# Remarks on pseudo-continuity

John Cotrina Universidad del Pacífico, Lima, Perú. Email<EMAIL_ADDRESS>
###### Abstract

In this work we extend a maximum theorem proposed by Morgan and Scalzo. We
also show some results of minimax inequalities which are equivalent to the
famous Ky Fan minimax inequality. Additionally, we prove that the existence
result of Nash equilibria proposed by Morgan and Scalzo is actually equivalent
to a classical result in the literature.

Keywords: Maximum theorem; Minimax inequality; Nash games; Pseudo-continuity

MSC (2010): 47H10; 54C60; 91A10

##  1 Introduction

Historically, the concept of pseudo-continuity was introduced by Morgan and
Scalzo [18], where they used first the name _sequential pseudo-continuity_.
They, in [17, 21], showed that pseudo-continuity for functions is equivalent
to continuity for preference relations, among other properties. They also
studied, in [20], the asymptotical behaviour of finite economies. Moreover,
they generalized the Tikhonov well-posed result for optimization problems in
[19].

The notion of pseudo-continuity has gained more attention, for instance Anh,
Khanh and Van [3] studied the well-posedness for quasi-equilibrium problems
and quasi-optimization problems. Wangkeeree, Bantaojai and Yimmuang [29] dealt
with continuity properties of the set-valued solution map for a special kind
of quasi-equilibrium problem. Al-Homidan, Hadjisavvas and Shaalan [1]
implicitly used the concept of pseudo-continuity in order to characterize
quasi-convex functions as a composition of two functions where one of them has
the property that every local minimum is a global minimum. Recently, in [5]
the authors used the pseudo-continuity assumption in order to establish the
existence of solution for a particular generalized Nash game proposed by Rosen
[26], which generalizes many existence results in the literature.

The Berge maximum theorem [2] is an important tool in the area of general
equilibrium theory and in mathematical economics. It establishes that the
argmax correspondence is upper hemicontinuous and its value function is
continuous, under certain continuity assumptions. In [17, 21], Morgan and
Scalzo presented a maximum theorem under a pseudo-continuity assumption.
However, they do not say anything about the value function. In that sense, we
extend the result proposed by Morgan and Scalzo showing that the value
function is pseudo-continuous.

Relaxing the continuity assumption allows us to deal with a large class of
problems in optimization and game theory. One of the first existence result of
Nash equilibria for discontinuous games is due Dasgupta and Maskin [9], which
generalizes the one given by Debreu [10] for continuous games. In a similar
way, Morgan and Scalzo [21] presented a generalization of Debreu’s result
using pseudo-continuity, despite their result being a consequence of the one
proposed by Reny [25]. However, we will show that their result can be obtain
from Debreu’s theorem thanks to a Scalzo’ result in [27]. On the other hand,
Qiu and Peng [23] showed a result of minimax inequality under the pseudo-
continuity assumption. We will show that this result is actually equivalent to
Ky Fan’s minimax inequality.

The remainder of the paper is organized as follows. In Section 2, we introduce
the concept of pseudo-continuity for functions and continuity correspondences
used in our study. Section 3 is devoted to the Berge maximum theorem. In
Section 4 we present some results about pseudo-continuity, which are
equivalent to the famous Ky Fan’s minimax inequality. Finally, in Section 5 we
show that the results on the existence of Nash equilibria proposed by Morgan
and Scalzo [17, 21], Debreu [10] and, Arrow and Debreu [4] are equivalent.

##  2 Definitions, Notations and Preliminary results

Given a topological space $X$, an extended real-valued function
$f:X\to\mathbb{R}\cup\\{\pm\infty\\}$ is said to be:

  * •

_upper semi-continuous_ if, for any $x\in X$ and any $\lambda\in\mathbb{R}$
such that $f(x)<\lambda$, there exists a neighbourhood $\mathscr{V}_{x}$ of
$x$ satisfying

$f(x^{\prime})<\lambda,\mbox{ for all }x^{\prime}\in\mathscr{V}_{x};$

  * •

_upper pseudo-continuous_ if, for any $x,y\in X$ such that $f(x)<f(y)$, there
exists a neighbourhood $\mathscr{V}_{x}$ of $x$ satisfying

$f(x^{\prime})<f(y),\mbox{ for all }x^{\prime}\in\mathscr{V}_{x};$

  * •

_transfer upper continuous_ if, for any $x,y\in X$ such that $f(x)<f(y)$,
there exist a neighbourhood $\mathscr{V}_{x}$ of $x$, and $y^{\prime}\in X$
satisfying

$f(x^{\prime})<f(y^{\prime}),\mbox{ for all }x^{\prime}\in\mathscr{V}_{x}.$

######  Remark 2.1.

Clearly, upper semi-continuity implies upper pseudo-continuity, which in turn
implies transfer upper continuity. The converse implications are not true in
general. The concept of pseudo-continuity was introduced by Morgan and Scalzo
in [21, 17], while the transfer continuity was studied by Tian and Zhou in
[28].

On the other hand, it is well known that the sum of two upper semi-continuous
function is upper semi-continuous. However, in [8] the authors showed that the
same does not hold for transfer upper continuity. In a similar way, it does
not hold for upper pseudo-continuity, a contra-example can be found in [29].

Associated to the function $f$ we consider the following set

$U_{f}(\lambda)=\\{x\in X:~{}f(x)\geq\lambda\\},$

where $\lambda\in\mathbb{R}$. It is clear that a function $f$ is upper semi-
continuous if, and only if, the set $U_{f}(\lambda)$ is closed, for all
$\lambda\in\mathbb{R}$. In a similar way, $f$ is upper pseudo-continuous if,
and only if, the set $U_{f}(\lambda)$ is closed, for all $\lambda\in f(X)$,
see [21, 17]. Moreover, in [28] the authors showed that $f$ is transfer upper
continuous if, and only if,

$\bigcap_{x\in X}U_{f}(x)=\bigcap_{x\in X}\overline{U_{f}(x)},$

where we use $U_{f}(x)$ instead $U_{f}(f(x))$.

The function $f$ is said to be lower (pseudo) semi-continuous, if $-f$ is
upper (pseudo) semi-continuous. Also, a real-valued function is (pseudo)
continuous if, and only if, it is both upper and lower (pseudo) semi-
continuous.

The following result is Theorem 3.2 in [27].

######  Proposition 2.2 (Scalzo).

Let $X$ be a connected topological space and $f:X\to\mathbb{R}$ be a function.
Then $f$ is pseudo-continuous if, and only if, there exist a continuous
function $u:X\to\mathbb{R}$ and a increasing function $h:u(X)\to\mathbb{R}$
such that

$f=h\circ u.$

######  Remark 2.3.

A similar result to the previous one is Proposition 4 in [11].

Thanks to Proposition 2.2 in [20], we can state Rader’s utility representation
[24] as follow.

######  Proposition 2.4 (Rader).

Let $X$ be a topological space with a countable base and $f:X\to\mathbb{R}$ be
a function. Then $f$ is upper pseudo-continuous if, and only if, there exist a
upper semi-continuous function $u:X\to\mathbb{R}$ and an increasing function
$h:u(X)\to\mathbb{R}$ such that

$f=h\circ u.$

Given a convex set $X$ of a vector space, a function
$f:X\to\mathbb{R}\cup\\{\pm\infty\\}$ is said to be _quasi-concave_ if, for
all $\lambda\in\mathbb{R}$, $U_{f}(\lambda)$ is convex. Also, $f$ is called
_quasi-convex_ if, $-f$ is quasi-concave.

As a consequence of Proposition 2.4, we give an answer to a question proposed
by Al-Homidan, Hadjisavvas and Shaalan in [1]. Before that we need to
introduce the following definition: A function
$f:\mathbb{R}^{n}\to\mathbb{R}\cup\\{-\infty\\}$ is said to be _neatly quasi-
concave_ [1] if it is quasi-concave and for every $x$ with $f(x)<\sup f$, the
sets $U_{f}(x)$ and $U_{f}^{<}(x)=\\{y\in\mathbb{R}^{n}:~{}f(y)>f(x)\\}$ have
the same closure.

The following result is Theorem 4.1 in [1], but we state it in terms of quasi-
concavity instead of quasi-convexity.

######  Theorem 2.5.

For every quasi-concave and upper pseudo-continuous function
$f:\mathbb{R}^{n}\to\mathbb{R}\cup\\{-\infty\\}$, there exist a neatly quasi-
concave and upper pseudo-continuous function $g:\mathbb{R}^{n}\to\mathbb{R}$,
and a nonincreasing function
$h:g(\mathbb{R}^{n})\to\mathbb{R}\cup\\{-\infty\\}$ such that $f=h\circ g$.

The question proposed by Al-Homidan, Hadjisavvas and Shaalan in [1] is the
following: _Is it possible to choose an upper semicontinuous function $g$ in
the previous theorem when $f$ is upper semicontinuous_? The following
corollary gives a positive answer to this question.

######  Corollary 2.6.

For every quasi-concave and upper pseudo-continuous function
$f:\mathbb{R}^{n}\to\mathbb{R}\cup\\{-\infty\\}$, there exist a neatly quasi-
concave and upper semicontinuous function $g:\mathbb{R}^{n}\to\mathbb{R}$, and
a nonincreasing function $h:g(\mathbb{R}^{n})\to\mathbb{R}\cup\\{-\infty\\}$
such that $f=h\circ g$.

###### Proof.

From Theorem 2.5, there exist a neatly quasi-concave function
$g_{1}:\mathbb{R}^{n}\to\mathbb{R}$, and a nonincreasing function
$h_{1}:g(\mathbb{R}^{n})\to\mathbb{R}\cup\\{-\infty\\}$ such that
$f=h_{1}\circ g_{1}$. Moreover, the function $g_{1}$ is upper pseudo-
continuous. Thus, Proposition 2.4 guarantees the existence of an upper semi-
continuous function $g:\mathbb{R}^{n}\to\mathbb{R}$ and an increasing function
$v:g(\mathbb{R}^{n})\to\mathbb{R}$ such that $g_{1}=v\circ g$. It is not
difficult to see that $g$ is neatly quasi-concave. Since $h_{1}\circ v$ is a
nonincreasing function, the result follows. ∎

Given a convex set $X$ of a vector space and a function
$f:X\to\mathbb{R}\cup\\{\pm\infty\\}$, the _quasi-concave regularization_
$f_{q}$ of $f$ is defined as

$f_{q}(x)=\sup\\{\lambda\in\mathbb{R}:~{}x\in\operatorname*{conv}(U_{f}(\lambda))\\},$

where $\operatorname*{conv}(U_{f}(\lambda))$ means the convex hull of
$U_{f}(\lambda)$. The function $f_{q}$ is the smallest quasi-concave function
which is lower bounded by $f$.

######  Lemma 2.7.

Let $X$ be a convex set of a vector space and $f,g:X\to\mathbb{R}$ be two
functions such that $f\leq g$. If $g$ is quasi-concave, then $f_{q}$ is a
real-valued function. Moreover, $f\leq f_{q}\leq g$.

###### Proof.

It follows from definition of quasi-concave regularization. ∎

We finish this section recalling continuity notions for correspondences.
Before that, we introduce the concept of correspondence.

Let $U,V$ be non-empty sets. A _correspondence_ or _set-valued map_
$T:U\rightrightarrows V$ is an application $T:U\to\mathcal{P}(V)$, that is,
for $u\in U$, $T(u)\subset V$. The _graph_ of $T$ is defined as

$\operatorname{gra}(T)=\big{\\{}(u,v)\in U\times V\>:\>v\in T(u)\big{\\}}.$

Let $T:X\rightrightarrows Y$ be a correspondence with $X$ and $Y$ two
topological spaces. The map $T$ [2] is said to be:

  * •

_closed_ , when $\operatorname{gra}(T)$ is a closed subset of $X\times Y$;

  * •

_lower hemicontinuous_ when for all $x\in X$ and any open set $V\subset Y$,
with $T(x)\cap V\neq\emptyset$, there exists $\mathscr{V}_{x}$ neighbourhood
of $x$ such that $T(x^{\prime})\cap V\neq\emptyset$ for all
$x^{\prime}\in\mathscr{V}_{x}$;

  * •

_upper hemicontinuous_ when for all $x\in X$ and any open set $V$, with
$T(x)\subset V$, there exists $\mathscr{V}_{x}$ neighbourhood of $x$ such that
$T(\mathscr{V}_{x})\subset V$;

  * •

_continuous_ when it is upper and lower hemicontinuous.

We state below the well known Kakutani’s fixed point theorem, see [14, 13].

######  Theorem 2.8 (Kakutani-Fan-Glicksberg).

Let $X$ be a non-empty convex and compact subset of a Hausdorff locally convex
topological vector space $Y$ and let $T:X\rightrightarrows X$ be a
correspondence. If $T$ is upper hemicontinuous with convex, closed and non-
empty values, then there exists $x_{0}\in X$ such that $x_{0}\in T(x_{0})$.

The following result is an extension to the previous one, see [16] for more
details.

######  Theorem 2.9 (Fan-Browder).

Let $X$ be a compact, convex and nonempty subset of a topological vector space
and $T:X\rightrightarrows X$ be a correspondence with convex values. If $T$
has open fibres, i.e. $\\{x\in X:y\in T(x)\\}$ is open for all $y\in X$, then
there exists $x_{0}\in X$ such that $T(x_{0})=\emptyset$ or $x_{0}\in
T(x_{0})$.

##  3 On the Morgan-Scalzo-Berge maximum theorem

The Berge maximum theorem can be stated as follows (see [2]).

######  Theorem 3.1.

Let $X$, $Y$ be two topological spaces, $S:X\rightrightarrows Y$ be a
continuous correspondence with non-empty and compact values, and $f:X\times
Y\to\mathbb{R}$ be a continuous function. Then the “argmax” correspondence
$M:X\rightrightarrows Y$, defined as

$\displaystyle M(x)=\\{y\in S(x):f(x,y)=m(x)\\}$ (1)

is upper hemicontinuous and has non-empty compact values. Moreover, the “value
function” $m:X\to\mathbb{R}$ defined as

$\displaystyle m(x)=\max_{y\in S(x)}f(x,y)$ (2)

is continuous.

Let $X$ and $Y$ be topological spaces and $T:X\rightrightarrows Y$ be a set-
valued map. A function $f:X\times Y\to\mathbb{R}$ is called _quasi-transfer
upper continuous_ [28] on $T$ if, for all
$(x,y),~{}(x,z)\in\operatorname{gra}(T)$ with $f(x,y)<f(x,z)$, there exists a
neighbourhood $\mathscr{V}_{(x,y)}$ such that for any
$(x^{\prime},y^{\prime})\in\mathscr{V}_{(x,y)}\cap\operatorname{gra}(T)$ there
is $z^{\prime}\in T(x^{\prime})$ satisfying

$f(x^{\prime},y^{\prime})<f(x^{\prime},z^{\prime}).$

Tian and Zhou [28] noticed that if $f$ is continuous and $T$ is lower
hemicontinuous, then $f$ is quasi-transfer upper continuous on $T$. In a
similar way, we have the same result with pseudo-continuity instead continuity
of $f$.

######  Proposition 3.2.

Let $X$ and $Y$ be topological spaces, $T:X\rightrightarrows Y$ be a set-
valued map and $f:X\times Y\to\mathbb{R}$ be a function. If $f$ is pseudo-
continuous and $T$ is lower hemicontinuous, then $f$ is quasi-transfer upper
continuous on $T$.

###### Proof.

Let $(x,y),~{}(x,z)\in\operatorname{gra}(T)$ such that $f(x,y)<f(x,z)$. If
there exists $(a,b)\in\operatorname{gra}(T)$ such that

$\displaystyle f(x,y)<f(a,b)<f(x,z)$ (3)

then by pseudo-continuity of $f$ there are neighbourhoods
$\mathscr{V}_{x},\mathscr{V}_{y}$ and $\mathscr{V}_{z}$ respectively of $x,y$
and $z$, such that

$\displaystyle f(x^{\prime},y^{\prime})<f(a,b)<f(x^{\prime},z^{\prime})\mbox{
for all
}(x^{\prime},y^{\prime},z^{\prime})\in\mathscr{V}_{x}\times\mathscr{V}_{y}\times\mathscr{V}_{z}.$
(4)

Since $\mathscr{V}_{y}\cap T(x)\neq\emptyset$ and $\mathscr{V}_{z}\cap
T(x)\neq\emptyset$, we deduce that there is a neighbourhood
$\hat{\mathscr{V}}_{x}$ of $x$ such that

$\mathscr{V}_{y}\cap T(x^{\prime})\neq\emptyset\mbox{ and }\mathscr{V}_{z}\cap
T(x^{\prime})\neq\emptyset,\mbox{ for all
}x^{\prime}\in\hat{\mathscr{V}}_{x},$

due to the lower hemicontinuity of $T$. We set
$\mathscr{U}_{x}=\mathscr{V}_{x}\cap\hat{\mathscr{V}}_{x}$, and we can see
that for all $(x^{\prime},y^{\prime})\in\mathscr{U}_{x}\times\mathscr{V}_{y}$,
there exists $z^{\prime}\in\mathscr{V}_{z}\cap T(x^{\prime})$ such that (4)
holds.

If there is not $(a,b)\in\operatorname{gra}(T)$ such that (3) holds, then
there exist neighbourhoods $\mathscr{V}_{x},\mathscr{V}_{y}$ and
$\mathscr{V}_{z}$ respectively of $x,y$ and $z$, such that

$\displaystyle f(x^{\prime},y^{\prime})<f(x,z)\mbox{ and
}f(x,y)<f(x^{\prime},z^{\prime})\mbox{ for all
}(x^{\prime},y^{\prime},z^{\prime})\in\mathscr{V}_{x}\times\mathscr{V}_{y}\times\mathscr{V}_{z},$

because $f$ is pseudo-continuous. This implies

$\displaystyle f(x^{\prime},y^{\prime})\leq f(x,y)\mbox{ and }f(x,z)\leq
f(x^{\prime},z^{\prime}),$ (5)

for all
$(x^{\prime},y^{\prime},z^{\prime})\in\mathscr{V}_{x}\times\mathscr{V}_{y}\times\mathscr{V}_{z}$
such that
$(x^{\prime},y^{\prime}),(x^{\prime},z^{\prime})\in\operatorname{gra}(T)$.
Now, following the same steps in the previous case, thanks the lower semi-
continuity of $T$, there is a neighbourhood $\hat{\mathscr{V}}_{x}$ of $x$
such that

$\mathscr{V}_{y}\cap T(x^{\prime})\neq\emptyset\mbox{ and }\mathscr{V}_{z}\cap
T(x^{\prime})\neq\emptyset,\mbox{ for all
}x^{\prime}\in\hat{\mathscr{V}}_{x}.$

Take $\mathscr{U}_{x}=\mathscr{V}_{x}\cap\hat{\mathscr{V}}_{x}$, thus for all
$(x^{\prime},y^{\prime})\in\mathscr{U}_{x}\times\mathscr{V}_{y}\cap\operatorname{gra}(T)$,
there exists $z^{\prime}\in\mathscr{V}_{z}\cap T(x^{\prime})$ such that (5)
holds. Hence $f(x^{\prime},y^{\prime})<f(x^{\prime},z^{\prime})$, and this
proves that $f$ is quasi-transfer upper continuous on $T$. ∎

We state below a generalization of Berge’s maximum theorem due to Tian and
Zhou [28].

######  Theorem 3.3.

Let $X$ and $Y$ be two topological spaces, $T:X\rightrightarrows Y$ be a non-
empty compact-valued closed correspondence and $f:X\times Y\to\mathbb{R}$ be a
function. Then the best response correspondence $M$, defined as in (1), is
non-empty, compact-valued and closed if, and only if, the function
$f(x,\cdot)$ is transfer upper continuous on $T(x)$, for every $x\in X$; and
$f$ is quasi-transfer upper continuous in $(x,y)$ with respect to $T$. If, in
addition $T$ is upper hemicontinuous, then so is $M$.

The following result is an extension of Theorem 3.1 in [17] and a
generalization of Theorem 3.1 in [21].

######  Theorem 3.4.

Let $X$ and $Y$ be two topological spaces, $T:X\rightrightarrows Y$ be a
correspondence and $f:X\times Y\to\mathbb{R}$ be a function. If $f$ is pseudo-
continuous and $T$ is continuous with non-empty and compact values, then the
argmax correspondence $M$, defined as in (1), is upper hemicontinuous and the
value function $m$, defined as in (2), is real-valued and pseudo-continuous.

###### Proof.

From Proposition 3.2 and Theorem 3.3, we deduce that the argmax correspondence
$M$ is upper hemicontinuous.

Since for each $x$, the set $T(x)$ is non-empty and compact, the function
$f(x,\cdot)$ attains it maximum. Thus $m$ is real-valued. Now, we will show
that $m$ is pseudo-continuous. First, we will prove that $m$ is lower pseudo-
continuous. Let $x_{1}$ and $x_{2}$ be two elements of $X$ such that
$m(x_{1})>m(x_{2})$. There exist $y_{1}\in T(x_{1})$ and $y_{2}\in T(x_{2})$
such that

$f(x_{1},y_{1})=m(x_{1})>m(x_{2})=f(x_{2},y_{2}).$

Since $f$ is lower pseudo-continuous, there are neighbourhoods
$\mathscr{V}_{x_{1}}$ and $\mathscr{V}_{y_{1}}$, respectively of $x_{1}$ and
$y_{1}$, such that

$f(x^{\prime},y^{\prime})>f(x_{2},y_{2}),\mbox{ for all
}(x^{\prime},y^{\prime})\in\mathscr{V}_{x_{1}}\times\mathscr{V}_{y_{1}}.$

On the other hand, as $\mathscr{V}_{y_{1}}\cap T(x_{1})\neq\emptyset$ there is
a neighbourhood $\hat{\mathscr{V}}_{x_{1}}$ of $x_{1}$ satisfying

$\mathscr{V}_{y_{1}}\cap T(x^{\prime})\neq\emptyset,\mbox{ for all
}x^{\prime}\in\hat{\mathscr{V}}_{x_{1}},$

this is a consequence of the lower semi-continuity of $T$. Therefore, we
deduce that for all
$(x^{\prime},y^{\prime})\in(\mathscr{U}_{x_{1}}\times\mathscr{V}_{y_{1}})\cap\operatorname{gra}(T)$,
where $\mathscr{U}_{x_{1}}=\hat{\mathscr{V}}_{x_{1}}\cap\mathscr{V}_{x_{1}}$,
the following holds

$m(x^{\prime})\geq f(x^{\prime},y^{\prime})>f(x_{2},y_{2})=m(x_{2}).$

Finally, we will prove that $m$ is upper pseudo-continuous. Let $x_{1}$ and
$x_{2}$ be two elements of $X$ such that $m(x_{1})<m(x_{2})$. There exist
$y_{1}\in T(x_{1})$ and $y_{2}\in T(x_{2})$ such that

$f(x_{1},y_{1})=m(x_{1})<m(x_{2})=f(x_{2},y_{2}).$

We distinguish the following two cases.

First, if there exists $(x_{0},y_{0})\in X\times Y$ such that
$m(x_{1})<f(x_{0},y_{0})<m(x_{2})$. We have that $f(x_{1},y)<f(x_{0},y_{0})$,
for all $y\in K(x_{1})$. Thus, there are open neighbourhoods $\mathscr{V}_{y}$
and $\mathscr{V}_{x_{1}}^{y}$, respectively of $y$ and $x_{1}$ such that

$\displaystyle f(x^{\prime},y^{\prime})<f(x_{0},y_{0}),\mbox{ for all
}(x^{\prime},y^{\prime})\in\mathscr{V}_{x_{1}}^{y}\times\mathscr{V}_{y}.$ (6)

The family of set $\\{\mathscr{V}_{y}\\}_{y\in K(x_{1})}$ is an open covering
of $K(x_{1})$, which is compact, thus it can be covered by $n$ neighbourhoods
$\mathscr{V}_{y_{i}}$. That means
$K(x_{1})\subset\bigcup_{i=1}^{n}\mathscr{V}_{y_{i}}$. By upper semicontinuity
of $T$, there exists open neighbourhood $\mathscr{V}_{x_{1}}^{0}$ such that

$\displaystyle K(x)\subset\bigcup_{i=1}^{n}\mathscr{V}_{y_{i}},\mbox{ for all
}x\in\mathscr{V}_{x_{1}}^{0}.$ (7)

For each $x\in\mathscr{V}_{x_{1}}=\bigcap_{i=0}^{n}\mathscr{V}_{x_{1}}^{i}$
and each $y\in K(x)$, we have from (6) and (7) the following

$f(x,y)<f(x_{0},y_{0}).$

Consequently, $m(x)\leq f(x_{0},y_{0})<m(x_{2})$.

Second, assume that there is not any $(x_{0},y_{0})\in X\times Y$ such that
$m(x_{1})<f(x_{0},y_{0})<m(x_{2})$. We have that $f(x_{1},y)<f(x_{2},y_{2})$,
for all $y\in K(x_{1})$. By the same steps given in the previous part, there
exists a neighbourhood $\mathscr{V}_{x_{1}}$ of $x_{1}$ such that

$f(x,y)<f(x_{2},y_{2}),\mbox{ for all }x\in\mathscr{V}_{x_{1}}\mbox{ and all
}y\in K(x).$

Since $f(x,\cdot)$ attains its maximum on $K(x)$, we deduce that
$m(x)<m(x_{2})$. ∎

##  4 Equivalent results of Ky Fan’s minimax inequality

The following result is the famous minimax inequality due to Ky Fan [15].

######  Theorem 4.1 (Ky Fan).

Let $X$ be a compact convex subset of a Hausdorff topological vector space.
Let $f$ be a real-valued function defined on $X\times X$ such that

  1. (i)

for each $y\in X$, $f(\cdot,y)$ is lower semicontinuous;

  2. (ii)

for each $x\in X$, $f(x,\cdot)$ is quasi-concave.

Then, the minimax inequality

$\min_{x\in X}\sup_{y\in X}f(x,y)\leq\sup_{x\in X}f(x,x)$

holds.

The Ky Fan minimax inequality is equivalent to the Fan-Browder theorem, we
suggest [16] in order to see this equivalence.

Now, we state a similar result where we use pseudo-continuity instead lower
semicontinuity.

######  Proposition 4.2.

Let $X$ be a compact convex subset of a Hausdorff topological vector space.
Let $f$ be a real-valued function defined on $X\times X$ such that

  1. (i)

$f$ is pseudo-continuous;

  2. (ii)

for each $x\in X$, $f(x,\cdot)$ is quasi-concave.

Then, the minimax inequality

$\min_{x\in X}\max_{y\in X}f(x,y)\leq\max_{x\in X}f(x,x)$

holds.

###### Proof.

By Theorem 3.4, the argmax correspondence $M:X\rightrightarrows X$ defined as

$M(x)=\left\\{y\in X:~{}f(x,y)=\max_{z\in X}f(x,z)\right\\}$

is upper hemicontinuous with compact, convex and nonempty values. Thus, there
exists $x_{0}\in X$ such that $x_{0}\in M(x_{0})$, due to Kakutani’s fixed
point theorem. That means $f(x_{0},x_{0})\geq f(x_{0},y)$, for all $y\in X$.
Therefore,

$\min_{x\in X}\max_{y\in X}f(x,y)\leq\max_{y\in
X}f(x_{0},y)=f(x_{0},x_{0})\leq\max_{x\in X}f(x,x).$

∎

As a direct consequence of the previous proposition we recover the following
result concerning the existence of fixed points.

######  Corollary 4.3 (Browder).

Let $X$ be a compact, convex and nonempty subset of $\mathbb{R}^{n}$ and
$h:X\to X$ be a continuous function. Then there exists $x_{0}\in X$ such that
$x_{0}=h(x_{0})$.

######  Remark 4.4.

Since Corollary 4.3 implies Browder-Fan’s theorem, so this last one is a
consequence of Proposition 4.2.

At first glance it seems that Theorem 4.1 and Proposition 4.2 are independent,
because pseudo-continuity does not imply lower semicontinuity, and conversely
lower semicontinuity in the second variable does not imply pseudo-continuity
in both variables. However, we will show that these are equivalent.

######  Proposition 4.5.

Theoren 4.1 and Proposition 4.2 are equivalent.

###### Proof.

First, we will prove that Theorem 4.1 implies Proposition 4.2. Since $X\times
X$ is a convex set, in particular it is connected. By Proposition 2.2, there
exists a continuous function $u:X\times X\to\mathbb{R}$ and an increasing
function $h:u(X\times X)\to\mathbb{R}$ such that $f=h\circ u$. Moreover, for
each $x$ we have $U_{f(x,\cdot)}(y)=U_{u(x,\cdot)}(y)$, for all $y\in X$. In
other words, $u(x,\cdot)$ is quasi-concave, for every $x\in X$. Thus, by
Theorem 4.1 we have

$\min_{x\in X}\max_{y\in X}u(x,y)\leq\max_{x\in X}u(x,x).$

Since $u$ is continuous, there exist $x_{0}$ and $x_{1}$ both in $X$ such that

$u(x_{0},y)\leq u(x_{1},x_{1}),$

for all $y\in X$. Thus, $f(x_{0},y)=h(u(x_{0},y))\leq h(u(x_{1},x_{1}))\leq
f(x_{1},x_{1})$ and consequently

$\min_{x\in X}\max_{y\in X}f(x,y)\leq\max_{x\in X}f(x,x).$

Conversely, since Proposition 4.2 implies Theorem 2.9 and this is equivalent
to Theorem 4.1, the result follows.

∎

Another consequence of Ky Fan’s minimax inequality is stated below.

######  Theorem 4.6.

Let $X$ be a compact convex subset of a Hausdorff topological vector space.
Let $f$ and $g$ be two real-valued function defined on $X\times X$ such that

  1. (i)

for all $x,y\in X$, $f(x,y)\leq g(x,y)$;

  2. (ii)

for each $y\in X$, the function $f(\cdot,y)$ is lower semicontinuous;

  3. (iii)

for each $x\in X$, the function $g(x,\cdot)$ is quasi-concave;

  4. (iv)

for all $x\in X$, $g(x,x)\leq 0$.

Then, there exists $x_{0}\in X$ such that $f(x_{0},y)\leq 0$, for all $y\in
X$.

###### Proof.

For each $x$, we denote by $f_{q}(x,\cdot)$ the quasi-concave regularization
of $f(x,\cdot)$. Since $g$ is quasi-concave in its second argument,
$f_{q}(x,\cdot)$ is real-valued, due to Lemma 2.7. Moreover, $f(x,y)\leq
f_{q}(x,y)\leq g(x,y)$, for all $x,y\in X$. By Proposition 3.10 in [7],
$f_{q}$ is lower semicontinuous in its first argument. Thus, there exists
$x_{0}\in X$ such that $f_{q}(x_{0},y)\leq 0$, for all $y\in X$, due to
Theorem 4.1. Therefore, the result follows. ∎

######  Remark 4.7.

The previous result is actually equivalent to Theorem 4.1. For that, it is
enough to show that Theorem 4.1 is a consequence of Theorem 4.6. In that
sense, we denote $\alpha=\sup_{x\in X}f(x,x)$. If $\alpha=+\infty$ there is
nothing to prove. Otherwise, we define the function $h:X\times X\to\mathbb{R}$
by $h(x,y)=f(x,y)-\alpha$, which satisfies all assumptions of Theorem 4.6.

Other similar result to the previous one was considered by Qiu and Peng [23],
where they used lower pseudo-continuity instead of lower semicontinuity, but
they need both functions to vanish on the diagonal. We will give a simple
proof in order to show that it is a consequence of Fan-Browder’s theorem which
is equivalent to Theorem 4.1.

######  Theorem 4.8 (Qiu and Peng).

Let $X$ be a compact convex subset of a Hausdorff topological vector space.
Let $f$ and $g$ be two real-valued function defined on $X\times X$ such that

  1. (i)

for all $x,y\in X$, $f(x,y)\leq g(x,y)$;

  2. (ii)

for each $y\in X$, the function $f(\cdot,y)$ is lower pseudo-continuous;

  3. (iii)

for each $x\in X$, the function $g(x,\cdot)$ is quasi-concave;

  4. (iv)

for all $x\in X$, $f(x,x)=g(x,x)=0$.

Then, there exists $x_{0}\in X$ such that $f(x_{0},y)\leq 0$, for all $y\in
X$.

###### Proof.

We consider two correspondences $F,G:X\rightrightarrows X$ defined by

$F(x)=\\{y\in X:~{}f(x,y)>0\\}\mbox{ and }G(x)=\\{y\in X:~{}g(x,y)>0\\}$

Clearly $G$ has convex values and does not have fixed points. Moreover,
$F(x)\subset G(x)$, for all $x\in X$. Also, $F$ has open fibres, due to $f$
being lower pseudo-continuous in its first argument and the fact that $f$
vanishes on the diagonal of $X\times X$. By Lemma 5.1 in [30] the
correspondence $\operatorname*{conv}(F):X\rightrightarrows X$ defined as
$\operatorname*{conv}(F)(x)=\operatorname*{conv}(F(x))$ has open fibres.
Furthermore, $\operatorname*{conv}(F)(x)\subset G(x)$, for all $x\in X$.
Hence, by Fan-Browder’s theorem there exists $x_{0}\in X$ such that
$\operatorname*{conv}(F(x_{0}))=\emptyset$. Consequently, $F(x_{0})=\emptyset$
and this means $f(x_{0},y)\leq 0$, for all $y\in X$. ∎

######  Remark 4.9.

Theorem 4.8 is equivalent to Theorem 4.1. In order to see that, it is enough
to show that Theorem 4.8 implies Theorem 4.2. Indeed, if $f$ is pseudo-
continuous then there exists a continuous function $u$ and an increasing
function $h$ such that $f=h\circ u$, due to Proposition 2.2. Moreover, $u$ is
quasi-concave in its second argument. Now, we consider the function $u_{0}$
defined by $u_{0}(x,y)=u(x,y)-u(x,x)$, which is continuous and it vanishes on
the diagonal. We now apply Theorem 4.8 to $u_{0}$ and recover Theorem 4.1.

##  5 Equivalence results in Nash games

A _Nash game_ , [22, Nash 1951], consists of $p$ players, each player $i$
controls the decision variable $x_{i}\in C_{i}$ where $C_{i}$ is a subset of a
Hausdorff locally convex topological vector space $E_{i}$. The “total strategy
vector” is $x$ which will be often denoted by

$x=(x_{1},x_{2},\dots,x_{i},\dots,x_{p}).$

Sometimes we write $(x_{i},x_{-i})$ instead of $x$ in order to emphasize the
$i$-th player’s variables within $x$, where $x_{-i}$ is the strategy vector of
the other players. Player $i$ has a payoff function
$\theta_{i}:C\to\mathbb{R}$ that depends on all player’s strategies, where
$C=\prod_{i=1}^{p}C_{i}$. Given the strategies $x_{-i}$ of the other players,
the aim of player $i$ is to choose a strategy $x_{i}$ solving the problem
$P_{i}(x_{-i})$:

$\displaystyle\max_{x_{i}}\theta_{i}(x_{i},x_{-i})~{}\mbox{ subject to
}~{}x_{i}\in C_{i}.$

A vector $\hat{x}\in C$ is a _Nash equilibrium_ if for all $i$, $\hat{x}_{i}$
solves $P_{i}(x_{-i})$.

Thanks to Debreu [10], Glicksberg [14] and Fan [13], we have the following
existence result of Nash equilibria.

######  Theorem 5.1.

For each $i$, $C_{i}$ is compact, convex and non-empty. If for all $i$, the
payoff function $\theta_{i}$ is continuous and quasi-concave in $x_{i}$, then
there exists at least one Nash equilibrium.

We now present an existence result of Nash equilibria for discontinuous games,
due to Morgan and Scalzo [21].

######  Theorem 5.2.

For each $i$, $C_{i}$ is compact, convex and non-empty. If for all $i$, the
payoff function $\theta_{i}$ is pseudo-continuous and quasi-concave in
$x_{i}$, then there exists at least one Nash equilibrium.

In a generalized Nash game, each player’s strategy must belong to a set
identified by the correspondence $K_{i}:C^{-i}\rightrightarrows C_{i}$ in the
sense that the strategy space of player $i$ is $K_{i}(x_{-i})$, which depends
on the rival player’s strategies $x_{-i}$. Given the strategy $x_{-i}$, player
$i$ chooses a strategy $x_{i}$ such that it solves the following problem

$\displaystyle\max_{x_{i}}\theta_{i}(x_{i},x_{-i})~{}\mbox{ subject to
}~{}x_{i}\in K_{i}(x_{-i}).$ ($GP_{i}(x_{-i})$)

Thus, a _generalized Nash equilibrium_ (GNEP) is a vector $\hat{x}\in C$ such
that the strategy $\hat{x}_{i}$ is a solution of the problem
($GP_{i}(x_{-i})$) associated to $\hat{x}_{-i}$, for any $i$.

The following result is due to Arrow and Debreu [4], but we state it as in
[12].

######  Theorem 5.3.

For each $i$, $C_{i}$ is compact, convex and non-empty. If for all $i$, the
following hold:

  1. 1.

the payoff function $\theta_{i}$ is continuous and quasi-concave in $x_{i}$,

  2. 2.

the correspondence $K_{i}$ is lower and upper hemicontinuous with convex,
closed and non-empty values;

then, there exists at least one generalized Nash equilibrium.

Below, an existence result of generalized Nash equilibria for discontinuous
generalized Nash games, due to Morgan and Scalzo [17].

######  Theorem 5.4.

For each $i$, $C_{i}$ is compact, convex and non-empty. If for all $i$, the
following hold:

  1. 1.

the payoff function $\theta_{i}$ is pseudo-continuous and quasi-concave in
$x_{i}$,

  2. 2.

the correspondence $K_{i}$ is lower and upper hemicontinuous with convex,
closed and non-empty values;

then, there exists at least a generalized Nash equilibrium.

Clearly, Theorem 5.4 implies Theorem 5.3. However, the next result says that
they are actually equivalent.

######  Theorem 5.5.

Theorems 5.4 and 5.3 are equivalent.

###### Proof.

Since any convex set is connected, by Proposition 2.2, we have that for each
$i$ there exists a continuous function $u_{i}:C\to\mathbb{R}$ and an
increasing function $h_{i}:u_{i}(C)\to\mathbb{R}$ such that

$\theta_{i}=h_{i}\circ u_{i}.$

As $\theta_{i}$ is quasi-concave in $x_{i}$, it is not difficult to show that
so is $u_{i}$. Now, the generalized Nash game defined by the functions $u_{i}$
and the correspondences $K_{i}$, admits a generalized Nash equilibrium, due to
Theorem 5.3, say $\hat{x}$. Since each function $h_{i}$ is increasing we have
that

$\theta_{i}(\hat{x})=h_{i}\circ u_{i}(\hat{x})\geq h_{i}\circ
u_{i}(x_{i},\hat{x}_{-i})=\theta_{i}(x_{i},\hat{x}_{-i}),\mbox{ for all
}x_{i}\in K_{i}(\hat{x}_{i}).$

The result follows. ∎

It is clear that any Nash game is a generalized Nash game. Moreover, Theorem
5.3 implies Theorem 5.1, and Theorem 5.4 implies Theorem 5.2. Thus, as a
direct consequence of the previous result we have that Theorem 5.1 is
equivalent to Theorem 5.2. Furthermore, the following result establishes that
Theorem 5.3 is equivalent to Theorem 5.1 on Banach spaces.

######  Theorem 5.6.

Theorem 5.1 implies Theorem 5.3 on Banach spaces.

In order to prove the previous result, we need the following lemma, which is
inspired by Theorem 4.1 in [6].

######  Lemma 5.7.

Theorem 5.1 implies Proposition 4.2 on Banach spaces.

###### Proof.

By Proposition 2.2, we assume without loss of generality that $f$ is
continuous. Now, consider the two-player game defined by the strategy sets and
the payoff functions as follows

$C_{1}=C_{2}=X,~{}\theta_{1}(x_{1},x_{2})=f(x_{2},x_{1})-f(x_{2},x_{2})\mbox{
and }\theta_{2}(x_{1},x_{2})=-\|x_{2}-x_{1}\|,$

where $\|\cdot\|$ is the norm. It is clear that this game satisfies all
assumption of Theorem 5.1, thus there exists $(\hat{x}_{1},\hat{x}_{2})\in
X\times X$ such that

$\theta_{1}(\hat{x}_{2},\hat{x}_{1})\geq\theta_{1}(\hat{x}_{2},x_{1})\mbox{
and }\|\hat{x}_{2}-\hat{x}_{1}\|\leq\|x_{2}-\hat{x}_{1}\|,~{}\mbox{for all
}x_{1},x_{2}\in X.$

From the second part we deduce that $\hat{x}_{1}=\hat{x}_{2}=\hat{x}$. Thus,
using this in the first part we obtain $0\geq\theta_{1}(\hat{x},x_{1})$ for
all $x_{1}\in X$. Therefore,

$\max_{z\in X}f(z,z)\geq f(\hat{x},\hat{x})\geq f(\hat{x},y),\mbox{ for all
}y\in X.$

∎

###### Proof of Theorem 5.6.

Since Theorem 5.3 is a consequence of Theorem 2.8 and this is consequence of
Fan-Browder’s theorem. The result follows from Lemma 5.7 and Remark 4.4. ∎

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#  T4DT: Tensorizing Time for Learning Temporal 3D Visual Data

Mikhail Usvyatsov  ETH Zurich, Switzerland, {mikhailu<EMAIL_ADDRESS>Rafael Ballester-Ripoll  IE University, Madrid, Spain<EMAIL_ADDRESS>Lina Bashaeva  Skolkovo Institute of Science and Technology, Moscow, Russia,
{lina.bashaeva, g.ferrer<EMAIL_ADDRESS>Konrad Schindler  ETH
Zurich, Switzerland, {mikhailu<EMAIL_ADDRESS>Gonzalo Ferrer  Skolkovo
Institute of Science and Technology, Moscow, Russia, {lina.bashaeva, g.ferrer,
<EMAIL_ADDRESS>Ivan Oseledets  Skolkovo Institute of Science and
Technology, Moscow, Russia, {lina.bashaeva, g.ferrer<EMAIL_ADDRESS>
###### Abstract

Unlike 2D raster images, there is no single dominant representation for 3D
visual data processing. Different formats like point clouds, meshes, or
implicit functions each have their strengths and weaknesses. Still, grid
representations such as signed distance functions have attractive properties
also in 3D. In particular, they offer constant-time random access and are
eminently suitable for modern machine learning. Unfortunately, the storage
size of a grid grows exponentially with its dimension. Hence they often exceed
memory limits even at moderate resolution. This work proposes using low-rank
tensor formats, including the Tucker, tensor train, and quantics tensor train
decompositions, to compress time-varying 3D data. Our method iteratively
computes, voxelizes, and compresses each frame’s truncated signed distance
function and applies tensor rank truncation to condense all frames into a
single, compressed tensor that represents the entire 4D scene. We show that
low-rank tensor compression is extremely compact to store and query time-
varying signed distance functions. It significantly reduces the memory
footprint of 4D scenes while remarkably preserving their geometric quality.
Unlike existing, iterative learning-based approaches like DeepSDF and NeRF,
our method uses a closed-form algorithm with theoretical guarantees.

(a) Original, $284\text{\,}\mathrm{GB}$

(b) 1:12345, $24\text{\,}\mathrm{MB}$

(c) 1:793, $368\text{\,}\mathrm{MB}$

(d) 1:227, $1.24\text{\,}\mathrm{GB}$

Figure 1: T4DT with different compression levels in OQTT format for a
longshort-flying-eagle scene of resolution $512^{3}$ with 284 time frames.
Only frames 1, 142, and 284 are depicted. The compression ratio is different
from the actual memory consumption due to the padding of the time dimension to
512. High compression is achieved with $r_{\max}=400$, $\text{MSDM2}=0.45$ in
Fig. 1(b), medium compression with $r_{\max}=1800$, $\text{MSDM2}=0.32$ in
Fig. 1(c), and low compression / high quality with $r_{\max}=4000$,
$\text{MSDM2}=0.29$ in Fig. 1(d).

##  1 Introduction

Recent advances in hardware development have made it possible to collect rich
depth datasets with commodity devices. For example, modern AR/VR hardware can
record videos of 3D data, thus capturing their temporal evolution to yield 4D
datasets. Nevertheless, working with 4D data presents computational challenges
due to the curse of dimensionality. Furthermore, while 2D data mostly come in
the form of raster images, there is an entire zoo of popular representations
for 3D data, ranging from point clouds and meshes to voxel grids and implicit
(neural) functions.

In this work, we address the case of representations based on regular grids.
Such representations are highly structured and allow for efficient
manipulation, e.g., they offer random access, slicing, and other operations in
constant time. However, their storage requirements become a bottleneck: a grid
of resolution $I$ along each dimension $D$ has $I^{D}$ elements. We propose
applying tensor decompositions to leverage the high temporal and spatial
correlation in grids that arise from time-evolving 3D data. Besides exploring
the general idea of compression via low-rank constraints, we compare different
tensor decomposition schemes and their potential for 4D video. For example,
although the Tucker format Tucker, (1963, 1966) is known to yield good results
for the 3D case, it still suffers from the curse of dimensionality since it
requires $O(r^{D})$ elements w.r.t. the Tucker rank $r$, while $r$ must
typically be chosen $\approx\frac{I}{\text{const.}}$ to achieve reasonable
accuracy. Therefore, we explore hybrid low-rank formats based on the tensor
train (TT) decomposition Oseledets, (2011).

The Signed Distance Function (SDF, sometimes also called Signed Distance
Field) and its truncated version, known as Truncated SDF or TSDF, is a
convenient format for data fusion: several methods convert partial 3D views,
even if initially in mesh format, to TSDFs to fuse them into complete models
Izadi2011KinectFusionR3. This pipeline is useful for multi-view stereo
acquisition, laser scanners, as well as resource-limited devices (e.g., mobile
augmented reality). TSDFs offer 1-to-1 correspondence across time, which is
not easy to achieve with meshes if the objects undergo non-rigid deformation;
while every mesh can be converted to a TSDF on the same voxel grid (with
appropriately chosen resolution to avoid loss of detail). We experimentally
analyze the effects of the low-rank constraint, using TSDF sequences of 3D
scenes from the CAPE dataset Ma et al., (2020); Pons-Moll et al., (2017) and
problem-specific quality metrics. Our method can compress time-varying grids
to a usable size, while the same scenes in uncompressed form would each
require hundreds of GigaBytes of memory.

In the following, Section 2 reviews applicable tensor methods and techniques
for 3D and 4D data compression. Section 4 gives a detailed outline of our
proposed approach, Section 5 demonstrates its the performance in an
experimental study, followed by a discussion of limitations in Section 6, and
concluding remarks in Section 7. Our contributions are:

  1. 1.

The first tensor decomposition framework for compressing temporal sequences in
3D voxel space;

  2. 2.

A benchmark and analysis of different decomposition schemes for both the
spatial and temporal dimensions;

  3. 3.

An open source implementation of our method based on the tntorch framework
Usvyatsov et al., (2022), available at https://github.com/prs-eth/T4DT.

##  2 Background and Related Work

In our work, we apply tensor methods to temporal sequences of 3D scenes. We
consider grid representations of the 3D data, i.e., a tensor
$\displaystyle{\bm{\mathsfit{A}}}\in\mathbb{R}^{I_{1}\times\dots\times I_{D}}$
constitutes a discrete sampling of a $D$-dimensional space on a grid
$\mathbb{I}=I_{1}\times\dots\times I_{D}$, with $I_{d}$ samples along
dimension $d$.

###  2.1 SDF and Truncated SDF

The SDF at position $p\in\mathbb{R}^{3}$ is defined as:

$\text{SDF}(p)=\begin{cases}\text{dist}(p,\partial\Omega)&\text{if
}p\in\Omega,\\\
\hfill-\text{dist}(p,\partial\Omega)&\text{otherwise},\end{cases}$ (1)

while the truncated SDF clamps the SDF as follows:

$\text{TSDF}(p)=\begin{cases}\hfill-\tau&\text{if }\text{SDF}(p)\leq-\tau,\\\
\hfill\tau&\text{if }\text{SDF}(p)\geq\tau,\\\
\hfill\text{SDF}(p)&\text{otherwise}.\end{cases}$ (2)

Depending on the available input data and the desired application, different
formats may be more or less suitable. For example, colored images can be
converted to NERF or NELF Mildenhall et al., (2020); Sitzmann et al., (2021),
which optimizes the reconstruction error via differentiable rendering. Depth
images can be fused into TSDFs as described in Curless and Levoy, (1996);
meshes can be converted into TSDFs, too, and also the inverse transformation
is possible with variants of marching cubes Lorensen and Cline, (1987); point
clouds can be turned into meshes or directly into TSDFs; etc.

###  2.2 Compression

Storing signed distance fields constitutes a computational and memory
bottleneck, especially for time-varying data. In Tang et al., (2018), a real-
time compression method was proposed that uses a learnable, implicit temporal
TSDF representation, combining independent color and texture encodings with
learnable geometry compression. Multi-dimensional visual data compression
based on the Tucker decomposition, combined with bit-plane coding, was applied
to volumetric data in Ballester-Ripoll et al., (2019). Later, Boyko et al.,
(2020) proposed compressing single-frame TSDFs with a block-based neural
network. They showed that, although TSDF tensors usually cannot be written
exactly as a low-rank expansion, in practice, one can store them in the low-
rank TT representation with low compression error. Here, we build on the idea
of tensor decompositions as a tool for compressing high-dimensional visual
data. We address the task of compressing dynamic 3D scenes, whose 4D grid
representations quickly exceed practical memory limits when stored in
uncompressed form.

Compressing TSDF with tensor methods not only allows one to work with data
that otherwise would not fit the memory, but it can also be useful for:

  1. 1.

Fusion, as demonstrated in Boyko et al., (2020);

  2. 2.

Constant-time TSDF random access,111The complexity of accessing a single
element depends on the choice of tensor format and rank but is independent of
the location of the query point within the tensor which may be useful to test
if a given point is inside or outside of a mesh, and potentially for ray-
casting;

  3. 3.

Constant-time computation of the TSDF gradient, used to compute surface
normals Sommer et al., (2022);

  4. 4.

Unlike neural network-based learning approaches, tensor methods rely on SVD-
based schemes, which provide theoretical guarantees Oseledets, (2011) and make
the framework more interpretable.

Throughout this work, we use the term compression ratio (or just compression,
if clear from the context) to denote the ratio between the number of
coefficients used for the compressed format and the number of coefficients
required to store the uncompressed tensor.

##  3 Tensor Formats

###  3.1 Notation

##### Tensor diagram notation.

To visualize tensor networks, we adopt Penrose’s graphical notation Penrose,
(1971). See Fig. 2 for visualization of a vector, a matrix and a matrix-vector
product.

(a) vector

(b) matrix

(c) matrix-vector product

Figure 2: Each tensor is denoted as a node whose edges represent its
dimensions. Whenever an edge is shared, tensor contraction is assumed: e.g.,
Fig. 2(c) depicts the matrix-vector product
$\sum_{j}{\bm{M}}_{ij}{\bm{v}}_{j}$.

###  3.2 Tucker

The Tucker decomposition Tucker, (1963) factors a tensor of dimension $D$ into
a $D$-dimensional _core_ tensor
$\displaystyle{\bm{\mathsfit{G}}}\in\mathbb{R}^{r_{1}\times\dots\times r_{D}}$
and $D$ matrices $\displaystyle\\{{\bm{A}}_{d}\\}_{d=1}^{D}$,
$\displaystyle{\bm{A}}_{d}\in\mathbb{R}^{r_{d}\times I_{d}}$. The format is
defined as

$\displaystyle{\bm{\mathsfit{A}}}[i_{1},\dots,i_{D}]={\bm{\mathsfit{G}}}{\bm{\mathsfit{A}}}_{1}[:,i_{1}]\dots\bm{\mathsfit{A}}_{D}[:,i_{D}],$
(3)

where $r_{d}$ are the Tucker-ranks. The Tucker decomposition has
$\mathcal{O}\big{(}(\max_{d}[r_{d}])^{D}+\max_{d}[r_{d}]\cdot\max_{d}[I_{d}]\big{)}$
storage cost. See Fig. 3(a) for visualization of Tucker format in graphical
notation.

(a) Tucker

(b) TT

Figure 3: Graphical representation of the Tucker Fig. 3(a) and TT Fig. 3(b)
decompositions.

###  3.3 Tensor Train

The TT decomposition Oseledets, (2011) factors a tensor of dimension $D$ into
a sequence of $D$ 3-dimensional tensors. The format is defined as

$\displaystyle{\bm{\mathsfit{A}}}[i_{1},\dots,i_{D}]={\bm{\mathsfit{Q}}}_{1}[0,i_{1},:]{\bm{\mathsfit{Q}}}_{2}[:,i_{2},:]\dots\bm{\mathsfit{Q}}_{D}[:,i_{D},0],$
(4)

where the tensors
$\displaystyle\\{{\bm{\mathsfit{Q}}}_{d}\\}_{d=1}^{D},{\bm{\mathsfit{Q}}}_{d}\in\mathbb{R}^{r_{d-1}\times
I_{d}\times r_{d}}$, are called TT-cores; and $r_{d}$ are the _TT-ranks_
($r_{0}\\!=\\!r_{D}\\!=\\!1$). The TT decomposition has
$\mathcal{O}\big{(}D\cdot(\max_{d}[r_{d}])^{2}\cdot\max_{d}[I_{d}]\big{)}$
storage cost, and leads to a linear tensor network; see Fig. 3(b) for a
graphical example.

###  3.4 QTT

Quantics TT (QTT) is an extension of TT that includes reshaping222With zero-
padding where needed the input into a tensor of shape $2\times 2\dots\times
2=2^{\sum_{d=1}^{D}\log_{2}(I_{d})}$, with $I_{j}$ the size of the $j$-th mode
Kazeev et al., (2017). The sub-dimensions $x,y,z$ are then grouped side-by-
side for each octet, which imitates the traversal of a $z$-space filling curve
and makes QTT similar to a tensorized octree. Last, the resulting tensor is
then subject to standard TT decomposition. This scheme is also connected to
the wavelet transform Kazeev and Oseledets, (2013).

##### Octet QTT.

We also propose to reuse a variant of QTT with a base dimension of size eight
instead of two by merging the three sub-dimensions of each octet into one.
This format was originally proposed by qttnf2022. In this way, the resulting
tensor has shape $8\times 2\times...\times 8\times 2$. We refer to this format
as OQTT for octet QTT and found it to reduce discontinuity artifacts (Section
5).

##  4 Proposed Method

We introduce T4DT, a method to compress high-resolution temporal TSDF fields
with tensor decompositions discussed in Sections 3.3 \- 3.4. We exploit that
individual TSDF frames can be compressed into a low-rank TT decomposition with
reasonable error Boyko et al., (2020). Under the low-rank constraint, a zero-
level set can be reconstructed with sufficiently good quality. However, this
becomes more challenging when considering time-evolving data since
uncompressed 4D grid scenes at fine spatial resolutions can range in the
hundreds of GBs. Algorithm 1 gives a high-level overview of our pipeline.

${\mathbb{I}}=I_{1}\times I_{2}\times I_{3}\times I_{4}$ – temporal 3D grid

${\bm{\mathsfit{X}}}\in\mathbb{R}^{{\mathbb{I}}}$ – input temporal TSDF

$R_{s}$ – maximal rank along spatial dimensions

$R_{t}$ – maximal rank along time dimension

$\displaystyle{\bm{\mathsfit{X}}}^{\prime}$ – Collection of compressed TSDF
frames

$\displaystyle{\bm{\mathsfit{Y}}}$ – Compressed temporal TSDF

for $\text{i}=1,2,\cdots,I_{4}-1,I_{4}$ do

$\displaystyle{\bm{\mathsfit{X}}}^{\prime}_{\text{i}}\leftarrow$
truncate($\displaystyle{\bm{\mathsfit{X}}}[\cdots,\text{i}],R_{s}$)

end for

while $\displaystyle{\bm{\mathsfit{X}}}^{\prime}\neq\varnothing$ and
$|{\bm{\mathsfit{Y}}}|\neq 1$ do

$\displaystyle{\bm{\mathsfit{Y}}}\leftarrow\text{truncate}({\bm{\mathsfit{X}}}_{0}^{\prime}\cup{\bm{\mathsfit{Y}}},R_{t})$

$\text{remove}({\bm{\mathsfit{X}}}_{0}^{\prime})$

end while

Algorithm 1 T4DT pipeline. Since the whole 4D scene is not expected to fit
into memory, we first compress each frame individually. Next, the collection
of individually compressed frames is assembled into a single compressed scene
using a tree-like merge procedure
${\bm{\mathsfit{X}}}_{0}^{\prime}\cup{\bm{\mathsfit{Y}}}$. See Section 4.2 for
the merge procedure details. Figure 4: We compress and merge individual frames
progressively into a compressed scene (for simplicity, all tensors are shown
in uncompressed form). Two of the previous level tensors are merged at each
tree level using the concatenation procedure (which increases tensor rank),
followed by rank truncation. On the first level, an additional core for the
temporal dimension is inserted into each frame.

###  4.1 Framewise Decomposition

The first technical choice is the base decomposition at the frame level. We
considered three variants: Tucker, TT, and QTT. In 3D, Tucker is an attractive
choice since it is equivalent to TT plus an additional compression step along
the second dimension. For the 4D case, we propose to combine the best of both
worlds by using a TT-Tucker blend, where spatial dimensions are stored in
Tucker formatted cores, and the temporal dimension is represented with a TT
core. Each core or factor is responsible for a single dimension in both the TT
and Tucker formats. However, the Tucker model shares the core tensor between
all dimensions.

Many real-world datasets are sparse in the sense that most of the occupied
volume is empty. An octree is an excellent choice to pack such sparse data
into a compact representation. We argue that QTT is a tensor analog of an
octree data structure. Indeed, it is easy to see that, padded and reshaped to
$2^{\sum_{d=1}^{D}\log_{2}(I_{d}))}$, the scene becomes a piece-wise
separated, reshaped set of octets after permutation of the corresponding sub-
dimensions. Rank truncation is a way to reduce the number of coefficients to
encode each octet sub-dimension.

###  4.2 Frame Concatenation

Since the original scene can by far exceed the available memory, we developed
a progressive procedure to merge compressed frames into a single compressed
scene. The algorithm is akin to the _pairwise summation_ method to reduce
round-off errors in numerical summation Higham, (2002): we stack the frames by
pairs along the time dimension, recompress each pair via tensor rank
truncation Oseledets, (2011), and repeat the procedure recursively in a
binary-tree fashion to ultimately yield a single 4D compressed tensor. See
Fig. 4 for an illustration.

(a) Chamfer distance

(b) Hausdorff distance

(c) IoU

(d) MSDM2

Figure 5: Ranks vs. performance for TT-Tucker compressed longshort-flying-
eagle scene of resolution $512^{3}$ with 284 time frames. Metrics are averaged
between frames 1, 142, and 284. Each data point is annotated with the
corresponding compression ratio scaled by $10^{6}$.

Note that this procedure is compatible with any format that supports rank
truncation, which includes the TT and Tucker formats and mixtures thereof that
allow for concatenation in compressed format. They are implemented via
concatenation of the cores, increasing the rank of the resultant cores to a
sum of the ranks of the operands. Afterward, rank truncation is applied to
reduce the rank back to the desired maximal value with an SVD-like algorithm
Oseledets, (2011).

##### QTT scene.

The scene stored in QTT format has the shape $x_{1}\times y_{1}\times
z_{1}\times t_{1}\times x_{2}\times y_{2}\times z_{2}\times
t_{2}\times\cdots\times x_{k}\times y_{k}\times z_{k}\times t_{k}$, where
$x_{i}=y_{i}=z_{i}=t_{i}=2$ and
$\prod_{i=1}^{\lceil\log_{2}I_{1}\rceil}x_{i}=W$,
$\prod_{i=1}^{\lceil\log_{2}I_{2}\rceil}y_{i}=D$,
$\prod_{i=1}^{\lceil\log_{2}I_{3}\rceil}z_{i}=H$,
$\prod_{i=1}^{\lceil\log_{2}I_{4}\rceil}t_{i}=T$, with uncompressed scene
shape $W\times D\times H\times T$. In order to reconstruct the $i$-th frame
from a scene compressed in this way, one must first compute the binary
representation of $i$ and then decompress the sub-tensor at
$[:,:,:,i_{\text{bit}_{k}},:,:,:,i_{\text{bit}_{k-1}},\cdots,:,:,:,i_{\text{bit}_{0}}]$.

##### OQTT scene.

The scene stored in OQTT format has the shape $x_{1}y_{1}z_{1}\times
t_{1}\times x_{2}y_{2}z_{2}\times t_{2}\times\cdots\times
x_{k}y_{k}z_{k}\times t_{k}$, where $x_{i}=y_{i}=z_{i}=t_{i}=2$ and
$\prod_{i=1}^{\lceil\log_{2}I_{1}\rceil}x_{i}=W$,
$\prod_{i=1}^{\lceil\log_{2}I_{2}\rceil}y_{i}=D$,
$\prod_{i=1}^{\lceil\log_{2}I_{3}\rceil}z_{i}=H$,
$\prod_{i=1}^{\lceil\log_{2}I_{4}\rceil}t_{i}=T$, with uncompressed scene
shape $W\times D\times H\times T$. In order to reconstruct the $i$-th frame,
one must again first compute the binary representation of $i$, then decompress
the sub-tensor at
$[:,i_{\text{bit}_{k}},:,i_{\text{bit}_{k-1}},\cdots,:,i_{\text{bit}_{0}}]$.

${\mathbb{I}}=I_{1}\times I_{2}\times I_{3}$ – 3D grid

${\bm{\mathsfit{X}}}\in\mathbb{R}^{{\mathbb{I}}}$ – input 3D volumetric frame

$R$ – maximal rank

$\displaystyle{\bm{\mathsfit{Y}}}$ – Compressed 3D volumetric frame

for $\text{d}=1,2,3$ do

${\bm{\mathsfit{X}}}\leftarrow\text{pad}(X,2^{\lceil\log_{2}I_{d}\rceil})$

end for

${\bm{\mathsfit{X}}}\leftarrow\text{reshape}(X,2^{\sum_{d=1}^{3}\lceil\log_{2}(I_{d})\rceil)})$

${\bm{\mathsfit{X}}}\leftarrow\text{permute}(X,(x_{1},y_{1},z_{1},\cdots,x_{\lceil\log_{2}I_{1}\rceil},y_{\lceil\log_{2}I_{2}\rceil},z_{\lceil\log_{2}I_{3}\rceil}))$,
$I_{1}=\prod_{i=1}^{\lceil\log_{2}I_{1}\rceil}x_{i}$,
$I_{2}=\prod_{i=1}^{\lceil\log_{2}I_{2}\rceil}y_{i}$ ,
$I_{3}=\prod_{i=1}^{\lceil\log_{2}I_{3}\rceil}z_{i}$

${\bm{\mathsfit{Y}}}\leftarrow\text{TT}({\bm{\mathsfit{X}}},R)$

Algorithm 2 Conversion of a single 3D volumetric frame into OQTT format.

##  5 Results

###  5.1 Data

For our experiments, we use selected scenes from the CAPE dataset Ma et al.,
(2020); Pons-Moll et al., (2017). The dataset consists of 3D meshes of 15 (10
male, 5 female) clothed people in motion. For each scene frame, we compute its
TSDF and discretize it at resolution $512^{3}$, using the PySDF library. We
use $\tau=0.05$ in Eq. 2 to allow $\approx 10$ voxels with distinct levels
near the TSDF zero level set. Since all scene frames must share a common
coordinates frame, we recover the scene’s bounding box and compute the SDF of
each frame within that box.

We further demonstrate the performance of T4DT on selected scenes from the
Articulated Mesh Animation dataset Vlasic et al., (2008). That real-world
dataset contains 10 mesh sequences depicting 3 different humans performing
various actions. For each scene frame, we compute its TSDF and discretize it
at resolution $512^{3}$, using the PySDF library. We use $\tau=0.05$ in Eq. 2
to allow $\approx 10$ voxels with distinct levels near the TSDF zero level
set.

###  5.2 Influence of Tensor Ranks

We provide several error metrics computed for the TT-Tucker format, averaged
across the first, middle, and last frames of the longshort-flying-eagle scene,
for different spatial and temporal ranks in Fig. 5. See Section A.1 for the
definition of the error metrics. Note that the time dimension is far from
full-rank in the TT-Tucker format since the error metrics saturate well below
the total sequence length of 284 frames.

The best compression/performance was obtained for the OQTT base format.
Selected visualizations are provided in Section A.2: Fig. 9 shows error
metrics for the TT format, as a function of the rank. Qualitative results are
shown in Fig. 6 for the TT-Tucker format, in Fig. 7 for the TT format, and in
Fig. 8 for QTT. For visual results of OQTT please refer to Fig. 1. In the
cases of TT and TT-Tucker, severe rank truncation smoothes the reconstructed
surface and erodes smaller features like fingers or facial details. In
contrast, QTT preserves more details at the cost of discontinuities due to the
separation and independent compression of octet sub-dimensions. OQTT reduces
these discontinuity artifacts by encoding the octets as a single dimension
without compressing the ranks between sub-dimensions of a single octet. See
Fig. 1 for qualitative results for OQTT. We also present quantitative OQTT
compression results in Table 1, at a rank that offers a good compromise
between visual quality and compression rate.

|  Crane |  Swing |  Handstand |  Samba
---|---|---|---|---
Resolution | $512^{3}\times 174$ | $512^{3}\times 174$ | $512^{3}\times 174$ | $512^{3}\times 149$
Metric |  |  |  |
L2 $\downarrow$ | 2.67 | 2.07 | 2.45 | 1.71
Chamfer distance $\downarrow$ | $5e^{-5}$ | $4e^{-5}$ | $5e^{-5}$ | $4e^{-5}$
Hausdorff distance $\downarrow$ | 0.190 | 0.015 | 0.012 | 0.013
MSDM2 $\downarrow$ | 0.36 | 0.38 | 0.35 | 0.36
IoU $\uparrow$ | 0.41 | 0.40 | 0.64 | 0.54
Compression | 1:954 | 1:1356 | 1:938 | 1:935
Size | $549\text{\,}\mathrm{MB}$ | $386\text{\,}\mathrm{MB}$ | $558\text{\,}\mathrm{MB}$ | $560\text{\,}\mathrm{MB}$
Table 1: Quantitative performance of OQTT for selected scenes from the
Articulated Mesh Animation dataset Vlasic et al., (2008). The metrics are
averaged across the first, middle, and last frames. For all scenes,
$r_{\max}=4000$, chosen to provide a good trade-off between visual quality and
compression rate.

##  6 Failure Cases and Limitations

We did not observe significant failures while working with the CAPE and AMA
datasets. However, how T4DT fares for 3D data that are not structurally sparse
(i.e., not mostly free space) remains to be tested. Unfortunately, there are
no such scenes in the two datasets.

We do note that TT decomposition is not invariant against the rotation of the
input. Hence, one could attack the scheme by rotating the scene w.r.t. the
grid axes in a way that maximizes the reconstruction error. A straightforward
measure to mitigate the influence of the grid orientation is to pre-rotate the
3D scene to its principal axes before grid sampling and OQTT compression.

##  7 Conclusions

We have presented T4DT, a scalable and interpretable compression pipeline for
3D time sequence data based on tensor decomposition. Our scheme improves over
the related TT-TSDF method in two ways: (i) We are able to process temporally
varying data. We can do so even though the TSDF field takes hundreds of GBs,
and our method works fully in-memory; (ii) We tested various tensorization
schemes, such as a TT-Tucker hybrid, the QTT format, and the new OQTT variant,
and found OQTT to outperform previous decompositions. OQTT is tailored
specifically to 3D volumetric scene representations, and our experiments
quantitatively and qualitatively support its special reshaping of the 3D grid.

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##  Appendix A Appendix

###  A.1 Metrics

Throughout this work, we convert temporal 3D data from mesh format to tensor
format for compression using sampled truncated SDF and from tensor format back
to temporal 3D data in the form of meshes for quantitative tests using the
marching cubes algorithm.

We selected a tensor-based l2 reconstruction metric for the comparison in
tensorial form. We use the metrics below to evaluate the quality of the
obtained mesh.

####  A.1.1 Intersection over Union (IoU)

IoU is a commonly used metric for comparing 3D shapes Rezatofighi et al.,
(2019). For two given 3D volumes ${\bm{\mathsfit{A}}}$ and
${\bm{\mathsfit{B}}}$:

$\text{IoU}({\bm{\mathsfit{A}}},{\bm{\mathsfit{B}}})=\frac{\|{\bm{\mathsfit{A}}}\cap{\bm{\mathsfit{B}}}\|}{\|{\bm{\mathsfit{A}}}\cup{\bm{\mathsfit{B}}}\|}.$
(5)

3D mesh is converted to an occupancy grid, and the operation is easily
performed with implementation from Fuji Tsang et al., (2022).

####  A.1.2 Hausdorff distance

One-sided Hausdorff distance is computed as:

$d_{\text{h}}({\bm{\mathsfit{A}}},{\bm{\mathsfit{B}}})=\max\\{\min\\{||a-b||:a\in{\bm{\mathsfit{A}}}\\}:b\in{\bm{\mathsfit{B}}}\\}.$
(6)

Note, that $d_{\text{h}}$ is not symmetric. The symmetric version is defined
as :

$d_{\text{H}}({\bm{\mathsfit{A}}},{\bm{\mathsfit{B}}})=\max\\{d_{\text{h}}({\bm{\mathsfit{A}}},{\bm{\mathsfit{B}}}),d_{\text{h}}({\bm{\mathsfit{B}}},{\bm{\mathsfit{A}}})\\}.$
(7)

We use the implementation from Jacobson et al., (2018).

####  A.1.3 Chamfer distance

Symmetric Chamfer distance is defined as:

$d_{\text{CD}}({\bm{\mathsfit{A}}},{\bm{\mathsfit{B}}})=\sum_{a\in{\bm{\mathsfit{A}}}}\min_{b\in{\bm{\mathsfit{B}}}}||x-y||^{2}+\sum_{b\in{\bm{\mathsfit{B}}}}\min_{a\in{\bm{\mathsfit{A}}}}||x-y||^{2}.$
(8)

Following Boyko et al., (2020), we sample 30’000 points from each mesh and
compute sampled symmetric Chamfer distance. Finally, we use the implementation
from Fuji Tsang et al., (2022).

####  A.1.4 Mesh Structural Distortion Measure (MSDM2)

Introduced in Lavoué, (2011) MSDM2 metric is used to estimate the correlation
with human visual perception. The metric assumes one mesh to be original and
the second distorted. See Algorithm 3 for an algorithmic view of MSDM2
computation.

Algorithm 3 MSDM2 high-level computation scheme

$M_{o}$ – original mesh, $M_{d}$ – distorted mesh, $\\{v_{i}\in M\\}$ – set of
mesh vertices

for $v_{i}\in M_{o}$ do

Find $v_{j}=\min_{v_{k}\in M_{d}}\text{dist}(v_{i},v_{k})$

end for

for $v_{i}\in M_{o}$ do

calculate curvature of $v_{i}$

end for

for $v_{i}\in M_{d}$ do

calculate curvature of $v_{i}$

end for

interpolate curvature per face for $M_{o}$ and $M_{d}$

for $v_{i}\in M_{d}$ do

calculate MSDM2 for $v_{i}$ based on curvature features and nearest neighbours
$v_{j}$ from $M_{o}$

end for

Integrate global asymmetric MSDM2

Refer to Lavoué et al., (2006); Lavoué, (2011) for the more details on metric
formulation. We use original implementation from Lavoué, (2011) and self
written pybind11 Jakob et al., (2017) interface to Python.

###  A.2 Selected Visualizations

(a) Original, $284\text{\,}\mathrm{GB}$

(b) Compression 1:769230, $0.2\text{\,}\mathrm{MB}$

(c) Compression 1:1333, $33\text{\,}\mathrm{MB}$

(d) Compression 1:116, $385\text{\,}\mathrm{MB}$

Figure 6: T4DT with different compression levels in TT-Tucker format for a
longshort-flying-eagle scene of resolution $512^{3}$ with 284 time frames.
Only frames 1, 142, and 284 are depicted.

(a) Original, $284\text{\,}\mathrm{GB}$

(b) compression 1:476190, $0.61\text{\,}\mathrm{MB}$

(c) compression 1:6250, $16\text{\,}\mathrm{MB}$

(d) compression 1:2000, $48.5\text{\,}\mathrm{MB}$

Figure 7: T4DT with different compression levels in TT format for a longshort-
flying-eagle scene of resolution $512^{3}$ with 284 time frames. Only frames
1, 142, and 284 are depicted. The compression ratio is different from the
actual memory consumption due to the padding of the time dimension to 512.

(a) Original, $284\text{\,}\mathrm{GB}$

(b) compression 1:6666, $0.08\text{\,}\mathrm{MB}$

(c) compression 1:1587, $332\text{\,}\mathrm{MB}$

(d) compression 1:244, $1.4\text{\,}\mathrm{GB}$

Figure 8: Different levels of compression in QTT format for a longshort-
flying-eagle scene of resolution $512^{3}$ with 284 time frames. Only frames
1, 142, and 284 are depicted. The compression ratio is different from the
actual memory consumption due to the padding of the time dimension to 512.

(a) Chamfer distance

(b) Hausdorff distance

(c) IoU

(d) MSDM2

Figure 9: Ranks vs. performance for TT compressed longshort-flying-eagle scene
of resolution $512^{3}$ with 284 time frames. Metrics are averaged between
frames 1, 142, and 284. Each data point is annotated with the corresponding
compression ratio scaled with 1e6.

(a) TT

(b) TT Tucker

Figure 10: Ranks vs. $L_{2}$ for longshort-flying-eagle scene of resolution
$512^{3}$ with 284 time frames. $L_{2}$ is averaged between frames 1, 142, and
284. Each data point is annotated with the corresponding compression ratio
scaled with 1e6.