together the average mass estimate (\ref{eq:AvgBHmassU}) into the BH boundary volume (\ref{eq:BHBounVol}), one gets the estimate for the Universe's boundary volume
\begin{align}
    \text{Vol} \left( \partial \mathcal{M} \right) \sim &  \mathcal{N}_{BH}10^{107.5^{+2.5}_{-3.1}} m^3.\label{eq:BounVol}
\end{align}
\subsection{Topological cosmological constant}
Combining the expression (\ref{eq:Topolambdabody}) with the evaluation of the average volume through the boundary volume (\ref{eq:BounVol}), we get the final result
\begin{align}
\Lambda \approx 10^{-52.9_{-1.3}^{+1.5}} \, k_E\mathcal{C},\label{eq:TopoLambdaKC}
\end{align}
which can be further approximated using $k_E\approx1$ into
\begin{align}
\Lambda \approx 10^{-52.9_{-1.3}^{+1.5}} \, \mathcal{C}.\label{eq:TopoLambdaWithC}
\end{align}

\subsubsection{Confrontation with observed cosmological constant}

From Planck observations \cite{Planck2018}, we currently evaluate $\Lambda_O=10^{-51.08\pm0.01}m^{-2}$. In the case of a 4-manifold with null sectional curvature, Ref.~\cite{ASENS, Hoffman} evaluated the isoperimetric constant $\mathcal{C} = 11.8$. Using it as an evaluation for the topological cosmological constant one obtains
\begin{align}
 \Lambda \approx 10^{-51.8^{+1.5}_{-1.3}} m^{-2},
\end{align}
which is compatible with the observed value. The topology of the Universe can therefore be considered as a serious candidate for the cosmological constant, without raising the usual cosmological constant fine tuning problem.

\subsubsection{Evaluation of the Universe isoperimetric constant}

Since the isoperimetric constant of the Universe remains unknown, although its order of magnitude is expected in the range of $\approx 10$, this evaluation (\ref{eq:TopoLambdaWithC}) through the average BH mass can be combined with cosmological constant observations \cite{Planck2018} to appraise its actual value, and therefore characterise the topology of the Universe with
\begin{align}
    \mathcal{C}=10^{1.82^{+1.31}_{-1.51}}.
\end{align}

\section{\label{sec:Conclusion}Conclusion}

In this work, we have discussed the application to the cosmological constant problem of the topological invariants examined in \citep{Lorca}, introduced as Lagrange multipliers, that are compatible with GR for cosmological manifolds. At the level of the Einstein-Hilbert action, this results in the emergence of a topological effective cosmological constant from the Euler number. To do so we restricted an Einstein-Cartan theory to solutions with GR-like behavior at the level of the field equations, a sufficient but not necessary condition to recover GR. As we linked the torsion to the topological invariants, the extremely strong attractor behaviour of the invariant solutions, fixing the manifold topology, allows to consider gravity as GR for most of the observable Universe.

The resulting effective topological cosmological constant (TCC), assuming the bare constant is set to zero by symmetry considerations, can then be computed as founded on the inverse square root of the volume of the manifold boundary, the volume of all black holes in the Universe history, and therefore yields an evaluation, that carries the caveats of our current knowledge on BH distribution in the universe, encompassing the observed value of the cosmological constant \citep{Hinshaw:2012aka,Bennett:2012zja,Aghanim:2012vda,Ade:2013sjv,Planck2018}. That value can then be considered seriously as sourced by the topology of the Universe, in particular its Euler number. This also solves the fine tuning problem of the cosmological constant \citep{Weinberg:1988cp,Weinberg:2008zzc}, since the inverse square root of the volume of the boundary of the Universe is quite naturally much smaller than the Planck density, and links the coincidence problem with the distribution of BHs in the Universe. Moreover, we used the observed value of the cosmological constant, combined with our evaluation, to provide a first evaluation of the isoperimetric constant $\mathcal{C}^2$ of the Universe, that is the ratio of the compact volume domain over its boundary hypersurface volume. In light of these results, we argue that the dynamical coupling zero-forms act as a kind of topological quintessence field, effecting the accelerated expansion of the Universe to comply with its BH boundary.

As this result proceeds from BH distribution evaluation that involved GW input, we can be confident that future GW detections and other improvement in our understanding of the BH population should improve our determination of the TCC and of the Universe's isoperimetric constant. A dynamic theory of the topology of the universe in the line of emerging geometry \citep{Brandenberger:2008nx,LevasseurPerreault:2011mw,Brandenberger:2012um,Afshordi:2014cia} should provide the framework for understanding this choice of $\mathcal{C}^2$ and perhaps for solving as well the coincidence problem
\citep{Amendola:1999er,TocchiniValentini:2001ty,Zimdahl:2001ar,Zimdahl:2002zb} by producing a mechanism of selection for the Betti numbers of the space-time manifold.

\section*{Acknowledgments}

The authors wish to thank M.Fontanini and E. Huguet for very useful discussions. The work of M.Le~D. has been supported by Lanzhou University starting fund and PNPD/CAPES20132029. M.Le~D. also wishes to acknowledge IFT/UNESP for hosting the beginning of this project.


\section*{Acknowledgement}
We would like to thank AI R\&D team of Kakao Enterprise for the helpful discussion.
In particular, we would like to thank Yunmo Park who designed the visual materials.

\clearpage
\bibliographystyle{splncs04}

\section{Proposed Method}
We describe the widely-adopted learning scheme in face recognition, and then illustrate the proposed BroadFace in detail.

\subsection{Typical Learning}

\noindent\textbf{Learning of Face Recognition.}
In general, a face recognition network is divided into two parts:
(1) an encoder network that extracts an embedding vector from a given image
and (2) a linear classifier that maps an embedding vector into probabilities of identities.
An evaluation is performed by comparing embedding vectors on images of unseen identities, so the classifier is discarded at the inference stage.
Here, $f$ is the encoder network that extracts a $D$-dimensional embedding vector $e$ from a given image $x$: $e = f(x; \theta)$ with a model parameter $\theta$.
The linear classifier performs classification for $C$ identities from an embedding vector
$e$ with a weight matrix $W$ of $C$$\times$$D$-dimensions.
For a mini-batch $\mathbf{X}$, the objective function such as a variant of angular softmax losses \cite{ArcFace,SphereFace,NormFace} is used to optimize the encoder and the classifier:
\begin{equation}
    \mathcal{L}(\mathbf{X}) = \frac{1}{|\mathbf{X}|}\sum_{i\in \mathbf{X}}{l(e_i, y_i)},
    \label{eq:general_form1}
\end{equation}
\begin{equation}
    l(e_i, y_i) = - \log \frac{\text{exp}(\hat{{W}}^{T}_{y_i}
    \hat{e}_i)}{\sum_{j=1}^{C}{\text{exp}(\hat{{W}}^{T}_{j}\hat{e}_j)}},
    \label{eq:general_form2}
\end{equation}
where ${y}_i$ is an labeled identity of ${x}_i$ and $\hat{\cdot}$ indicates that a given vector is $L_2$ normalized (\textit{e.g.}, $\norm{\hat{{e}}}_2=1$).
In Eq. \ref{eq:general_form2}, $\hat{W}_{y_i}$ acts as an representative instance of the given identity $y_i$ that maximizes the cosine similarity with an embedding vector $e_i$.
Thus, $\hat{W}_{y}$ can be regarded as the identity-representative vector, which is the expectation of instances belong to $y$:
\begin{equation}
\hat{W}_y = E_{x}\left[\hat{e}_{i}\big|y_i = y\right].
\label{eq:expected}
\end{equation}

\noindent\textbf{Limitations of Mini-batch.}
The parameters of the model are updated in an iterative process that considers a mini-batch that contains only a small portion of the entire dataset for each step (Fig. \ref{fig:concept}).
However, use of a small mini-batch may not represent the entire distribution of training datasets.
Moreover, in face recognition, the number of identities is very large and each mini-batch only contains few of them; for example, MSCeleb-1M \cite{msceleb1m} has 10M images of 100k celebrities.
Therefore, each parameter update of a model can be biased on a small number of identities, and this restriction complicates the task of finding optimal decision boundaries.
Enlarging the mini-batch size may mitigate the problem, but this solution requires heavy computation of the encoder, proportional to the batch-size.

\subsection{BroadFace}

We introduce BroadFace, which is a simple yet effective way to cover a large number of instances and identities.
BroadFace learns globally well-optimized identity-representative vectors from a massive number of embedding vectors (Fig. \ref{fig:concept}).
For example, on a single Nvidia V100 GPU, the size of a mini-batch for ResNet-100 is at most 256, whereas BroadFace can utilize more than 8k instances at once.
The following describes each step.

\noindent\textbf{(1) Queuing Past Embedding Vectors.}
BroadFace has two queues of pre-defined size: $\mathbb{E}$ stores embedding vectors; $\mathbb{W}$ stores identity-representative vectors from the past.
For each iteration, after model update, embedding vectors of a given mini-batch $\{e_i\}_{i\in\mathbf{X}}$ are enqueued to $\mathbb{E}$, and corresponding identity-representative vectors $\{W_{y_i}\}_{i\in\mathbf{X}}$ of each instance are enqueued to $\mathbb{W}$.
By referring to the past embedding vectors in the queue to compute the loss $\mathcal{L}(\mathbf{X}\cup\mathbb{E})$, the network increase the number of instances and identities explored at each update.

\begin{figure}[t]
	\begin{center}
		\subfloat[without Compensation]{
			\label{fig:compensate1}
			\includegraphics[width=5.3cm]{images/figure2a.pdf}
		}
		\subfloat[with Compensation]{
			\label{fig:compensate2}
			\includegraphics[width=5.3cm]{images/figure2b.pdf}
		}
	\end{center}
	\caption{
 	(a) enqueued embedding vectors (gray circles) at the past are further away from the embedding vectors (blue circles) at the current iteration due to the parameter update and this indicates significant errors.
 	(b) the compensated embedding vectors (orange circles) closely approach to the embedding vectors at the current iteration, by considering the difference between the identity-representative vectors (class centers) at the past and the current iteration.
	}
	\label{fig:compensate}
\end{figure}

\noindent\textbf{(2) Compensating Past Embedding Vectors. }
As the model parameter $\theta$ of the encoder is updated over iterations, past embedding vectors $e^{\text{-}}\in\mathbb{E}$ conflict with the embedding space of the current parameter (Fig. \ref{fig:compensate1}); $\epsilon = e - e^{\text{-}}$ where $\theta^{\text{-}}$ is the past parameter of the encoder and $e^{\text{-}}=f(x;\theta^{\text{-}})$.
The magnitude of the error $\epsilon$ is relatively small when few iterations have been passed from the past.
However, the error is gradually accumulated over iterations and the error hinders appropriate training.
We introduce a compensation function $\rho(y)$ for each identity to reduce the errors as an additive model; $e_{i}^{*} = e^{\text{-}}_i + \rho(y)$ where $e_{i}^{*}$ is a compensated past embedding vector to a current embedding vector (Fig. \ref{fig:compensate2}).
The compensation function should minimize an expected squared error $J$ between the current embedding vectors and the compensated past embedding vectors that belong to $y$: \begin{equation}
\begin{aligned}
    \text{minimize } J\big(\rho(y)\big) = & E_x \left[ \big(e_i^{*} - e_i\big)^2 \big | y_i = y \right], \\
    = & E_x \left[ \big(e_i^{\text{-}} + \rho(y) - e_i\big)^2 \big | y_i = y \right].
\end{aligned}
\end{equation}
The partial derivative of $J$ with respect to $\rho(y)$ is:
\begin{equation}
    \frac{\partial J}{\partial \rho(y)}=
    E_x \Big[ 2\big(e_i^{\text{-}} + \rho(y) - e_i\big) \big| y_i = y\Big].
\end{equation}
Thus, the optimal compensation function is the difference between the expectations of current embedding vectors and past embedding vectors:
\begin{equation}
\begin{aligned}
    \rho(y) & = E_x \left[ e_i \big | y_i = y \right] - E_x\left[e_i^{\text{-}} \big | y_i = y \right], \\
            & \approx \lambda (W_{y} - W^{\text{-}}_{y}),
\end{aligned}
\end{equation}
where $W^{\text{-}}_y\in\mathbb{W}$ is an identity-representative vector, which is enqueued to the queue during the same iteration as $e^{\text{-}}_{i}\in\mathbb{E}$.
As explained in Eq. \ref{eq:expected}, the identity-representative vector and the expected embedding vector point in the same direction when the vectors are projected onto a hyper-sphere, but the vectors are different in scale.
Thus, we deploy a simple normalization term per each instance to adjust these scales: $\lambda={\norm{e^{\text{-}}_i}}/\norm{W^{\text{-}}_{y_i}}$.
Then the compensated embedding vector $e^{*}$ is computed as:
\begin{equation}
    e_{i}^{*} = e_{i}^{\text{-}} + \frac{\norm{e^{\text{-}}_i}}{\norm{W^{\text{-}}_{y_i}}} (W_{y_i} - W^{\text{-}}_{y_i}).
\end{equation}
In empirical studies, the compensation function reduces the error significantly.

\begin{figure}[t]
    \centering
    \includegraphics[width=12.3cm]{images/figure3_fm.pdf}
	\caption{
    Learning process of the proposed method.
    BroadFace deploys large queues to store embedding vectors and their corresponding identity-representative vectors per iteration.
    The embedding vectors of the past instances stored in the queues are used to compute loss for identity-representative vectors.
    BroadFace effectively learns from tens of thousands of instances for each iteration.
	}
	\label{fig:network}
\end{figure}

\noindent\textbf{(3) Learning from Numerous Embedding Vectors.}
By executing the preceding two steps, BroadFace generates additional large-scale embedding vectors from the past.
In our method, the encoder is trained on a mini-batch as before and the classifier is trained on both a mini-batch and the additional embedding vectors.
The objective functions for the encoder and the classifier are defined as:
\begin{equation}
    \mathcal{L}_{\text{encoder}}(\mathbf{X}) = \frac{1}{|\mathbf{X}|}
    \left \{
    \sum_{i\in \mathbf{X}}{l(e_i)}
    \right \},\\
    \label{eq:divide_objective1}
\end{equation}
\begin{equation}
     \mathcal{L}_{\text{classifier}}(\mathbf{X}\cup\mathbb{E}) =  \frac{1}{|\mathbf{X} \cup \mathbb{E}|}
    \left \{
    \sum_{i\in \mathbf{X}}{l(e_i)}
    +
    \sum_{j\in \mathbb{E}}{l(e^{*}_j )}
    \right \}.
    \label{eq:divide_objective2}
\end{equation}
The parameter $\theta$ of the encoder is updated \textit{w.r.t.}  $\mathcal{L}_{\text{encoder}}(\mathbf{X})$ while the parameter of the classifier $W$ is updated \textit{w.r.t.}  $\mathcal{L}_{\text{classifier}}(\mathbf{X}\cup\mathbb{E})$.
The large number of embedding vectors in the queue helps to learn highly precise identity-representative vectors that show reduced bias on a mini-batch and increased optimality on the entire dataset.
The precise identity-representative vectors can accelerate the learning procedure.
Moreover, our method can be easily implemented by adding several queues in the learning process (Fig. \ref{fig:network}) and significantly improves accuracy in face recognition without any computational cost at inference stage.

\subsection{Discussion}
\begin{figure}[t]
	\begin{center}
		\subfloat[Averaged Error]{
			\label{fig:vis_com1}
			\includegraphics[width=5.5cm]{images/figure4a.pdf}
		}
		\subfloat[Scatter Plot]{
			\label{fig:vis_com2}
			\includegraphics[width=5.5cm]{images/figure4b.pdf}
		}
	\end{center}
	\caption{
	(a) the average of the cosine errors between the embedding vectors at the current iteration and the past iteration with and without compensation.
	The errors are computed with randomly sampled instances over 64 iterations.
	(b) the scatter plot of the past (before 64 iterations), current and compensated embedding vectors for 64 instances.
	}
	\label{fig:vis_com}
\end{figure}

\noindent\textbf{Effectiveness of Compensation. }
We show that the compensation method is empirically effective.
After a small number of iterations, the error of the enqueued embedding vectors is also small and the compensation method is not necessary.
However, after a large number of iterations, the error increases and the compensation method becomes necessary to keep a large number of embedding vectors (Fig. \ref{fig:vis_com1}).
A large accumulated error may degrade the training process of the network (Fig. \ref{fig:img1} and Fig. \ref{fig:img2}).
We illustrate how the compensation function reduces the difference between past and current embedding vectors in 2-dimensional space by t-SNE \cite{t_SNE} (Fig. \ref{fig:vis_com2}).
The past embedding vectors approach to current embedding vectors after applying compensation.
This shows that the proposed compensation function works properly in practice.

\begin{figure}[t]
	\begin{center}
	    \includegraphics[width=5.5cm]{images/figure5.pdf}
	\end{center}
	\caption{
	Illustration on memory consumption of a conventional mini-batch learning (blue line) and the proposed BroadFace (red line) depending on the size of a mini-batch.
	The blue dotted line indicates memory consumption that is estimated for a large size of mini-batch by linear regression.
	}
	\label{fig:memory}
\end{figure}

\noindent\textbf{Memory Efficiency.}
We compare BroadFace with enlarging the size of a mini-batch in terms of memory consumption (Fig. \ref{fig:memory}).
A na\"ive mini-batch learning requires a huge amount of memory to forward and backward the entire network.
The maximum size of a mini-batch is about $240$ instances when a model based on ResNet-100 is trained on NVidia V100 of 32 GB.
However, BroadFace requires only a matrix multiplication between the embedding vectors in $\mathbb{E}$ and the weight matrix $W$ (Eq.~\ref{eq:general_form2}).
The marginal computational cost of BroadFace enables a classifier to learn decision boundaries from a massive set of instances, \textit{e.g.}, 8192 instances for a single GPU.
Note that, enlarging the size of a mini-batch to 8192 requires about 952 GB of memory which is infeasibly large for a single GPU.

\section{Conclusion}

We introduce a new way called BroadFace that allows an embedding space to distinguish numerous identities in a broad perspective by increasing the optimality of constructed identity-representative vectors.
BroadFace is significantly effective for face recognition and image retrieval where their datasets consist of numerous identities and instances.
BroadFace can be easily applied on many existing face recognition methods to obtain a significant improvement without any extra computational cost in the inference stage.

\section{Related Works}

Recent studies of face recognition tend to introduce a new objective function that learns an embedding space by exploiting an identity-representative vector.
NormFace \cite{NormFace} reveal that optimization using cosine similarity between identity-representative vectors and embedding vectors is more effective than optimization using the inner product.
To increase the discriminative abilities of learned features, SphereFace \cite{SphereFace}, CosFace \cite{CosFace} and ArcFace \cite{ArcFace} adopted different kinds of margin into the embedding space.
Futhermore, some works adopted an additional loss function to regulate the identity-representative vectors.
RegularFace \cite{regularface} minimized a cosine similarity between identity-representative vectors, and UniformFace \cite{uniformface} equalized distances between all the cluster centers.
However, those methods can suffer from an enormous number of identities and instances because they are based on a mini-batch learning.
Our BroadFace overcomes the limitation of a mini-batch learning, and, it can be easily applied on those face recognition methods.

In terms of preserving knowledge of model on previously visited data, the continual learning \cite{hou2019learning,li2017learning} shares the similar concept with BroadFace.
However, BroadFace is different from continual learning, as BroadFace preserves knowledge of previous data from the same dataset while continual learning preserves knowledge of previous data from different datasets.

\section{Introduction}

Face recognition is a key technique for many applications of biometric authentication such as electronic payment, lock screen of smartphones, and video surveillance.
The main tasks of face recognition are categorized into face verification and face identification.
In face verification, a pair of faces are compared to verify whether their identities are the same or different.
In face identification, the identity of a given face is determined by comparing it to a pre-registered gallery of identities.
Many researches \cite{10.1007/978-3-540-24670-1_36,6619233,joint_bayesian,5459250,CSML,fisherface,eigenface,WHT:ECCVW08:DBMW,QiYin:2011:AMF:2191740.2192084} on face recognition have been conducted for decades. The recent adoption \cite{ArcFace,marginal_loss,SphereFace,FaceNet,deepface,NormFace,CosFace,centerloss} of Convolutional Neural Networks (CNNs) has dramatically increased recognition accuracy.
However, many difficulties of face recognition still remain to be solved.

\begin{figure}[t]
	\begin{center}
		\subfloat[Typical Mini-batch Learning]{
			\label{fig:teaser_a}
			\includegraphics[height=4.5cm]{images/figure1a.pdf}
		}
		\subfloat[The Proposed Learning]{
			\label{fig:teaser_b}
			\includegraphics[height=4.5cm]{images/figure1b.pdf}
		}
	\end{center}
	\caption{
	(a) In typical mini-batch learning, the parameter $\theta$ of encoder $f$ and the parameter $W$ of linear classifier are optimized on a small mini-batch $\mathbf{X}$.
	(b) In the proposed method, the parameter of the encoder is optimized on a small mini-batch, but the parameter of the classifier is optimized on both the mini-batch and the large queue $\mathbb{E}$ that contains embedding vectors $e^{\text{-}}$ of past iterations.
	}
	\label{fig:teaser}
\end{figure}

Most previous studies focus on improving the discriminative power of an embedding space, because face recognition models are evaluated on independent datasets that include unseen identities.
The mainstream of recent studies \cite{ArcFace,SphereFace,NormFace,CosFace} is to introduce a new objective function to maximize inter-class discriminability and intra-class compactness;
they try to consider all identities by referring an identity-representative vector, which is the weight vector of the last fully-connected layer for identity classification.

However, conventional methods still have difficulty in covering a massive set of identities at once, because these methods use a small mini-batch (Fig. \ref{fig:teaser_a}) much less than the number of identities due to memory constraints.
Inspecting tens of thousands of identities with the mini-batch of the small size requires numerous iterations, and this complicates the task of learning optimal decision boundaries in an embedding space while considering all of the identities, comprehensively.
Increasing the size of the mini-batch may alleviate some of the problem, but in general, this solution is impractical because of memory constraints; it also does not guarantee improved accuracy \cite{goyal2017accurate,hoffer2017train,keskar2016large,you2017scaling}.

We propose a novel method, called \textit{BroadFace}, which is a learning process to consider a massive set of identities, comprehensively (Fig. \ref{fig:teaser_b}).
BroadFace has a large queue to keep a massive number of embedding vectors accumulated over past iterations.
Our learning process increases the optimality of decision boundaries of the classifier by considering the embedding vectors of both a given mini-batch and the large queue for each iteration.
The parameters of the model are updated iteratively, so after a few iterations the error of enqueued embedding vectors gradually increases.
Therefore, we introduce a compensation method that reduces the expected error between the current and enqueued embedding vectors by referencing the difference of the identity-representative vectors of current and past iterations.
Our BroadFace has several advantages:
(1) the identity-representative vectors are updated with a large number of embedding vectors to increase the portion of the training set that is considered for each iteration,
(2) the optimality of the model is increased on the entire dataset by referring to the globally well-optimized identity-representative vectors,
(3) the learning process is accelerated.
We summarize the contributions as follows:
\begin{itemize}[leftmargin=+.2in,label=$\bullet$]
    \item
    We propose a new way that allows an embedding space to distinguish numerous identities in a broad perspective by learning identity-representative vectors from a massive number of instances.
    \item
    We perform extensive ablation studies on its behaviors, and experiments on various datasets to show the effectiveness of the proposed method, and to empirically prove the validity of our compensation method.
    \item
    BroadFace can be easily applied on many existing face recognition methods to obtain a significant improvement.
    Moreover, during inference time, it does not require any extra computational burden.
\end{itemize}

\section{Experiments}

\subsection{Implementation Details}

\noindent\textbf{Experimental Setting.}
As pre-processing, we normalize a face image to $112 \times 112$ by warping a face-region using five facial points from two eyes, nose and two corners of mouth \cite{ArcFace,SphereFace,CosFace}.
A backbone network is ResNet-100 \cite{resnet} that is used in the recent works \cite{ArcFace,AFRN}.
After the \texttt{res5c} layer of ResNet-100, a block of batch normalization, fully-connected and batch normalization layers is deployed to compute a 512-dimensional embedding vector.
The computed embedding vectors and the weight vectors of the linear classifier are $L_2$-normalized and trained by the ArcFace~\cite{ArcFace}.
Our model is trained on $4$ synchronized NVidia V100 GPUs and a mini-batch of 128 images is assigned for each GPU.
The queue of BroadFace stores up to 8,192 embedding vectors accumulated over 64 iterations for each GPU, thus the total size of the queues is 32,768 for 4 GPUs.
To avoid abrupt changes in the embedding space, the network of BroadFace is trained from the pre-trained network that is trained by the softmax based loss \cite{ArcFace}.
We adopted stochastic gradient descent (SGD) optimizer, and a learning rate is set to $5\cdot10^{-3}$ for the first 50k, $5\cdot10^{-4}$ for the 20k, and $5\cdot10^{-5}$ for the 10k with a weight decay of $5\cdot10^{-4}$ and a momentum of $0.9$.

\noindent\textbf{Datasets.}
All the models are trained on MSCeleb-1M \cite{msceleb1m}, which is composed of about 10M images for 100k identities.
We use the refined version \cite{ArcFace}, which contains 3.8M images for 85k identities by removing the noisy labels of MSCeleb-1M.
For the test, we perform evaluations on the following various datasets:
\begin{itemize}[label=$\bullet$]
    \item
    Labeled Faces in the Wild (LFW) \cite{LFW} contains 13k images of faces that are collected from web for 5,749 different individuals.
    Cross-Age LFW (CALFW) \cite{CALFW} provides pairs with age variation, and Cross-Pose (CPLFW) \cite{CPLFWTech} provides pairs with pose variation from the images of LFW.
    \item
    YouTube Faces (YTF) \cite{YTF} contains 3,425 videos of 1,595 different people.
    \item
    MegaFace \cite{kemelmacher2016megaface} contains more than 1M images from 690k identities to evaluate recognition-accuracy with enormous distractors.
    \item
    Celebrities in Frontal-Profile (CFP) \cite{CFP_FP}
    contains 500 subjects; each subject has 10 frontal and 4 profile images.
    \item
    AgeDB-30 \cite{age_db}, which contains 12,240 images of 440 identities with age variations, is suitable to evaluate the sensitivity of a given method in age variation.
    \item
    IARPA Janus Benchmark (IJB) \cite{IJB_C,IJB_B}, which is designed to evaluate unconstrained face recognition systems,
    is one of the most challenging datasets in public.
    IJB-B~\cite{IJB_B} is composed of 67k face images, 7k face videos and 10k non-face images.
    IJB-C~\cite{IJB_C}, which adds additional new subjects with increased occlusion and diversity of geographic origin to IJB-B, is composed of 138k face images, 11k face videos and 10k non-face images.
\end{itemize}

\subsection{Evaluations on Face Recognition}
\label{sec:eval_fr}
We conduct experiments on the described various datasets to show the effectiveness of the proposed method.

\begingroup
\newcolumntype{C}[1]{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
 \begin{table}[th!]
    \footnotesize
    \begin{center}
     \caption{Verification accuracy ($\%$) on LFW and YTF.}
	 \label{tab:lfw}
     \end{center}
     \centering
	\def1.1{1.1}
    \begin{tabular}{p{3cm}|C{1.3cm}|C{1.3cm}||p{3cm}|C{1.3cm}|C{1.3cm}}
    \hline
    Method & LFW & YTF & Method & LFW & YTF \\
    \hline
    \hline
    DeepID \cite{contrastive_loss}      & 99.47 & 93.2 &
    DeepFace \cite{deepface}            & 97.35 & 91.4 \\
    VGGFace \cite{vggface}              & 98.95 & 97.3 &
    FaceNet \cite{FaceNet}              & 99.64 & 95.1 \\
    CenterLoss \cite{centerloss}        & 99.28 & 94.9 &
    RangeLoss \cite{range_loss}         & 99.52 & 93.7 \\
    MarginalLoss \cite{marginal_loss}   & 99.48 & 95.9 &
    SphereFace \cite{SphereFace}        & 99.42 & 95.0 \\
    RegularFace \cite{regularface}      & 99.61 & 96.7 &
    CosFace \cite{CosFace}              & 99.81 & 97.6 \\
    UniformFace \cite{uniformface}      & 99.80 & 97.7 &
    AFRN \cite{AFRN}                    & 99.85 & 97.1 \\
    ArcFace \cite{ArcFace}              & 99.83 & 97.7 &
    BroadFace                           & $\mathbf{99.85}$ & $\mathbf{98.0}$ \\
    \hline
    \end{tabular}
\end{table}
\endgroup
\begingroup
\newcolumntype{C}[1]{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
\begin{table}[th!]
    \footnotesize
    \begin{center}
     \caption{Verification accuracy ($\%$) on CALFW, CPLFW, CFP-FP and AgeDB-30.}
	 \label{tab:agedb}
	 \end{center}
	 \centering
	 \small
	\def1.1{1.1}
    \begin{tabular}{p{3cm}|C{1.5cm}|C{1.5cm}|C{1.5cm}|C{1.5cm}}
    \hline
    {Method} & {CALFW} & {CPLFW} & {CFP-FP} & {AgeDB-30} \\
    \hline
    \hline
    CenterLoss \cite{centerloss}    & 85.48 & 77.48 & - & -\\
    SphereFace \cite{SphereFace}    & 90.30 & 81.40 & - & -\\
    VGGFace2 \cite{vggface2}        & 90.57 & 84.00 & - & -\\
    CosFace \cite{CosFace}          & 95.76 & 92.28 & 98.12 & 98.11\\
    ArcFace \cite{ArcFace}          & 95.45 & 92.08 & 98.27 & 98.28 \\
    BroadFace                       & $\mathbf{96.20}$ & $\mathbf{93.17}$ & $\mathbf{98.63}$ & $\mathbf{98.38}$\\
    \hline
    \end{tabular}
 \end{table}
 \endgroup

\begingroup
\newcolumntype{C}[1]{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
\begin{table}[th!]
    \footnotesize
    \begin{center}
     \caption{Identification and verification evaluation on MegaFace \cite{kemelmacher2016megaface}.
     Ident indicates rank-1 identification accuracy ($\%$) and Verif indicates a true accept rate ($\%$) at a false accept rate of 1e-6.
     }
	 \label{tab:mega}
	 \end{center}
	 \centering
	\def1.1{1.1}
    \begin{tabular}{p{3cm}|C{1.6cm}|C{1.6cm}|C{1.6cm}|C{1.6cm}}
    \hline
    \multirow{2}{*}{Method} &  \multicolumn{2}{c|}{MF-Large} &  \multicolumn{2}{c}{MF-Large-Refined \cite{ArcFace}}  \\
    \cline{2-5}
    & {Ident} & {Verif} & {Ident} & {Verif} \\
    \hline
    \hline
    RegularFace \cite{regularface}      & 75.61 & 91.13 & - & - \\
    UniformFace \cite{uniformface}      & 79.98 & 95.36 & - & - \\
    SphereFace \cite{SphereFace}        & - & - & 97.91 & 97.91 \\
    AdaptiveFace \cite{adaptiveface}    & - & - & 95.02 & 95.61 \\
    CosFace \cite{CosFace}              & 80.56 & 96.56 & 97.91 & 97.91 \\
    ArcFace \cite{ArcFace}              & 81.03 & 96.98 & 98.35 & 98.49 \\
    BroadFace                           & $\mathbf{81.33}$ & $\mathbf{97.56}$ & $\mathbf{98.70}$ & $\mathbf{98.95}$ \\
    \hline
    \end{tabular}
\end{table}
\endgroup

 \begingroup
\newcolumntype{C}[1]{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
\begin{table}[th!]
    \footnotesize
   \begin{center}
    \caption{Verification evaluation with a True Accept Rate at a certain False Accept Rate (TAR@FAR) from 1e-4 to 1e-6 on IJB-B and IJB-C.
    $^{\dag}$ denotes BroadFace trained by CosFace \cite{CosFace}.}
    \label{tab:ijb}
    \end{center}
    \centering
	\def1.1{1.1}
   \begin{tabular}{p{3cm}|c|c|c|c|c|c}
   \hline
   \multirow{2}{*}{Method} &  \multicolumn{3}{c|}{IJB-B} &  \multicolumn{3}{c}{IJB-C}  \\
   \cline{2-4}\cline{5-7}
    & \scalebox{0.9}{FAR=1e-6} & \scalebox{0.9}{FAR=1e-5} & \scalebox{0.9}{FAR=1e-4} & \scalebox{0.9}{FAR=1e-6} & \scalebox{0.9}{FAR=1e-5} & \scalebox{0.9}{FAR=1e-4}  \\
    \hline\hline
    VGGFace2 \cite{vggface2}            & - & 0.671 & 0.800 &  - & 0.747 & 0.840 \\
    CenterFace \cite{centerloss}        & - & - & - & - & 0.781 & 0.853 \\
    ComparatorNet \cite{ComparatorNet}  & - & -     & 0.849 &  - & - & 0.885 \\
    PRN \cite{PRN}                      & - & 0.721 & 0.845 &  - & - & - \\
    AFRN \cite{AFRN}                    & - & 0.771 & 0.885 &  - & 0.884 & 0.931 \\
    \hline
    CosFace \cite{CosFace}              & 0.3649 & 0.8811 & 0.9480 & 0.8591 & 0.9410 & 0.9637 \\
    BroadFace$^\dag$                 & 0.4092 & 0.8997 & $\mathbf{0.9497}$ & 0.8596 & $\mathbf{0.9459}$ & $\mathbf{0.9638}$ \\
    \hline
    ArcFace \cite{ArcFace}              & 0.3828 & 0.8933 & 0.9425 & 0.8906 & 0.9394 & 0.9603 \\
    BroadFace                 & $\mathbf{0.4653}$ & $\mathbf{0.9081}$ & 0.9461 & $\mathbf{0.9041}$ & 0.9411 & 0.9603 \\
   \hline
   \end{tabular}
\end{table}
\endgroup

\noindent\textbf{LFW and YTF}
are widely used to evaluate verification performance under the unrestricted environments.
LFW, which contains pairs of images, evaluates a model by comparing two embedding vectors of a given pair.
YTF contains videos that are sets of images; from the shortest clip of 48 frames to the longest clip of 6,070 frames.
To compare a pair of videos, YTF compares a pair of video-representative embedding vectors that are averaged embedding vectors of images collected from each video.
Even though both datasets are highly-saturated in accuracy, our BroadFace outperforms other recent methods (Table \ref{tab:lfw}).

\noindent\textbf{CALFW, CPLFW, CFP-FP and AgeDB-30}
are also widely used to verify that methods are robust to pose and age variation.
CALFW and AgeDB-30 have multiple instances for same identity of different ages
and CPLFW and CFP-FP have multiple instances for same identity of different poses (frontal and profile faces).
BroadFace shows better verification-accuracy on all datasets (Table \ref{tab:agedb}).

\noindent\textbf{MegaFace}
is designed to evaluate both face identification and verification tasks under difficulty caused by a huge number of distractors.
We evaluate our BroadFace on Megaface Challenge 1 where the training dataset is more than 0.5 million images.
BroadFace outperforms the other top-ranked face recognition models for both face identification and verification tasks (Table \ref{tab:mega}).
On the refined MegaFace \cite{ArcFace}, where noisy labels are removed, BroadFace also surpasses the other models.

\noindent\textbf{IJB-B and IJB-C}
are the most challenging datasets to evaluate unconstrained face recognition.
We report BroadFace with CosFace \cite{CosFace} and BroadFace with ArcFace \cite{ArcFace} in verification task without any augmentations such as horizontal flipping in test time.
Our BroadFace shows significant improvements on all FAR criteria (Table \ref{tab:ijb}).
In IJB-B \cite{IJB_B}, BroadFace improves 8.25 percentage points on FAR=1e-6, 1.48 percentage points on FAR=1e-5 and 0.36 percentage points on FAR=1e-4 comparing to the results of ArcFace \cite{ArcFace}.

\subsection{Evaluations on Image Retrieval.}
\label{sec:eval_ir}

Both face recognition and image retrieval have the same goal that learn an optimal embedding space to compare a given pair of items such as face, clothes or industrial products.
To show that BroadFace is widely applicable on other applications, we compare BroadFace with recently proposed metric learning methods for image retrieval.

\noindent\textbf{Experimental Settings.}
We use ResNet-50 \cite{resnet} that is pre-trained on ILSVRC 2012-CLS~\cite{ILSVRC15} as a backbone network.
We use ArcFace \cite{ArcFace} as a baseline objective function and set the size of the queue to 32k for BroadFace.
We follow the standard input augmentation and evaluation protocol \cite{song2016deep}.
We evaluate on two large datasets with a large number of classes similar to face-recognition:
In-Shop Clothes Retrieval (In-Shop) \cite{liuLQWTcvpr16DeepFashion} and Stanford Online Products (SOP) \cite{song2016deep}.

\begingroup
\newcolumntype{C}[1]{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
\begin{table}[t]
    \footnotesize
    \begin{center}
    \caption{Recall@K comparison with state-of-the-art methods. For fair comparison, we divide methods according to the dimension (Dim.) of an embedding vector.
    The numbers under the datasets refer to recall at K.
    }
    \label{tab:image_retrieval}
    \end{center}
    \centering
	\def1.1{1.1}
	\begin{tabular}{p{3.5cm}|C{1cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}|C{0.8cm}}
   \hline
    \multirow{2}{*}{Methods} &  \multirow{2}{*}{Dim.} & \multicolumn{4}{c|}{In-Shop} & \multicolumn{4}{c}{SOP} \\
   \cline{3-6}\cline{7-10}
       & & 1 & 10 & 20 & 30 & 1 & 10 & $10^2$ & $10^3$  \\
    \hline\hline
    Margin~\cite{wu2017sampling} & 128 & - & - & - & - & 72.7 & 86.2 & 93.8 & 98.0\\
      MIC+Margin~\cite{Roth_2019_ICCV} & 128 & 88.2 & 97.0 & - & - & 77.2 & 89.4 & 95.6 & - \\
      DC~\cite{sanakoyeu2019divide} & 128 & 85.7 & 95.5 & 96.9 & 97.5 & 75.9 & 88.4 & 94.9 & 98.1  \\
      ArcFace \cite{ArcFace} & 128 & 84.1 & 94.9 & 96.2 & 96.9 & 73.3 & 86.4 & 93.2 & 97.1 \\
      BroadFace & 128 & $\mathbf{89.8}$ & $\mathbf{97.4}$ & $\mathbf{98.1}$ & $\mathbf{98.4}$ & $\mathbf{79.7}$ & $\mathbf{90.7}$ & $\mathbf{95.7}$ & $\mathbf{98.4}$ \\
      \cline{1-10}
      TML~\cite{Yu_2019_ICCV} & 512 & - & - & - & - & 78.0 & $\mathbf{91.2}$ & $\mathbf{96.7}$ & $\mathbf{99.0}$ \\
      NSM~\cite{zhai2018making} & 512 & 88.6 & $\mathbf{97.5}$ & $\mathbf{98.4}$ & $\mathbf{98.8}$ & 78.2 & 90.6 & 96.2 & -\\
      ArcFace \cite{ArcFace}  & 512 & 87.3 & 96.3 & 97.3 & 97.9 & 76.9 & 89.1 & 95.0 & 98.2 \\
      BroadFace & 512 & $\mathbf{90.1}$ & 97.4 & 98.1 & 98.4 & $\mathbf{80.2}$ & 91.0 & 95.9 & 98.4 \\
   \hline
   \end{tabular}
\end{table}
\endgroup

\noindent\textbf{In-Shop and SOP}
are the standard datasets in image retrieval.
In-Shop contains 11,735 classes of clothes.
For training, the first 3,997 classes with 25,882 images are used, and the remaining 7,970 classes with 26,830 images are split into query set gallery set for evaluation.
SOP contains 22,634 classes of industrial products.
For training, the first 11,318 classes with 59,551 images are used and the remaining 11,316 classes with 60,499 images are used for evaluation.
Our baseline models that are trained with ArcFace \cite{ArcFace} underperform comparing to the other state-of-the-art methods.
BroadFace significantly improves the recall of the baseline models and the improved model even outperforms the other methods (Table \ref{tab:image_retrieval}).

\subsection{Analysis of BroadFace}

\begingroup
\newcolumntype{C}[1]{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
\begin{table}[t]
    \footnotesize
	\begin{center}
    	\caption{
            Effects of BroadFace varying the size of the queue and the type of the backbone network on IJB-B dataset in face recognition.
            \\
    	}
    	\label{tab:abl}
		\centering
		\subfloat[Total Size of Queue\label{tab:abl_a} ]
		{
			\centering
			\def1.1{1.1}
            \begin{tabular}[t]{p{3.6cm}|C{1.9cm}|C{1.9cm}|C{1.9cm}|C{1.9cm}}
			\hline
			 \multirow{3}{*}{Size of Queue}  &   \multicolumn{4}{c}{TAR} \\
			    \cline {2-5}
				&   \multicolumn{2}{c|}{FAR=1e-6} & \multicolumn{2}{c}{FAR=1e-5} \\
			    \cline {2-5}
				& \scalebox{0.8}{\parbox[][0.8cm][c]{1.9cm}{\centering without \newline Compensation}} & \scalebox{0.8}{\parbox[][0.8cm][c]{1.9cm}{\centering with \newline Compensation}} & \scalebox{0.8}{\parbox[][0.8cm][c]{1.9cm}{\centering without \newline Compensation}} & \scalebox{0.8}{\parbox[][0.8cm][c]{1.9cm}{\centering with \newline Compensation}} \\
				\hline
				\hline
				{0 (Baseline)}	& 0.3828 & 0.3828 & 0.8933 & 0.8933 \\
				{2048 ($512 \times 4$ GPUs) }	    	& 0.4310 & 0.4255  & 0.9061 & 0.9077 \\
				{8192 ($2048 \times 4$ GPUs) } 		& 0.4346 & 0.4394 & 0.9071 & 0.9085 \\
				{32768 ($8192\times4$ GPUs) } & 0.4259 & 0.4653 & 0.9078 & 0.9081 \\
				\hline
			\end{tabular}
		}
		\\
		\centering
		\subfloat[Backbone Network\label{tab:abl_b}]
		{
			\centering
			\def1.1{1.1}
			\begin{tabular}[t]{p{2.8cm}|C{1.0cm}|C{1.5cm}|C{1.5cm}|C{1.5cm}|C{1.5cm}|C{1.2cm}}
            \hline
				& \multirow{3}{*}{Dim.} &   \multicolumn{4}{c|}{TAR} & \multirow{3}{*}{GFlops}   \\
			    \cline{3-6}
				&  &   \multicolumn{2}{c|}{FAR=1e-6} & \multicolumn{2}{c|}{FAR=1e-5} &   \\
				 \cline{3-6}
				 &  &   \scalebox{0.9}{ArcFace} & \scalebox{0.9}{BroadFace} & \scalebox{0.9}{ArcFace} & \scalebox{0.9}{BroadFace} &   \\
				\hline\hline
				{MobileFaceNet \cite{MobileFaceNet}} & 128 & 0.3552 & 0.3665 & 0.8456 & 0.8458  & 0.9G \\
				{ResNet-18 \cite{resnet}}	& 128 & 0.3678 & 0.3808 & 0.8588 & 0.8638 & 5.2G \\
				{ResNet-34 \cite{resnet}}	& 512 & 0.3981 & 0.4325 & 0.8798 & 0.8828 & 8.9G \\
				ResNet-100 \cite{resnet}	& 512 & 0.3828 & 0.4653 & 0.8933 & 0.9081 & 24.1G \\
                \hline
			\end{tabular}
		}
		\end{center}
\end{table}
\endgroup

\noindent\textbf{Size of Queue.}
BroadFace has only one hyper-parameter, the size of the queue, to determine the maximum number of embedding vectors accumulated over past iterations.
Using the single parameter makes our method easy to tune and the parameter plays a very important role in determining recognition-accuracy.
As the size of the queue grows, the performance increases steadily (Table \ref{tab:abl_a}).
Especially, without our compensation method, accuracy degradation occurs when the size of the queue is significantly large.
However, our compensation method alleviates the degradation by correcting the enqueued embedding vectors.
We show another experiment on the enormous size of the queue from $0$ to $32,000$ in image retrieval (Fig. \ref{fig:img1}).
With the proposed compensation, the recall is consistently improved as the size of the queue is increased.
However, without the proposed compensation, the recall of the models degrades when the size of the queue is more than 16k.

\noindent\textbf{Generalization Ability.}
Our BroadFace is generally applicable to any objective functions and any backbone networks.
We apply BroadFace to two widely used objective functions of CosFace \cite{CosFace} and ArcFace \cite{ArcFace}.
For both CosFace and ArcFace, BroadFace increases recognition-accuracy (Table \ref{tab:ijb}).
We also apply BroadFace to several backbone networks such as MobileFaceNet \cite{MobileFaceNet}, ResNet-18 \cite{resnet} and ResNet-34 \cite{resnet}.
We set the dimensions of embedding vector to 128 for light backbone networks such as MobileFaceNet and ResNet-18, and 512 for heavy backbone networks such as ResNet-34 and ResNet-100.
BroadFace is significantly effective for all backbone networks (Table \ref{tab:abl_b}).
In particular, ResNet-34 trained with BroadFace achieves comparable performance to ResNet-100 trained only with ArcFace, even though ResNet-34 has much less GFlops.

\begin{figure}[t]
	\begin{center}
		\subfloat[Size of Queue]{
			\label{fig:img1}
			\includegraphics[width=5.5cm]{images/figure6a.pdf}
		}
		\subfloat[Learning Acceleration]{
			\label{fig:img2}
			\includegraphics[width=5.5cm]{images/figure6b.pdf}
		}
	\end{center}
	\caption{
	(a) the recall depending on the size of the queue in BroadFace with and without our compensation function; the red line indicates the recall of ArcFace (baseline) on the test set.
(b) the learning curve for the test set when the size of the queue is 32k; ArcFace reaches the highest recall at the $45^{th}$ epoch, our BroadFace reaches the highest recall at the $10^{th}$ epoch, and the learning process collapses without our compensation function.
	}
	\label{fig:img}
\end{figure}

\noindent\textbf{Learning Acceleration.}
Our BroadFace accelerates the learning process of both face recognition and image retrieval.
In face recognition, many iterations are still needed to overcome a small gap of performance among the methods on the highly-saturated datasets.
Thus, we experiment the acceleration of the learning process in image retrieval to clearly show the effectiveness (Fig. \ref{fig:img2}).
Our BroadFace reaches peak performance much faster and higher than the baseline model.
Without our compensation method, the model gradually collapses.