# Pull-back of singular Levi-flat hypersurfaces

Andrés Beltrán  Dpto. Ciencias - Sección Matemáticas, Pontifícia Universidad
Católica del Perú. Av. Universitaria 1801, San Miguel, Lima 32, Peru
<EMAIL_ADDRESS>,  Arturo Fernández-Pérez  Departamento de Matemática - ICEX,
Universidade Federal de Minas Gerais, UFMG Av. Antônio Carlos 6627, 31270-901,
Belo Horizonte-MG, Brasil<EMAIL_ADDRESS>and  Hernán Neciosup  Dpto.
Ciencias - Sección Matemáticas, Pontifícia Universidad Católica del Perú. Av.
Universitaria 1801, San Miguel, Lima 32, Peru<EMAIL_ADDRESS>
###### Abstract.

We study singular real analytic Levi-flat subsets invariant by singular
holomorphic foliations in complex projective spaces. We give sufficient
conditions for a real analytic Levi-flat subset to be the pull-back of a
semianalytic Levi-flat hypersurface in a complex projective surface under a
rational map or to be the pull-back of a real algebraic curve under a
meromorphic function. In particular, we give an application to the case of a
singular real analytic Levi-flat hypersurface. Our results improve previous
ones due to Lebl and Bretas – Fernández-Pérez – Mol.

###### Key words and phrases:

Levi-flat subsets - Holomorphic foliations

###### 2010 Mathematics Subject Classification:

Primary 32V40 - 32S65

This work was supported by the Pontifícia Universidad Católica del Perú
project VRI-DGI-2018-0024. The second author is partially supported by CNPq-
Brazil Grant Number 302790/2019-5

##  1\. Introduction and statement of the results

Let $M$ be a complex manifold of $\dim_{\mathbb{C}}M=N\geq 2$, a closed subset
$H\subset M$ is a real analytic subvariety if for every $p\in H$, there are
real analytic functions with real values $\varphi_{1},\ldots,\varphi_{k}$
defined in a neighborhood $U\subset M$ of $p$, such that $H\cap U$ is equal to
the set where all $\varphi_{1},\ldots,\varphi_{k}$ vanish. A complex
subvariety is precisely the same notion, considering holomorphic functions
instead of real analytic functions. We say that a real analytic subvariety $H$
is irreducible if whenever we write $H=H_{1}\cup H_{2}$ for two subvarieties
$H_{1}$ and $H_{2}$ of $M$, then either $H_{1}=H$ or $H_{2}=H$. If $H$ is
irreducible, it has a well-defined dimension $\dim_{\mathbb{R}}H$. Let
$H_{reg}$ denote its regular part, i.e., the subset of points near which $H$
is a real analytic submanifold of dimension equal to $\dim_{\mathbb{R}}H$. A
set is semianalytic if it is locally constructed from real analytic sets by
finite union, finite intersection, and complement. For a real analytic
subvariety $H$, the set $\overline{H_{reg}}$ is a semianalytic subset where
the closure is with the standard topology. In general, the inclusion
$\overline{H_{reg}}\subset H$ is proper, which happens, for instance in the
Whitney umbrella. We really only study the set $\overline{H_{reg}}$, in this
sense, we consider $\textsf{Sing}(H):=\overline{H_{reg}}\setminus H_{reg}$ as
the singular set of $H$, this is not the usual definition of the singular set
in the literature, see for instance [15].

If $H\subset M$ is a real analytic hypersurface i.e., a real analytic
subvariety of real codimension one, then for each $p\in H_{reg}$, there is a
unique complex hyperplane ${\mathscr{L}}_{p}\subset T_{p}H_{reg}$. This
defines a real analytic distribution $p\mapsto{\mathscr{L}}_{p}$ of complex
hyperplanes in $TH_{reg}$. When this distribution is integrable in the sense
of Frobenius, we say that $H$ is Levi-flat. Here, $H_{reg}$ is foliated by
codimension one immersed complex submanifolds. This foliation, denoted by
${\mathscr{L}}$, is known as Levi foliation. According to Cartan [4],
${\mathscr{L}}$ can be extended to a non-singular holomorphic foliation in a
neighborhood of $H_{reg}$ in $M$, but in general, it is not possible to extend
${\mathscr{L}}$ to a singular holomorphic foliation in a neighborhood of $H$.
There are examples of singular Levi-flat hypersurfaces whose Levi foliations
extend to singular holomorphic webs in the ambient space, see for instance [8]
and [21]. When there is a singular holomorphic foliation $\mathscr{F}$ in the
ambient space $M$ that coincide with the Levi foliation on $H_{reg}$, we say
either that $H$ is invariant by $\mathscr{F}$ or that $\mathscr{F}$ is tangent
to $H$. Cerveau and Lins Neto [6] proved that germs of singular foliations of
codimension one at $(\mathbb{C}^{N},0)$ tangent to real analytic Levi-flat
hypersurfaces have meromorphic (possibly holomorphic) first integrals. We
recall that a non-constant function $f$ is the first integral for a foliation
$\mathscr{F}$ if each leaf of $\mathscr{F}$ is contained in a level set of
$f$. In the global context, the same problem has been studied in [1] and [9].

The aim of this paper is to study holomorphic foliations tangent to real
analytic Levi-flat subsets in complex manifolds. An irreducible real analytic
subvariety $H\subset M$, where $M$ is an $N$-dimensional complex manifold,
$N\geq 2$, is a Levi-flat subset if it has real dimension $2n+1$ and its
regular part $H_{reg}$ is foliated by immersed complex manifolds of complex
dimension $n$. Similarly to the case of hypersurfaces, this foliation is
called Levi foliation of $H$ and will be denoted by ${\mathscr{L}}$. The
number $n$ is the Levi dimension of $H$. We use the qualifier “ _Levi_ ” for
the foliation, its leaves, and its dimension. Since we deal with real analytic
Levi-flat subsets in complex manifolds we shall consider that $H$ is coherent.
Coherence implies that $H$ admits a global complexification [11, p. 40]. Here
coherent means that its ideal sheaf $\mathcal{I}(H)$ in
$\mathcal{A}_{\mathbb{R},M}$, the sheaf of germs of real analytic functions
with real values in $M$, is a coherent sheaf of
$\mathcal{A}_{\mathbb{R},M}$-modules. It follows from Oka’s theorem [17, p. 94
Proposition 5] that $H$ is coherent if the sheaf $\mathcal{I}(H)$ is locally
finitely generated, the latter means that for every point $p\in H$ there
exists an open neighborhood $U\subset M$ and a finite number of functions
$\varphi_{j}$, real analytic in $U$ and vanishing on $H$, such that for any
$q\in U$, the germs of $\varphi_{j}$ at $q$ generate the ideal
$\mathcal{I}(H_{q})$, where $H_{q}$ is the germ of $H$ at $q$. We remark that
not every real analytic subset is coherent as we shall see in Section 3 of
this paper.

In [3], singular Levi-flat subsets appear in the result of the lifting of a
real analytic Levi-flat hypersurface to the projectivized cotangent bundle of
the ambient space through the Levi foliation and in [20], the authors gave a
complete characterization of dicritical singularities of local Levi-flat
subsets in terms of their Segre varieties.

Let $Y$ be a complex projective surface, $T\subset Y$ be a real analytic Levi-
flat hypersurface, $X\subset\mathbb{P}^{N}$, $N\geq 3$, be a complex
projective subvariety of complex dimension $k<N$ and $\rho:X\dashrightarrow Y$
be a dominant rational map. Then it is easy to show that
$H=\overline{\rho^{-1}(T)}$ is a real analytic Levi-flat subset in
$\mathbb{P}^{N}$ and so $H$ is a Levi-flat subset defined via pull-back.
Therefore, one natural question is:

Given a real analytic Levi-flat subset $H\subset\mathbb{P}^{N}$. Under what
condition, $H$ is given by the pull-back of a Levi-flat hypersurface in a
projective complex surface via a rational map?

In [14], Lebl gave sufficient conditions for a real analytic Levi-flat
hypersurface in $\mathbb{P}^{N}$ to be a pull-back of a real algebraic curve
in $\mathbb{C}$ via a meromorphic function. In [2], Bretas et al. proved an
analogous result for real analytic Levi-flat subsets in $\mathbb{P}^{N}$. The
main hypothesis in these articles is that the Levi foliation has infinitely
many algebraic leaves. In this paper, we give an answer to the question,
assuming that $H$ is invariant by a singular holomorphic foliation on
$\mathbb{P}^{N}$ with quasi-invariant subvarieties (see Section 2). An
irreducible complex subvariety $S\subset X$ of complex dimension $n$ is quasi-
invariant by a global $n$-dimensional foliation $\mathscr{F}$ on a complex
projective manifold $X$ if it is not $\mathscr{F}$-invariant, but the
restriction to the foliation $\mathscr{F}$ to $S$ is an algebraically
integrable foliation of dimension $n-1$, i.e. every leaf of $\mathscr{F}|_{S}$
is algebraic. The concept of quasi-invariant subvarieties was introduced by
Pereira-Spicer [19] for codimension one holomorphic foliations on complex
projective manifolds to prove a variant of the classical Darboux-Jouanolou
Theorem. Here we shall use this concept for Levi foliations to prove our main
result:

######  Theorem 1.

Let $H\subset\mathbb{P}^{N}$, $N\geq 3$, be an irreducible real analytic Levi-
flat subset of Levi dimension $n$ invariant by an $n$-dimensional singular
holomorphic foliation $\mathscr{F}$ on $\mathbb{P}^{N}$. Suppose that $H$ is
coherent and $n>N/2$. If the Levi foliation has infinitely many quasi-
invariant subvarieties of complex dimension $n$, then there exists a unique
projective subvariety $X$ of complex dimension $n+1$ containing $H$ such that
either there exists a rational map $R:X\dashrightarrow\mathbb{P}^{1}$, and
real algebraic curve $C\subset\mathbb{P}^{1}$ such that
$\overline{H_{reg}}\subset\overline{R^{-1}(C)}$ or there exists a dominant
rational map $\rho:X\dashrightarrow Y$ on a projective surface $Y$ and a
semianalytic Levi-flat subset $T\subset Y$ such that
$\overline{H_{reg}}\subset\overline{\rho^{-1}(T)}$.

We emphasize that the hypothesis $n>N/2$ implies that $H$ is necessarily a
real analytic subvariety with singularities. In fact, Ni-Wolfson [18, Theorem
2.4] proved that no nonsingular real analytic Levi-flat subset of the Levi
dimension $n$ exist in $\mathbb{P}^{N}$, $n>N/2$.

Applying Theorem 1 to $n=N-1$, we get the following corollary:

######  Corollary 1.

Let $H\subset\mathbb{P}^{N}$, $N\geq 3$, be an irreducible coherent real
analytic Levi-flat hypersurface invariant by a codimension one holomorphic
foliation $\mathscr{F}$ on $\mathbb{P}^{N}$. If the Levi foliation has
infinitely many quasi-invariant complex hypersurfaces, then either there
exists a rational map $R:\mathbb{P}^{N}\dashrightarrow\mathbb{P}^{1}$, and
real algebraic curve $C\subset\mathbb{P}^{1}$ such that
$\overline{H_{reg}}\subset\overline{R^{-1}(C)}$ or there exists a dominant
rational map $\rho:\mathbb{P}^{N}\dashrightarrow Y$ on a projective surface
$Y$ and a semianalytic Levi-flat subset $T\subset Y$ such that
$\overline{H_{reg}}\subset\overline{\rho^{-1}(T)}$.

When $H$ is a real analytic hypersurface, the above corollary gives a nice
characterization of coherent real analytic Levi-flat hypersurfaces in
$\mathbb{P}^{N}$, $N\geq 3$, invariant by codimension one holomorphic
foliations which admit infinitely many quasi-invariant complex hypersurfaces.
Observe that, in order to improve our results, we need to extend the Levi
foliation of a Levi-flat subset to a holomorphic foliation in the ambient
space. Therefore, another interesting question is:

Given a real analytic Levi-flat subset $H\subset\mathbb{P}^{N}$ with Levi
foliation $\mathcal{L}$. Under what condition, $\mathcal{L}$ extend to a
singular holomorphic foliation on $\mathbb{P}^{N}$?

When $H$ is a local real analytic Levi-flat hypersurface, Lebl solved the
above question in the non-dicritical case in [15].

The paper is organized as follows: in Section 2, we define the concept of
quasi-invariant subvarieties of a foliation with complex leaves and state the
main result of [19], such a result is key to prove Theorem 1. Section 3 is
devoted to the study of real analytic Levi-flat subset in complex manifolds,
using some results of [3] and [2], we prove the algebraic extension of the
intrinsic complexification of $H$. In Section 4, we prove Theorem 1 and in
Section 5 we prove Corollary 1. Finally, in Section 6, we give two examples.
The first is an example of a Levi-flat hypersurface where Theorem 1 applies.
In the second example, we construct a Levi-flat hypersurface in
$\mathbb{P}^{3}$ that is not a pull-back of a Levi-flat hypersurface of
$\mathbb{P}^{2}$ under a rational map. Moreover, this example also is not a
pull-back of a real algebraic curve under a meromorphic function.

##  2\. Foliations with complex leaves and quasi-invariant subvarieties

###  2.1. Foliations with complex leaves

A foliation with complex leaves of complex dimension $n$ is a smooth foliation
$\mathcal{G}$ of dimension $2n$ whose local models are domains $U=W\times B$
of $\mathbb{C}^{n}\times\mathbb{R}^{k}$, $W\subset\mathbb{C}^{n}$,
$B\subset\mathbb{R}^{k}$ and whose local transformations are of the form

(1) $\displaystyle\varphi(z,t)=(f(z,t),h(t)),$

where $f$ is holomorphic with respect to $z$. A domain $U$ as above is said to
be a distinguished coordinate domain of $\mathcal{G}$ and
$z=(z_{1},\ldots,z_{n})$, $t=(t_{1},\ldots,t_{k})$ are said to be
distinguished local coordinates. As examples of such foliations we have the
Levi foliations of Levi-flat hypersurfaces of $\mathbb{C}^{n}$, see for
instance [5] and [10].

If we replace $\mathbb{R}^{k}$ by $\mathbb{C}^{k}$ and in (1) we assume
$t\in\mathbb{C}^{k}$ and that $f,h$ are holomorphic with respect to $z$, $t$
then we get the notion of holomorphic foliation of complex codimension $k$.

Now we define foliations with singularities. Let $M$ be a complex manifold. A
singular foliation with complex leaves $\mathcal{G}$ of dimension $n$ on $M$
is a foliation with complex leaves of dimension $n$ on $M\setminus E$, where
$E$ is a real analytic subvariety of $M$ of real dimension $<2n$. A point
$p\in E$ is called a removable singularity of $\mathcal{G}$ of there is a
chart $(U,\varphi)$ around $p$, compatible with the atlas $\mathcal{A}$ of
$\mathcal{G}$ restricted to $M\setminus E$, in the sense that
$\varphi\circ\varphi^{-1}_{i}$ and $\varphi_{i}\circ\varphi^{-1}$ have the
form (1) for all $(U_{i},\varphi_{i})\in\mathcal{A}$ with $U\cap
U_{i}\neq\emptyset$. The set of non-removable singularities of $\mathcal{G}$
in $E$ is called the singular set of $\mathcal{G}$, and is denoted by
$\textsf{Sing}(\mathcal{G})$.

###  2.2. Quasi-invariant subvarieties

Let $Z$ be a projective manifold of complex dimension $N\geq 2$ and let
$\mathscr{G}$ be a foliation with complex leaves of dimension $n$ on $Z$.

######  Definition 2.1.

We say that $\mathscr{G}$ is an algebraically integrable foliation on $Z$ if
every leaf of $\mathscr{G}$ is algebraic, i.e. every leaf of $\mathscr{G}$ is
a projective complex subvariety in $Z$.

Motivated by [19], we define the concept of a subvariety quasi-invariant by a
real analytic foliation with complex leaves.

######  Definition 2.2.

An irreducible subvariety $S\subset Z$ of complex dimension $n$ is quasi-
invariant by a foliation $\mathscr{G}$ if it is not $\mathscr{G}$-invariant,
but the restriction of the foliation $\mathscr{G}$ to $S$ is an algebraically
integrable foliation.

We note that the restriction foliation $\mathscr{G}|_{S}$ is a codimension one
foliation on $S$ and when $\mathscr{G}|_{S}$ is an algebraically integrable
foliation, we have that every leaf of $\mathscr{G}|_{S}$ are projective
complex hypersurfaces in $S$. Codimension one holomorphic foliations on $Z$
which admit infinitely many quasi-invariant hypersurfaces have been studied in
[19] and its main result is the following.

######  Theorem 2.1 (Pereira-Spicer [19]).

Let $\mathscr{F}$ be a codimension one holomorphic foliation on a projective
manifold $Z$. If $\mathscr{F}$ admits infinitely many quasi-invariant
hypersurfaces then either $\mathscr{F}$ is an algebraically integrable
foliation, or $\mathscr{F}$ is a pull-back of a foliation of dimension one on
a projective surface under a dominant rational map.

##  3\. Real analytic subsets

###  3.1. Coherent real analytic subsets.

We present some of the fundamental results concerning coherent real analytic
subsets.

Let $H$ be a real analytic subset in an open set $U\subset\mathbb{C}^{n}$ and
let $\mathcal{I}(H)$ be its ideal sheaf, it is the sheaf of germs of real
analytic functions with real values vanishing on $H$.

######  Definition 3.1.

$H$ is said to be coherent if $\mathcal{I}(H)$ is a coherent sheaf of
$\mathcal{A}_{\mathbb{R},U}$-modules, where $\mathcal{A}_{\mathbb{R},U}$ is
the sheaf of germs of real analytic functions with real values in $U$.

######  Proposition 3.1.

[17, p. 95] If $H$ is a coherent real analytic subset and the germ $H_{p}$ of
$H$ at $p$ is irreducible, then for $q$ near $p$, we have

$\dim_{\mathbb{R}}H_{p}=\dim_{\mathbb{R}}H_{q}.$

It is well known that locally, a real analytic subset always admits a
complexification (see for instance [11, p. 40]) and it is not true for global
real analytic subsets. It is shown in [11, p. 54] that the global
complexification of a coherent real analytic subset in a complex manifold
always exists.

######  Theorem 3.2.

[11, p. 54] A real analytic subset in a complex manifold is coherent if and
only if it admits a global complexification.

Now we build an irreducible real analytic hypersurface in $\mathbb{P}^{3}$
which is not coherent. Let $[z_{0}:z_{1}:z_{2}:z_{3}]$ be the homogeneous
coordinates in $\mathbb{P}^{3}$ and set $H\subset\mathbb{P}^{3}$ be the
complex cone whose equation is

$H=\\{(z_{3}\bar{z}_{0}+\bar{z}_{3}z_{0})\left((z_{1}\bar{z}_{0}+\bar{z}_{1}z_{0})^{2}+(z_{2}\bar{z}_{0}+\bar{z}_{2}z_{0})^{2}\right)-(z_{1}\bar{z}_{0}+\bar{z}_{1}z_{0})^{3}=0\\}.$

The germ $H_{p}$ of $H$ at $p=[1:0:0:0]$ is irreducible and of real dimension
5 at $p$. However, in a neighborhood of $[1:0:0:z]$, $z\neq 0$, $H$ reduces to
the complex line $z_{1}=z_{2}=0$, which is of real dimension 2. By Proposition
3.1, it follows that $H$ is not coherent.

###  3.2. Levi-flat subset in complex manifolds.

We give a brief resume of definitions and some known results about real
analytic Levi-flat subsets in complex manifolds. Let $H$ be an irreducible
real analytic Levi-flat subset of Levi dimension $n$ in an $N$-dimensional
complex manifold $M$. The notion of Levi-flat subset germifies and, in
general, we do not distinguish a germ at $(\mathbb{C}^{N},0)$ from its
realization in some neighborhood $U$ of $0\in\mathbb{C}^{N}$. If $p\in
H_{reg}$ then, according to [2, Proposition 3.1], there exists a holomorphic
coordinate system
$z=(z^{\prime},z^{\prime\prime})\in\mathbb{C}^{n+1}\times\mathbb{C}^{N-n-1}$
such that $z(p)=0\in\mathbb{C}^{N}$ and the germ of $H$ at $p$ is defined by

(2)
$H=\\{z=(z^{\prime},z^{\prime\prime})\in\mathbb{C}^{n+1}\times\mathbb{C}^{N-n-1}:\
\text{Im}(z_{n+1})=0,\,\,\,\,z^{\prime\prime}=0\\},$

where $z^{\prime}=(z_{1},...,z_{n+1})$ and
$z^{\prime\prime}=(z_{n+2},...,z_{N})$ and the Levi foliation is given by

$\\{z=(z^{\prime},z^{\prime\prime})\in\mathbb{C}^{n+1}\times\mathbb{C}^{N-n-1}:\
z_{n+1}=c,\,\,\,\,z^{\prime\prime}=0,\ \text{with}\ c\in\mathbb{R}\\}.$

This trivial model is, in fact, a local form for a non-singular real analytic
Levi-flat subset. Note that in the local form (2), $\\{z^{\prime\prime}=0\\}$
corresponds to the unique local $(n+1)-$dimensional complex subvariety of the
ambient space containing the germ of $H_{reg}$ at $p$. These local
subvarieties glue together forming a complex variety defined in a whole
neighborhood of $H_{reg}$. It is analytically extendable to a neighborhood of
$\overline{H_{reg}}$ by the following theorem:

######  Theorem 3.3 (Brunella [3]).

Let $M$ be an $N-$dimensional complex manifold and $H\subset M$ be a real
analytic Levi-flat subset of Levi dimension $n$. Then, there exists a
neighborhood $V\subset M$ of $\overline{H_{reg}}$ and a unique complex variety
$X\subset V$ of dimension $n+1$ containing $H$.

The variety $X$ is the realization in the neighborhood $V$ of a germ of
complex analytic variety around $H$. We denote it — or its germ — by
$H^{\imath}$ and call it _intrinsic complexification_ or _$\imath$
-complexification_ of $H.$ It plays a central role in the theory of real
analytic Levi-flat subsets. The notion of intrinsic complexification also
appears in [22] with the name of the _Segre envelope_. If $H$ is invariant by
a holomorphic foliation on $M$, the same holds for its
$\imath$-complexification, see for instance [2, Proposition 3.3].

######  Proposition 3.4.

Let $H\subset M$ be a real analytic Levi-flat subset of Levi dimension $n$,
where $M$ is a complex manifold of dimension $N$. If $H$ is invariant by an
$n$-dimensional holomorphic foliation $\mathcal{F}$ on $M$, then its
$\imath$-complexification $H^{\imath}$ is also invariant by $\mathcal{F}$.

As a consequence, if we denote by
$\mathscr{F}^{\imath}:=\mathscr{F}|_{H^{\imath}}$ (the restriction of
$\mathscr{F}$ to $H^{\imath}$), we have $\mathscr{F}^{\imath}$ has codimension
one in $H^{\imath}$. The following proposition shows the importance of the
assumption of the coherence of a Levi-flat subset.

######  Proposition 3.5.

[2, Proposition 3.6] Let $M$ be an $N$-dimensional complex manifold and
$H\subset M$ be an irreducible real analytic Levi-flat subset of Levi
dimension $n$. Suppose that $H$ is coherent. Then, there exist an open
neighborhood $V\subset M$ of $H$ and a unique irreducible complex subvariety
$X$ of $V$ of complex dimension $n+1$ containing $H$.

The variety $X$ is the small variety of complex dimension $n+1$ that contains
$H$. Again, let us denote this variety by $H^{\imath}$, the intrinsic
complexification of $H$.

###  3.3. Levi-flat subsets in complex projective spaces

In this subsection, we state some results of real analytic Levi-flat subset in
$\mathbb{P}^{N}$. Let $\sigma:\mathbb{C}^{N+1}\to\mathbb{P}^{N}$ be the
natural projection. Suppose that $H$ is a real-analytic subvariety of
$\mathbb{P}^{N}$. Define the set $\tau(H)$ to be the set of points
$z\in\mathbb{C}^{N+1}$ such that $\sigma(z)\in H$ or $z=0$. A real analytic
subvariety $H\subset\mathbb{P}^{N}$ is said to be algebraic if $H=\sigma(V)$
for some real algebraic complex cone $V$ in $\mathbb{C}^{N+1}$. A set $V$ is a
complex cone when $p\in V$ implies $\lambda p\in V$ for all
$\lambda\in\mathbb{C}$.

The following construction offers several examples of Levi-flat subsets in
$\mathbb{P}^{N}$.

######  Proposition 3.6.

[2, Proposition 6.1] Let $X\subset\mathbb{P}^{N}$ be an irreducible
$(n+1)$-dimensional algebraic variety, $R$ be a rational function in $X$ and
$C\subset\mathbb{P}^{1}$ be a real algebraic one-dimensional subvariety. Then
the set $\overline{R^{-1}(C)}$ is a real algebraic Levi-flat subset of Levi
dimension $n$ whose $\imath$-complexification is $X$

When we add the hypothesis that the Levi-flat subset is invariant by a
singular holomorphic foliation in the ambient space, we can state a reciprocal
result.

######  Proposition 3.7.

[2, Proposition 6.3] Let $\mathscr{F}$ be a singular holomorphic foliation in
$\mathbb{P}^{N}$ tangent to a real analytic Levi-flat subset $H$ of Levi
dimension $n$. Suppose that $H$ is coherent and its $\imath$-complexification
extends to an algebraic subvariety $H^{\imath}$ in $\mathbb{P}^{N}$. If
$\mathscr{F}^{\imath}$ has a rational first integral $R$, then there exists a
real algebraic one-dimensional subvariety $C\subset\mathbb{P}^{1}$ such that
$\overline{H_{reg}}\subset\overline{R^{-1}(C)}$.

Now, since $H$ is coherent, the intrinsic complexification $H^{\imath}$ is
well-defined as a complex subvariety in a neighborhood of $H$. Our aim is to
extend $H^{\imath}$ to an algebraic subvariety in $\mathbb{P}^{N}$. To get
this, we use the following extension theorem.

######  Theorem 3.8 (Chow [7]).

Let $Z\subset\mathbb{P}^{N}$ be a complex algebraic subvariety of dimension
$k$ and $V$ be a connected neighborhood of $Z$ in $\mathbb{P}^{N}$. Then any
complex analytic subvariety of dimension higher than $N-k$ in $V$ that
intersects $Z$ extends algebraically to $\mathbb{P}^{N}$.

Under certain hypotheses, we can prove that the $\imath$-complexification
$H^{\imath}$ can be extended to $\mathbb{P}^{N}$.

######  Proposition 3.9.

Let $H\subset\mathbb{P}^{N}$, $N\geq 3$, be an irreducible coherent real
analytic Levi-flat subset of Levi dimension $n$ such that $n>N/2$. If the Levi
foliation ${\mathscr{L}}$ has a quasi-invariant complex algebraic subvariety
of complex dimension $n$, then $H^{\imath}$ extends algebraically to
$\mathbb{P}^{N}$.

###### Proof.

Denote by $L$ such quasi-invariant algebraic complex subvariety with
$\dim_{\mathbb{C}}L=n-1$. Since $L$ algebraic with $L\subset H^{\imath}$ and
$\dim_{\mathbb{C}}H^{\imath}=n+1>N-(n-1)$, we can apply Theorem 3.8 to prove
that $H^{\imath}$ extends algebraically to $\mathbb{P}^{N}$. ∎

To end this section, we shall prove the following proposition.

######  Proposition 3.10.

Let $H\subset\mathbb{P}^{N}$ be an irreducible coherent real analytic Levi-
flat subset of Levi dimension $n$ invariant by an $n$-dimensional singular
holomorphic foliation $\mathscr{F}$ in $\mathbb{P}^{N}$. Suppose that the
$\imath$-complexification $H^{\imath}$ extends to an algebraic variety in
$\mathbb{P}^{N}$. If the Levi-foliation $\mathcal{L}$ has infinitely many
quasi-invariant algebraic subvarieties of complex dimension $n-1$. Then,
either the foliation $\mathscr{F}^{\iota}=\mathscr{F}|_{H^{\imath}}$ has a
rational first integral in $H^{\imath}$, or $\mathscr{F}^{\iota}$ is a pull-
back of a foliation on a projective surface under a dominant rational map.

###### Proof.

First of all, we need to desingularize the $\imath$-complexification
$H^{\imath}$. According to Hironaka desingularization theorem, there exist a
complex manifold $\widetilde{H^{\imath}}$ and a proper bimeromorphic morphism
$\pi:\widetilde{H^{\imath}}\to H^{\imath}$ such that

  1. (1)

$\pi:\widetilde{H^{\imath}}\setminus(\pi^{-1}(\textsf{Sing}(H^{\imath}))\to
H^{\imath}\setminus\textsf{Sing}(H^{\imath})$ is a biholomorphism,

  2. (2)

$\pi^{-1}(\textsf{Sing}(H^{\imath}))$ is a simple normal crossing divisor.

Since $H^{\imath}$ is compact then $\widetilde{H^{\imath}}$ is too. We lift
$\mathscr{F}^{\imath}$ to an $n$-dimensional singular holomorphic foliation
$\widetilde{\mathscr{F}^{\imath}}$ on $\widetilde{H^{\imath}}$. Since
$\dim_{\mathbb{C}}\widetilde{H^{\imath}}=n+1$, we have
$\widetilde{\mathscr{F}^{\imath}}$ has codimension one on
$\widetilde{H^{\imath}}$ and the tangency condition between
$\mathscr{F}^{\imath}$ and $H$ implies that $\mathscr{F}^{\imath}$ has
infinitely many quasi-invariant closed subvarieties (these are algebraic and
of codimension one in $\widetilde{H^{\imath}}$). Thus the same holds for
$\widetilde{\mathscr{F}^{\imath}}$. By Theorem 2.1, either
$\widetilde{\mathscr{F}^{\imath}}$ has a rational first integral or there
exist a dominant rational map
$\tilde{\rho}:\widetilde{H^{\imath}}\dashrightarrow Y$, where $Y$ is a
projective complex surface, $\mathcal{G}$ is a foliation by curves on $Y$ and
$\widetilde{\mathscr{F}^{\imath}}=\tilde{\rho}^{*}(\mathcal{G})$. If
$\widetilde{\mathscr{F}^{\imath}}$ admits a rational first integral in
$\widetilde{H^{\imath}}$, then all leaves of
$\widetilde{\mathscr{F}^{\imath}}$ are compact and so their $\pi$-images are
compact leaves of $\mathscr{F}^{\imath}$ in $H^{\imath}$. Applying Gómez-
Mont’s theorem [12], we have that there exists a one-dimensional projective
manifold $S$ and a rational map $f:H^{\imath}\dashrightarrow S$ whose fibers
contain the leaves of $\mathscr{F}^{\imath}$. A rational first integral is
obtained by composing $f$ with any non-constant rational map
$r:S\dashrightarrow\mathbb{P}^{1}$. If $\widetilde{\mathscr{F}^{\imath}}$ is a
pull-back of a foliation $\mathcal{G}$ on a projective complex surface $Y$
under a dominant rational map
$\tilde{\rho}:\widetilde{H^{\imath}}\dashrightarrow Y$ then
$\mathscr{F}^{\imath}$ is the pull-back of $\mathcal{G}$ under
$\rho:=\tilde{\rho}\circ\pi^{-1}:H^{\imath}\dashrightarrow Y$, since $\pi$ is
a birational map. ∎

##  4\. Proof of Theorem 1

With all the above results, we can prove Theorem 1.

######  Theorem 1.

Let $H\subset\mathbb{P}^{N}$, $N\geq 3$, be an irreducible real analytic Levi-
flat subset of Levi dimension $n$ invariant by an $n$-dimensional singular
holomorphic foliation $\mathscr{F}$ on $\mathbb{P}^{N}$. Suppose that $H$ is
coherent and $n>N/2$. If the Levi foliation has infinitely many quasi-
invariant subvarieties of complex dimension $n$, then there exists a unique
projective subvariety $X$ of complex dimension $n+1$ containing $H$ such that
either there exists a rational map $R:X\dashrightarrow\mathbb{P}^{1}$, and
real algebraic curve $C\subset\mathbb{P}^{1}$ such that
$\overline{H_{reg}}\subset\overline{R^{-1}(C)}$ or there exists a dominant
rational map $\rho:X\dashrightarrow Y$ on a projective surface $Y$ and a
semianalytic Levi-flat subset $T\subset Y$ such that
$\overline{H_{reg}}\subset\overline{\rho^{-1}(T)}$.

###### Proof.

By Proposition 3.5, there exist an open neighborhood $V\subset\mathbb{P}^{N}$
of $H$ and a unique irreducible complex subvariety $H^{\imath}$ of $V$ of
complex dimension $n+1$ containing $H$. The Proposition 3.4 implies that
$H^{\imath}$ is invariant by $\mathscr{F}$ and moreover it extends
algebraically to $\mathbb{P}^{N}$ by Proposition 3.9. We denote
$\mathscr{F}^{\imath}:=\mathscr{F}|_{H^{\imath}}$ the restrict foliation to
$H^{\imath}$. Observe now that $\mathscr{F}^{\imath}$ is a foliation of
codimension one on $H^{\imath}$ which admit infinitely many quasi-invariant
subvarieties of complex dimension $n-1$. Therefore, either
$\mathscr{F}^{\iota}$ has a rational first integral in $H^{\imath}$, or
$\mathscr{F}^{\iota}$ is a pull-back of a foliation on a projective surface
under a dominant rational map by Proposition 3.10.

If $\mathscr{F}^{\imath}$ has a first integral $R$ then there exists a real
algebraic curve $C\subset\mathbb{P}^{1}$ such that
$\overline{H_{reg}}\subset\overline{R^{-1}(C)}$ by Proposition 3.7. Now if we
assume that $\mathscr{F}^{\imath}$ is a pull-back of a foliation $\mathcal{G}$
on a projective complex surface $Y$ under a dominant rational map
$\rho:H^{\imath}\dashrightarrow Y$. Then we can take $X=H^{\imath}$. Let us
prove that there exists a semianalytic Levi-flat subset $T\subset Y$. Indeed,
let $z\in H_{reg}\setminus Ind(\rho)$ (here $Ind(\rho)$ denotes the
indeterminacy set of $\rho$). Then there exists a neighborhood $U\subset
H^{\imath}\setminus Ind(\rho)$ of $z$ and a non-singular real analytic curve
$\gamma:(-\epsilon,\epsilon)\to U$ such that $\gamma(0)=z$,
$\\{\gamma\\}\subset H_{reg}$, and such that $\gamma$ is transverse to the
Levi foliation ${\mathscr{L}}$ on $H_{reg}$. Let $L_{\gamma(t)}$ be the leaf
of ${\mathscr{L}}$ through $\gamma(t)$. Since $L_{\gamma(t)}$ is also a leaf
of $\mathscr{F}^{\imath}$ and $\mathscr{F}^{\imath}=\rho^{*}(\mathcal{G})$,
then $\rho(L_{\gamma(t)})$ is a leaf of $\mathcal{G}$. Let us denote
$A_{t}=\overline{\rho(L_{\gamma(t)})}\subset Y$ and define

$T_{z}:=\bigcup_{t\in(-\epsilon,\epsilon)}A_{t}\subset V_{z},$

where $V_{z}$ is a neighborhood of $T_{z}$ on $Y$. Note that $T_{z}$ is a
union of complex subvarieties parametrized by $t$ such that each $A_{t}$
contains leaves of $\mathcal{G}$, thus $T_{z}$ is a semianalytic Levi-flat
subset on $V_{z}$. These local constructions are sufficiently canonical to be
patched together when $z$ varies on $H_{reg}$: if $T_{z_{1}}\subset V_{z_{1}}$
and $T_{z_{2}}\subset V_{z_{2}}$ are as above, with $V_{z_{1}}\cap
V_{z_{2}}\neq\emptyset$, then $T_{z_{1}}\cap V_{z_{1}}\cap V_{z_{2}}$ and
$T_{z_{2}}\cap V_{z_{1}}\cap V_{z_{2}}$ have some common leaves of
$\mathcal{G}$ because $\mathcal{G}$ is a global foliation defined on $Y$, so
$T_{z_{1}}$ and $T_{z_{2}}$ can be glued by identifying these leaves. In this
way, we get a semianalytic Levi-flat subset $T$ in $Y$.

Finally, we assert that $\overline{H_{reg}}\subset\overline{\rho^{-1}(T)}$. In
fact, let $w\in\overline{H_{reg}}$, then there exists a sequence $z_{k}\to w$,
$z_{k}\in H_{reg}$, so $\rho(z_{k})\in T$ which imply that
$z_{k}\in\rho^{-1}(T)$ and $w\in\overline{\rho^{-1}(T)}$. This finishes the
proof. ∎

##  5\. Proof of Corollary 1

######  Corollary 1.

Let $H\subset\mathbb{P}^{N}$, $N\geq 3$, be an irreducible coherent real
analytic Levi-flat hypersurface invariant by a codimension one holomorphic
foliation $\mathscr{F}$ on $\mathbb{P}^{N}$. If the Levi foliation has
infinitely many quasi-invariant complex hypersurfaces, then either there
exists a rational map $R:\mathbb{P}^{N}\dashrightarrow\mathbb{P}^{1}$, and
real algebraic curve $C\subset\mathbb{P}^{1}$ such that
$\overline{H_{reg}}\subset\overline{R^{-1}(C)}$ or there exists a dominant
rational map $\rho:\mathbb{P}^{N}\dashrightarrow Y$ on a projective complex
surface $Y$ and a semianalytic Levi-flat subset $T\subset Y$ such that
$\overline{H_{reg}}\subset\overline{\rho^{-1}(T)}$.

###### Proof.

If $H$ is an irreducible real analytic Levi-flat hypersurface in
$\mathbb{P}^{N}$, $N\geq 3$, then the Levi dimension of $H$ is $N-1$. Moreover

$N-1>N/2\iff N>2.$

Thus, we can apply Theorem 1 to $H$, so there exist a unique projective
subvariety $X$ of complex dimension $N$ containing $H$ such that either there
exists a rational map $R:X\dashrightarrow\mathbb{C}$, and real algebraic curve
$C\subset\mathbb{C}$ such that $\overline{H_{reg}}\subset\overline{R^{-1}(C)}$
or there exists a dominant rational map $\rho:X\dashrightarrow Y$ on a
projective complex surface $Y$ and a semianalytic Levi-flat subset $T\subset
Y$ such that $\overline{H_{reg}}\subset\overline{\rho^{-1}(T)}$. Since
$X\subset\mathbb{P}^{N}$ has complex dimension $N$, we must have
$X=\mathbb{P}^{N}$ and hence we conclude the proof. ∎

##  6\. Examples

######  Example 6.1.

We give an example of a real analytic Levi-flat hypersurface in
$\mathbb{P}^{3}$ where Theorem 1 applies. Let

$H=\\{[z_{0}:z_{1}:z_{2}:z_{3}]\in\mathbb{P}^{3}:z_{0}z_{1}\bar{z}_{2}\bar{z}_{3}-z_{2}z_{3}\bar{z}_{0}\bar{z}_{1}=0\\},$

then $H$ is Levi-flat because it is foliated by the complex hypersurfaces

(3) $\displaystyle z_{0}z_{1}=cz_{2}z_{3},\quad\text{where}\quad
c\in\mathbb{R}.$

Let $\mathscr{F}$ be the codimension one holomorphic foliation on
$\mathbb{P}^{3}$ of degree two defined by

$\omega=z_{1}z_{2}z_{3}dz_{0}+z_{0}z_{2}z_{3}dz_{1}-z_{0}z_{1}z_{3}dz_{2}-z_{0}z_{1}z_{2}dz_{3},$

then $\mathscr{F}$ has a rational first integral
$R:\mathbb{P}^{3}\dashrightarrow\mathbb{P}^{1}$ given by

$R[z_{0}:z_{1}:z_{2}:z_{3}]=[z_{0}z_{1}:z_{2}z_{3}].$

Since the leaves of $\mathscr{F}|_{H}$ coincide with the leaves of the Levi
foliation (3), $H$ must be invariant by $\mathscr{F}$. On the other hand, note
that $H=\overline{R^{-1}(C)}$, where

$C=\\{[t:u]\in\mathbb{P}^{1}:t\bar{u}-u\bar{t}=0\\}.$

######  Example 6.2.

In the following example, we construct a real analytic Levi-flat hypersurface
$H$ in $\mathbb{P}^{3}$ that is not a pull-back of a Levi-flat hypersurface of
$\mathbb{P}^{2}$ under a rational map, furthermore, $H$ also is not a pull-
back of a real algebraic curve under a meromorphic function.

Consider $z=(z_{0},z_{1},z_{2},z_{3})$,
$\bar{z}=(\bar{z}_{0},\bar{z}_{1},\bar{z}_{2},\bar{z}_{3})$ and

$F(z,\bar{z})=\det\left(\begin{array}[]{cccccc}z_{0}&z_{1}&z_{2}&z_{3}&0&0\\\
0&z_{0}&z_{1}&z_{2}&z_{3}&0\\\ 0&0&z_{0}&z_{1}&z_{2}&z_{3}\\\
\bar{z}_{0}&\bar{z}_{1}&\bar{z}_{2}&\bar{z}_{3}&0&0\\\
0&\bar{z}_{0}&\bar{z}_{1}&\bar{z}_{2}&\bar{z}_{3}&0\\\
0&0&\bar{z}_{0}&\bar{z}_{1}&\bar{z}_{2}&\bar{z}_{3}\end{array}\right)$

Define $H=\\{[z_{0}:z_{1}:z_{2}:z_{3}]\in\mathbb{P}^{3}:F(z,\bar{z})=0\\}$,
$H$ is a real analytic hypersurface well defined since $F$ is a bihomogeneous
polynomial of bi-degree $(3,3)$. Moreover, $H$ is Levi-flat, because it is
foliated by the complex hyperplanes

(4) $\displaystyle z_{0}+cz_{1}+c^{2}z_{2}+c^{3}z_{3}=0,\quad\text{where}\quad
c\in\mathbb{R}.$

Let $\mathscr{W}$ be the codimension one holomorphic $3$-web on
$\mathbb{P}^{3}$ given by the implicit differential equation $\Omega=0$,

$\Omega=\det\left(\begin{array}[]{cccccc}z_{0}&z_{1}&z_{2}&z_{3}&0&0\\\
0&z_{0}&z_{1}&z_{2}&z_{3}&0\\\ 0&0&z_{0}&z_{1}&z_{2}&z_{3}\\\
dz_{0}&dz_{1}&dz_{2}&dz_{3}&0&0\\\ 0&dz_{0}&dz_{1}&dz_{2}&dz_{3}&0\\\
0&0&dz_{0}&dz_{1}&dz_{2}&dz_{3}\end{array}\right)$

Since the leaves of $\mathscr{W}|_{H}$ and ${\mathscr{L}}$ are the same, we
get $H$ is invariant by $\mathscr{W}$.

Now, we prove that $H$ is not a pull-back of a Levi-flat hypersurface of
$\mathbb{P}^{2}$. To prove this fact, we use the following result of [13,
Proposition 4.4]:

######  Proposition 6.1.

Let $\omega_{1}$, $\omega_{2}$ and $\omega_{3}$ be independent germs of
integrable 1-forms at $(\mathbb{C}^{3},0)$ with singular sets of codimension
at least two. Suppose that there exists a non-zero holomorphic 2-form $\eta$,
locally decomposable outside its singular set, that is tangent to each
$\omega_{i}$, for $i=1,2,3$. Then $\omega_{1}$, $\omega_{2}$ and $\omega_{3}$
define foliations that are in a pencil. Furthermore, $\eta$ is integrable,
defining the axis foliation of this pencil.

Suppose by contradiction that $H$ is a pull-back of a Levi-flat hypersurface
under a dominant rational map
$\rho:\mathbb{P}^{3}\dashrightarrow\mathbb{P}^{2}$. Then pick a point $p\in
U_{0}$, where $U_{0}$ is an open subset in $\mathbb{P}^{3}$ such that
$\rho|_{U_{0}}:U_{0}\subset\mathbb{C}^{3}\to\mathbb{C}^{2}$ is a holomorphic
submersion. We may have needed to perhaps move to yet another point
$p^{\prime}\in U_{0}$ such that $U_{0}$ does not intersect the discriminant
set of the web $\mathscr{W}$. We set $p=p^{\prime}$ and works in a
neighborhood of $U_{0}$. Therefore, the germ of $\mathscr{W}$ at $p$ is a
decomposable 3-web, defined by the superposition of three independent
foliations $\mathscr{F}_{1}$, $\mathscr{F}_{2}$, and $\mathscr{F}_{3}$. We can
assume that these foliations are defined by independent germs of integrable
1-forms $\omega_{1}$, $\omega_{2}$, and $\omega_{3}$ respectively. Since $H$
is given by a pull-back, all the leaves of ${\mathscr{L}}$ and, hence the
leaves of $\mathscr{W}$ in $H\cap U_{0}$ are tangent to the fibers of
$\rho|_{U_{0}}$, these fibers define a non-zero holomorphic 2-form
$\eta_{\rho}$ that is tangent to each $\omega_{i}$, for $i=1,2,3$. Then,
according to Proposition 6.1, $\omega_{1}$, $\omega_{2}$, and $\omega_{3}$
define foliations that are in a pencil, an absurd. Hence, the assertion is
proved.

Now we assert that $H$ is not a pull-back of a real algebraic curve under a
meromorphic function. In fact, $H$ is a Levi-flat hypersurface in
$\mathbb{P}^{3}$ such that there does not exist a point contained in
infinitely many leaves of ${\mathscr{L}}$, because, the leaves of
${\mathscr{L}}$ are given by the equation (4) and through at a point only pass
three leaves. If $H$ is defined by a pull-back of a meromorphic function,
there has to exist a point $p$ of indeterminacy since the dimension is at
least 2. Then through at $p$ pass infinitely many leaves of ${\mathscr{L}}$.
Since $H$ does not satisfy this property, we finish the proof of the
assertion.

Acknowledgments. The authors wish to express his gratitude to Maria Aparecida
Soares Ruas (ICMC - USP, São Carlos) and Judith Brinkschulte (Universität
Leipzig) for many valuable conversations and suggestions.

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# Lossy Compression with Distortion Constrained Optimization

Ties van Rozendaal, Guillaume Sautière, Taco S. Cohen
Qualcomm AI Research, Qualcomm Technologies Netherlands B.V.
{ties, gsautie<EMAIL_ADDRESS>Qualcomm AI Research is an initiative
of Qualcomm Technologies, Inc. and/or its subsidiaries

###### Abstract

When training end-to-end learned models for lossy compression, one has to
balance the rate and distortion losses. This is typically done by manually
setting a tradeoff parameter $\beta$, an approach called $\beta$-VAE. Using
this approach it is difficult to target a specific rate or distortion value,
because the result can be very sensitive to $\beta$, and the appropriate value
for $\beta$ depends on the model and problem setup. As a result, model
comparison requires extensive per-model $\beta$-tuning, and producing a whole
rate-distortion curve (by varying $\beta$) for each model to be compared.

We argue that the constrained optimization method of Rezende and Viola, 2018
[29] is a lot more appropriate for training lossy compression models because
it allows us to obtain the best possible rate subject to a distortion
constraint. This enables pointwise model comparisons, by training two models
with the same distortion target and comparing their rate. We show that the
method does manage to satisfy the constraint on a realistic image compression
task, outperforms a constrained optimization method based on a hinge-loss, and
is more practical to use for model selection than a $\beta$-VAE.

(a)  $R$/$D$ performance for constrained optimization and hinge loss
baselines. Dashed lines indicate the distortion target value $c_{D}$.  (b)
$R$/$D$ performance for constrained optimization and $\beta$-VAE.

Figure 1: Training rate / distortion performance for the same model trained
using different optimization methods.

##  1 Introduction

Deep latent variable models have started to outperform conventional baselines
on lossy compression of images [4, 7, 25, 14, 15, 24, 23, 33, 36], video [19,
8, 21, 31, 37, 20, 27, 6, 12], and audio [39, 36]. Nearly all of these methods
use a loss function of the form $D+\beta R$, where $D$ measures distortion,
$R$ measures bitrate, and $\beta$ is a fixed tradeoff parameter. We refer to
this approach as $\beta$-VAE [13], because this loss can be motivated from a
variational perspective [12].

Despite its popularity, $\beta$-VAE has several drawbacks. Firstly, setting
$\beta$ to target a specific point in the $R$/$D$ plane can be tricky. One can
show that a model trained with a given $\beta$ should end up at that point on
the $R$/$D$ curve where the slope $\frac{\partial R}{\partial D}$ equals
$\beta$ [1]. However, because the shape of the $R$/$D$ curve depends on the
model and hyperparameters, and because the $R$/$D$ curve can be very steep or
flat in the low or high bitrate regime, choosing $\beta$ can be difficult.

Secondly, in order to compare models it is not sufficient to train one
instance of each model because the converged models would likely differ in
both rate and distortion, which yields inconclusive results unless one model
dominates the other on both metrics. Instead, to compare models we need to
train both at several $\beta$ values to generate $R$/$D$ curves that can be
compared, which is computationally costly and slows down the research
iteration cycle.

A more natural way to target different regions of the $R$/$D$ plane is to set
a distortion _constraint_ and find our model parameters through constrained
optimization:

$\displaystyle\min_{\boldsymbol{\theta}}R(\boldsymbol{\theta})\text{ s.t.
}D(\boldsymbol{\theta})\leq c_{D},$ (1)

where $\boldsymbol{\theta}$ refers to the joint parameters of the encoder,
decoder and prior, and $c_{D}$ is a distortion target.

We can control the rate-distortion tradeoff by setting the distortion target
value $c_{D}$. Setting this value is more intuitive than setting $\beta$, as
it is independent of the slope of the $R$/$D$ curve, and hence independent of
model and hyperparameters.

As a result, we can easily compare two different models trained with the same
distortion constraint; as we have fixed the $D$ axis we only have to look at
the $R$ performance for each model.

Note that one could also minimize the distortion subject to a rate constraint.
This is less straightforward as putting too much emphasis on the rate loss at
the beginning of training can lead to posterior collapse [3, 11, 40, 28, 32].

There is a large literature on constrained optimization, but most of it does
not consider stochastic optimization and is limited to convex loss functions.
In this paper we evaluate, in addition to $\beta$-VAE, two constrained
optimization methods that are compatible with stochastic gradient descent
training of deep networks: A simple method based on the hinge loss (free bits
[17, 5, 1] but applied to distortion rather than rate), and the Lagrangian
distortion-constrained optimization method of [29] ($D$-CO).

We evaluate these methods on a modern image compression system applied to a
realistic compression benchmark. We report on suitable hyperparameters and
practical considerations that are relevant in this domain. We show that $D$-CO
outperforms the hinge method, and reaches a similar performance to
$\beta$-VAE. At the same time, $D$-CO is easier to work with and allows for
pointwise model selection.

##  2 Related Work

###  2.1 Constrained Optimization

Several works have proposed algorithms to train deep networks under equality
or inequality constraints [22, 10, 9, 29]. We deploy the algorithm of [29] as
the VAE context is most similar to our setup.

The focus of [29] is on generative modelling rather than data compression, and
there are a number of reasons why the models trained in [29] are not directly
applicable to data compression. Firstly, their models contain a stochastic
encoder which is not suitable for lossy compression, where bits-back coding is
inapplicable. Secondly, [29] do not report $R$/$D$ performance but instead
report log-likelihood. Furthermore, their latent space is continuous while
most compression papers use a discrete latent space that allows for entropy
coding of the latents under the prior. Lastly, they use a fixed Guassian prior
whereas in lossy compression powerful learnable priors are used to decrease
the bitrate as much as possible. In this paper we focus on the implementation
and evaluation of constrained optimization for practical lossy image
compression.

###  2.2 Hinge Loss

Another approach that was proposed for constrained optimization (in the
context of avoiding posterior collapse) is free-bits, where the rate loss is
hinged $D+\beta\max(R-c_{R},0)$ [17, 5, 1]. Like constrained optimization,
this loss allows us to set a target value, and as such has been used in lossy
compression [23]. However, we find that this method is inferior to constrained
optimization in terms of $R$/$D$ performance and has difficulty converging to
the target value.

###  2.3 Variable Bitrate Models

A different approach of dealing with the rate-distortion tradeoff is to train
a single model that can compress at different bitrates [34, 30, 7, 38].
However, some of these works do not meet the performance achieved with
specialized models [38] or require disjoint training of autoencoder and prior
[34]. Other methods could benefit from constrained optimization (e.g. [38]
still uses $\beta$ multipliers that could be replaced by a distortion target),
an exercise that is left for future research.

(a)  $D$-CO: $\min_{\boldsymbol{\theta}}R(\boldsymbol{\theta})$ s.t.
$D(\boldsymbol{\theta})\leq c_{d}$ (b)  $\beta$-VAE:
$\min_{\boldsymbol{\theta}}D(\boldsymbol{\theta})+\beta
R(\boldsymbol{\theta})$

Figure 2: Model selection using (a) distortion constrained optimization and
(b) $\beta$-VAE. The baseline model (dark line) is altered by halving the
number of latent channels (light line). Models that are trained using the same
distortion target (a) or the same value of $\beta$ (b) are connected using a
dotted line.

##  3 Method

###  3.1 Constrained Optimization

The Lagrangian of the primal problem in equation 1 is:

$\displaystyle\mathcal{L}_{\text{Lagrangian}}(\boldsymbol{\theta},\lambda^{D})=R(\boldsymbol{\theta})+\lambda^{D}\left(D(\boldsymbol{\theta})-c_{D}\right)$
(2)

For a convex problem, we would find the minimum of the dual at
$\max_{\lambda^{D}}\min_{\boldsymbol{\theta}}\mathcal{L}_{\text{Lagrangian}}(\boldsymbol{\theta},\lambda^{D})$.

For non-convex deep learning models, we deploy the algorithm proposed by [29]
and iteratively update $\boldsymbol{\theta}$ and $\lambda^{D}$ using
stochastic gradient descent and ascent respectively.

Note that the $\beta$-VAE loss is the Lagrangian of a rate-constrained
optimization. However, the multiplier $\lambda^{R}=\beta$ is either fixed or
updated according to a heuristic schedule [3, 11, 40, 28, 32], and thus no
constrained optimization is performed.

Because we found that the optimal $D$-CO hyperparameters were different
depending on the target value, we normalize our constraint function by the
target value. Our loss function thus becomes:

$\displaystyle\mathcal{L}_{D-\text{CO}}(\boldsymbol{\theta},\lambda^{D})=R(\boldsymbol{\theta})+\lambda^{D}\left(\dfrac{D(\boldsymbol{\theta})}{c_{D}}-1\right)$
(3)

####  3.1.1 Weight and Multiplier Updates

For each minibatch, we update $\boldsymbol{\theta}$ using the Adam optimizer,
and $\lambda^{D}$ using SGD with momentum, to respectively minimize and
maximize the batchwise Lagrangian (Eq. 3).

Like [29], we reparametrize $\mu^{D}=\log\lambda^{D}$ in order to enforce the
positivity of $\lambda^{D}$ (to satisfy the K.K.T. [18, 16] conditions for
inequality constraints). We also follow them in updating
$\Delta\mu^{D}_{t+1}\propto\frac{\partial\mathcal{L}}{\partial\lambda^{D}_{t}}=\frac{D_{t}}{c_{D}}-1$
as this resulted in smoother updates of our multipliers than using the actual
gradient $(\frac{D_{t}}{c_{D}}-1)\exp{(\mu^{D}_{t})}$.

We use a high momentum ($\alpha=0.99$) for our multiplier updates, to ensure a
smooth multiplier trajectory despite the high variance of the MSE loss. As we
use the PyTorch [26] SGD implementation, we make sure to set dampening to be
equal to momentum. We clip our log-multiplier s.t. $\lambda^{D}\leq 10^{3}$
for stability.

Unlike [29] we choose to set our initial value of $\lambda^{D}$ to the clip
value $\lambda^{D}_{0}=10^{3}$. This way, we focus on training the autoencoder
for distortion at the beginning of training, which we found to be essential
for high performance. The final multiplier trajectories are shown in Figure 3.

Figure 3: Trajectory of Lagrangian multipliers during training. Each curve
represents a model with a different constraint.

##  4 Experiments

We conduct a series of experiments to show how constrained optimization is
more suitable for training lossy compression models than $\beta$-VAE or
distortion hinge baselines.

###  4.1 General Setup

We use the autoencoder architecture of [23] but without the mask. Our prior is
the gated pixelCNN [35] as used in [12]. Like [12] we jointly train our code-
model and autoencoder, without any detaching of the gradients. We use scalar
quantization with a learned codebook and a straight-trough estimator (hardmax
during forward pass and softmax gradient during backward) [2, 23, 12].

We train our model on random 160x160 crops of ImageNet Train, and evaluate on
160x160 center crops of ImageNet Validation. Like [23] we resize the smallest
side of all images to 256 to reduce compression artifacts.

We train using the rate loss expressed in bits per pixel (bpp) and using the
distortion loss expressed in average MSE computed on unnormalized images on a
0-255 scale.

We update our parameters using Adam with a learning rate of $2\cdot 10^{-3}$
for the autoencoder and $1\cdot 10^{-4}$ for the prior. We decay both learning
rates every 3 epochs (120087 iterations) by a factor of $0.1$. For the
multiplier updates, we use SGD with a learning rate of $5\cdot 10^{-3}$. We
use a batch size of 32.

###  4.2 $D$-CO vs. $D$-hinge

For this experiment we choose exponentially spaced $D$-constraint values (60,
65, 70, 80, 100, 125, 150, 200, 300 MSE) and look at how well the methods
converge to the set target. We compare our $D$-CO training with the simpler
$D$-hinge baselines of the form:

$\displaystyle\mathcal{L}_{D\text{-hinge}}(\boldsymbol{\theta})=R(\boldsymbol{\theta})+\lambda^{D}\max\left(\dfrac{D(\boldsymbol{\theta})}{c_{D}}-1,0\right)$
(4)

Unlike $D$-CO, $\lambda^{D}$ is fixed during training, but we train models
with different values (0.01, 0.1, 1, 10, 100). In line with $D$-CO, we use the
normalized constraint function as we verified that it worked better than the
unnormalized one.

Results are shown in Figure 1a. Observe that the $D$-CO models converge very
closely to the set target (within 1 MSE point for achievable constraints). For
the hinge models, the constraint is not satisfied reliably and overall $R$/$D$
performance is worse (some models converged to $R$/$D$ values outside of the
chosen display range). Furthermore, the hinge models are sensitive to the
value of $\lambda_{D}$, and the optimal value differs per target.

Figure 3 shows the trajectories of the $D$-CO multipliers. For stricter
constraints, it takes longer before the multiplier starts to drop, changing
emphasis from $D$ to $R$. In the limit of an unachievable constraint (MSE
$<62$), the multiplier remains constant at the clip value. All multipliers
converge to a relatively stable final value, which is dependent on the target
(as expected since the $\frac{\partial R}{\partial D}$ slope is different).

###  4.3 $D$-CO vs. $\beta$-VAE

In the next experiment, we compare the $R$/$D$ performance of $D$-CO to the
$\beta$-VAE baseline. We first train $\beta$-VAE models for exponentially
spaced $\beta$ values (0.1, 10, 50, 100, 200, 250, 500, 750). For each
$\beta$-VAE, we use the distortion loss over the last training epoch as the
target for training a $D$-CO model.

Results are shown in Figure 1b (PSNR results in Figure A.1). The $R$/$D$
performance of the $D$-CO models is similar to that of the $\beta$-VAEs. For
bitrates higher than 0.4 bpp, we see a slight advantage for the $\beta$-VAE.
For these target values, the $D$-CO multipliers are almost constant (see the
strict constraints in Figure 3) and we thus attribute this difference to the
optimization hyperparameters being fine-tuned for the scale of the $\beta$-VAE
loss.

###  4.4 Model Selection

In the final experiment we highlight how constrained optimization can simplify
the model selection process. We adapt our architecture by changing the number
of latent channels from $32$ to $16$, effectively halving the maximum channel
capacity from 1.29 bpp to 0.64 bpp. We train $\beta$-VAE models for the
$\beta$s from Section 4.3 and $D$-CO models using the targets from Section
4.2.

Results are shown in Figure 2. For both optimization methods, the lowest
achievable distortion has increased from $\sim 70$ MSE to $\sim 130$ MSE for
the model with decreased channel capacity.

For the $\beta$-VAE optimization, points with the same $\beta$ now end up at
very different points on the $R$/$D$ plane. For the half-capacity model, we
cover a narrow range of 240-128 MSE.

In contrast, $D$-CO produces two comparable $R$/$D$ curves. Distortion targets
below 130 MSE are unachievable for the half-capacity model and are all
collapsed into a single point. However, for any achievable distortion target,
both models end up with a similar distortion which allows us to do a pointwise
comparison.

##  5 Conclusion

We present distortion constrained optimization ($D$-CO) as an alternative to
$\beta$-VAE training for lossy compression. We report suitable hyperparameters
and propose to normalize the constraint function for better performance. We
demonstrate that $D$-CO gives similar performance to $\beta$-VAE on a
realistic image compression task, while at the same time providing a more
intuitive way to balance the rate and distortion losses. Finally, we show how
$D$-CO can facilitate the model selection process by allowing pointwise model
comparisons.

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##  Appendix A Supplementary Material

###  A.1 PSNR Results

Figure A.1: Rate / PSNR performance for $D$ constrained optimization and
$\beta$-VAE. Figure A.2: Validation Rate / PSNR performance for $D$
constrained optimization and $\beta$-VAE.