# Conditional expectation operators on $C(X)$

Mohamed Amine BEN AMOR †

† Research Laboratory of Algebra, Topology, Arithmetic, and Order
Department of Mathematics, Faculty of Mathematical, Physical and Natural
Sciences of Tunis
Tunis-El Manar University, 2092-El Manar, Tunisia

###### Abstract

At the COSAEF conference in 2021, several participants asked the question
whether a conditional expectation operator in the sense of Kuo, Labaushagne
and Watson could be constructed in vector lattices other than
$\mathcal{L}_{p}$ spaces and in particular in $C(X)$. This work answers
positively to this question and participates in an old discussion on integrals
in $C(X)$ space.

##  1 Introduction

The study of probabilities in Riesz spaces is a very active research area.
Several research schools are involved in this effort. We can mention in
particular [1, 2, 3, 4, 5, 6, 7]. Generally, when it was necessary to
illustrate this theory, the authors always give examples on $\mathcal{L}_{p}$
spaces. The question of existence of a conditional expectation operator on
spaces other than $\mathcal{L}_{p}$ ones then became legitimate. In this work,
we will give sufficient conditions for the existence of such operator in the
space $C(X)$. In a future work, the authors will characterize this type of
operators in finite dimensional spaces with the considerable contributions of
Professors Kuo and Watson.

This work will be divided into three parts. A preliminary part which will list
the useful notions for the sequel. A second part devoted to a dual approach
and in which we will be interested in the dual of the space $C^{\infty}(X)$.
In the third part, we will construct a non-trivial operator of the conditional
expectation. Throughout this paper, all topological spaces are assumed to be
Hausdorff.

##  2 Preliminaries

We recall here some basic properties of Riesz spaces and for details and
terminology see [8, 9].

Throughout this paper, $X$ denotes a Tychonoff space; that is, a completely
regular Hausdorff space.

A Riesz space is an ordered vector space $(E,\leq)$ such that each pair of
elements of $E$ has a supremum and an infimum element compatible with the
order. We say that a Riesz space $E$ is Dedekind complete if every non-empty
subset of $E$ which is bounded from above has a supremum. In particular, the
space $C(X)$ is Dedekind complete if and only if $X$ is extremally
disconnected (see for example [9]) i.e the closure of every open subset of $X$
is open-and-closed.

The study of the universal completion of $C(X)$ spaces has attracted the
interest of several researchers. We find constructions from the early 40’s
(see [10]) until very recently [11]. This notion plays an important role
throughout the paper and so we recall the definition and some elementary
properties for the reader’s convenience.

A Riesz space $E$ is said to be universally complete if E is Dedekind complete
and laterally complete i.e every subset of $E$ which consists of mutually
disjoint elements, has a supremum in $E$. $E^{u}$ is a universal completion of
$E$, if $E^{u}$ is universally complete and $E^{u}$ contains $E$ as an order
dense Riesz subspace.

Every Archimedean Riesz space has, up to a Riesz isomorphism, a unique
universal completion and if $e$ is a weak order unit for $E$ then $e$ is a
weak order unit for $E^{u}$,

From now, $X$ will denote an extremally disconnected compact space.

We recall the definition of a conditional expectation operator on a Riesz
space from [4].

######  Definition 2.1.

Let $E$ be a Riesz space with weak order unit. A positive order continuous
projection $T\colon E\rightarrow E$, with range, $\mathcal{R}\left(T\right)$,
a Dedekind complete Riesz subspace of $E$, is called a conditional expectation
operator on $E$ if $Te$ is a weak order unit of $E$ for each weak order unit
$e$ of $E$.

##  3 A Dual approach

In [12], the authors proved that a Dedekind complete Riesz space $E$ with a
weak order unit $e$ and who admits a strictly positive order continuous linear
functional $\varphi$ can be in a certain way identified as $\mathcal{L}_{1}$
space. The point is that it is well known that the order continuous dual of
$C(X)$, denoted by $C(X)_{n}^{\sim}$, didn’t contain any strictly positive
element if $X$ is a separable compact space that has no isolated points, (see
for example [13]).

We decide thus to focus on a larger space and we choose the universal
completion of $C(X)$. Indeed, the universal completion of a Riesz space plays
a key role in the probability theory in Riesz spaces (See for example [14]).

It has been shown by Azouzi and Trabelsi in [2] that a conditional expectation
$T$ on a Dedekind complete Riesz space $E$ can be extended to a largest Riesz
subspace of the universal completion $E^{u}$ of $E$, called the natural domain
of $T$ and denoted by $L^{1}(T)$, to which $T$ extends uniquely to a
conditional expectation. That is

$E\subset L^{1}(T)\subset E^{u}.$

The natural domain of $T$ enjoys several important properties. In fact,
$L^{1}(T)$ is an order dense ideal of $E^{u}$. It is also a Dedekind complete
Riesz space with $e$ as weak unit and it is $T$-universally complete (i.e.,
every increasing net $(x_{\alpha})$ in $L^{1}(T)^{+}$ with $(Tx_{\alpha})$
order bounded in $L^{1}(T)$ is order convergent).

In the particular case where $T$ is the identity function, one can take

$L^{1}(T)=E^{u}.$

It has been proved in [11] that the universal completion of $C(X)$, where $X$
is an extremally disconnected compact Hausdorff space, is the space
$C^{\infty}(X)$ of all extended real continuous functions $f$ on $X$ with the
property that $\\{x\in X\text{ such that }|f(x)|<\infty\\}$ is dense in $X$.

Our problem could then be rephrased as asking for the existence of a strictly
positive order continuous functional on $C^{\infty}(X)$.

######  Proposition 3.1.

Let $X$ be an extremally disconnected compact space and $\varphi$ be an
element of $(C^{\infty}(X)^{\sim}_{n})_{+}$. Then the following assertions are
equivalent:

  * a)

$\varphi(1_{A})=0$ for every open-closed set $A\subset X$

  * b)

$\varphi\equiv 0$

###### Proof.

Only (a $\Longrightarrow$ b) needs details. Suppose that there exists $0<f\in
C^{\infty}(X)$ such that $\varphi(f)>0$. Put

$A^{\lambda}_{f}=\\{x\in X\;\text{such that}\;\lambda 1_{X}(x)-f(x)>0\\}.$

We note that $A^{\lambda}_{f}$ is an open subset of $X$ for all
$\lambda\in\mathbb{R}_{+}$ and $\overline{A^{\lambda}_{f}}$ is an open-closed
set since $X$ is an extremally disconnected.

By the assumption above, $\varphi(1_{\overline{A^{\lambda}_{f}}})=0$ for every
$\lambda\in\mathbb{R}_{+}$ and by Freudenthal Spectral Theorem (Theorem 1.3.9
in [15]), we obtain $\varphi(f)=0$. ∎

The next two technical lemmas will be useful for the sequel. However, they are
of interest independently of the rest of the article.

######  Lemma 3.2.

Let $X$ be an extremally disconnected compact space and $A$ a subset of $X$.
If $A$ verify

$A\cap C\neq\varnothing\text{ implies that }A\subset C$ (3.1)

for every clopen set $C$, then $A$ is connected.

###### Proof.

Suppose by the way of contradiction that $A$ verify 3.1 and that $A$ is
disconnected. Then there exist two open sets $O_{1}$ and $O_{2}$ such that
$O_{1}\cap A\neq\varnothing$, $O_{2}\cap A\neq\varnothing$, $(O_{1}\cap
A)\cap(O_{2}\cap A)=\varnothing$ and $(O_{1}\cap A)\cup(O_{2}\cap A)=A$. Since
$X$ is extremally disconnected and compact, then by Theorem 16.17 in [16], the
clopen sets form a base for the topology. That is $\displaystyle
O_{1}=\bigcup\limits_{\alpha\in\Gamma}C_{\alpha}$, where $C_{\alpha}$ are
clopen sets which gives the existence of $\alpha_{0}$ in $\Gamma$ such that
$A\subset C_{\alpha_{0}}$. This yields to $A\subset O_{1}$ which contradicts
the fact that $(O_{2}\cap A)\neq\varnothing$. ∎

The following lemma is an easy consequence of lemma 3.2.

######  Lemma 3.3.

Let $X$ be an extremally disconnected compact space and $A$ a subset of $X$.
If $A$ is disconnected then there exists a clopen set $C$ such that $A\cap
C\neq\varnothing$ and $A\not\subset C$.

The next proposition will play a key role in the proof of the main theorem.

######  Proposition 3.4.

Let $X$ be an extremally disconnected compact space without isolated points
and $\varphi$ be a non null element of $(C^{\infty}(X)^{\sim}_{n})^{+}$. Then
there exists a disjoint composition of clopen sets $X=K_{1}\sqcup K_{2}$ such
that $\varphi(1_{K_{1}})=\alpha_{1}>0$ and $\varphi(1_{K_{2}})=\alpha_{2}>0.$

###### Proof.

Let us verify first that $\varphi(1_{X})=\alpha>0$. Suppose that this is not
the case. Then $\varphi(1_{X})=0$ which implies $\varphi(1_{A})=0$ for every
open-closed subset $A\subset X$ and Proposition 3.1 gives $\varphi=0$ which
contradicts the fact that $\varphi$ is strictly positive. Now, observe that,
$X$ being extremally disconnected, there exist two disjoint clopen sets
$K_{1}$ and $K_{2}$ such that $X=K_{1}\sqcup K_{2}$. In order to prove that
$\varphi(1_{K_{1}})=\alpha_{1}>0$ and $\varphi(1_{K_{2}})=\alpha_{2}>0.$, We
will suppose by the way of contradiction that for every clopen set $K$, we
have $\varphi(1_{K})=\alpha$ or $\varphi(1_{K})=0$. Put

$\mathfrak{K}_{\alpha}=\\{K\subset X;\ K\text{ is a clopen set and
}\varphi(1_{K})=\alpha\\}.$

If $K^{\prime}$ and $K"$ in $\mathfrak{K}_{\alpha}$, then it would
immmediately follow that $K^{\prime}\cap K"$ in $\mathfrak{K}_{\alpha}$.
Indeed, if $\varphi(1_{K^{\prime}\cap K"})=0$, then $\varphi(1_{K^{\prime}\cup
K"})=2\alpha>\varphi(1_{X})$ which is impossible. Now let

$K^{\circ}=\bigcap\limits_{K\in\mathfrak{K}_{\alpha}}K.$

If $K^{\circ}$ is disconnected then lemma 3.3 implies the existence of some
clopen set $C_{1}$ such that $K^{\circ}\cap C_{1}\neq\varnothing$ and
$K^{\circ}\not\subset C_{1}$. It follows that there is a clopen set
$C_{2}=X\backslash C_{1}$ such that $X=C_{1}\sqcup C_{2}$. Therefore, either
$C_{1}$ or $C_{2}$ is in $\mathfrak{K}_{\alpha}$. Suppose first that
$C_{1}\in\mathfrak{K}_{\alpha}$. It follows that $K^{\circ}\subset C_{1}$
which is impossible. Moreover, if $C_{2}\in\mathfrak{K}_{\alpha}$, then
$K^{\circ}\subset C_{2}$ which contradicts the fact that $C_{1}\cap
C_{2}=\emptyset$. It follows that $K^{\circ}$ is connected. This later fact
together with the total discontinuity of $X$ yield to the existence of
$x^{\star}$ in $X$ such that $K^{\circ}=\\{x^{\star}\\}$. From the fact that
$X$ has no isolated point, follows that

$\inf_{K\in\mathfrak{K}_{\alpha}}\\{1_{K}\\}\equiv 0.$

This is impossible since $\varphi(1_{K})=\alpha$ for all $K$ in
$\mathfrak{K}_{\alpha}$ and $\varphi$ is order continuous. ∎

Although the results of the following theorem seem to be known, we are not
able to find a precise reference. However, we believe that the approach and
the proof are new.

######  Theorem 3.5.

Let $X$ be an extremally disconnected compact space without isolated points
and $card(X)\geq\mathfrak{\aleph}_{0}$. Then
$C^{\infty}(X)^{\sim}_{n}=\\{0\\}.$

###### Proof.

From Proposition 3.4 We claim that there exists a disjoint composition
$X=K_{1}\sqcup K_{2}$ such that $\varphi(1_{K_{1}})=\alpha_{1}>0$ and
$\varphi(1_{K_{2}})=\alpha_{2}>0.$ Since $K_{1}$ is an open set in an
extremally disconnected space then $K_{1}$ is extremally disconnected.
Moreover, $K_{1}$ is closed in a compact space gives that $K_{1}$ is compact.
Applying 3.4 to $K_{1}$ gives the existence of a disjoint composition
$K_{1}=K_{1,1}\sqcup K_{1,2}$ such that $\varphi(1_{K_{1,1}})=\alpha_{1,1}>0$
and $\varphi(1_{K_{1,2}})=\alpha_{1,2}>0.$ Proceeding further, we can find a
sequence $(K_{i_{1},...,i_{k}})_{k\in{\mathbb{N}}}$ with $i_{l}\in\\{1,2\\}$
for every $l\in\\{1,...,k\\}$ such that
$\varphi(1_{K_{i_{1},...,i_{k}}})=\alpha_{i_{1},...,i_{k}}>0.$ Then is defined
an element

$f=\sup_{k\in{\mathbb{N}}}\frac{k}{\alpha_{i_{1},...,i_{k}}}1_{K_{i_{1},...,i_{k}}}$

satisfying $\varphi(f)>M$ for every $M\in\mathbb{R}_{+}$. This trivial
contradiction ends the proof. ∎

As an immediate corollary, we can state an analogous result of Theorem 5.2 in
[13].

######  Corollary 3.6.

Let $X$ be an extremally disconnected compact space with
$card(X)\geq\mathfrak{\aleph}_{0}$. Then the following assertions are
equivalent:

  * a)

$X$ has no isolated point.

  * b)

$C^{\infty}(X)^{\sim}_{n}=\\{0\\}.$

###### Proof.

Only a) $\Rightarrow$ b) needs some details. In order to prove this
implication we will assume that the space $X$ admits isolated points. So, if
$x^{\star}$ is an isolated point in $X$ then any open dense subset of $X$
contains $x^{\star}$. Since for every $f$ in $C^{\infty}(X)$, the set

$\mathfrak{F}_{f}=\\{x\in X\text{ such that }|f(x)|<\infty\\}$

is an open dense subset of $X$, it follows that for any $f$ in
$C^{\infty}(X)$, $f(x^{\star})$ is finite. Now, put $\delta_{x^{\star}}$ as

$\begin{array}[]{lcll}\delta_{x^{\star}}:&C^{\infty}(X)^{\sim}_{n}&\longrightarrow&\mathbb{R}\\\
&f&\longmapsto&f(x^{\star})\end{array}$

It is clear that $\delta_{x^{\star}}$ is in $C^{\infty}(X)^{\sim}_{n}$, which
makes an end to our proof. ∎

Recently, Mozo Carollo proved in [17], that if $X$ is an extremally
disconnected $P$-space then $C(X)$ is universally complete. The next corollary
follows immediately.

######  Corollary 3.7.

Let $X$ be an extremally disconnected compact $P$-space with
$card(X)\geq\mathfrak{\aleph}_{0}$. Then the following assertions are
equivalent:

  * a)

$X$ has no isolated point.

  * b)

$C(X)^{\sim}_{n}=\\{0\\}.$

##  4 Conditional expectation on $C(X)$

In this section, We will provide an affirmative answer of the question of
existence of non trivial strictly positive conditional expectation on spaces
bigger than $\mathcal{L}_{p}$ ones which is the most typical space $C(X)$.

We will recall throughout this section definitions from [18, 19].

######  Definition 4.1.

Let $X$ be a Tychonoff space. We call a point $x\in X$ perfectly disconnected
in $X$ if it is not simultaneously an accumulation point of two disjoint
subsets. If every point of $X$ is perfectly disconnected, then $X$ called
perfectly disconnected.

Notice that every perfectly disconnected space is extremally disconnected.

Following the notation in [18], we recall the definition of the Alexandroff
duplicate of $X$.

######  Definition 4.2.

Let $X$ be a topological space. The Alexandroff duplicate of $X$ is the space
created by taking two (disjoint) copies of $X$, say $A(X)=X\cup X^{\prime}$.

Recently, the authors in [18], proved in Theorem 2.4 that if $X$ is perfectly
disconnected and the set of isolated points of $X$ is clopen then $A(X)$ is
extremally disconnected.

We are able now, to prove the main result of this section.

######  Theorem 4.3.

Let $X$ be perfectly disconnected and the set of isolated points of $X$ is
clopen then there exists a non trivial conditional expectation operator on
$C(A(X))$.

###### Proof.

Our prove will be constructive. Consider the map

$\begin{array}[]{lrll}T\ :&C(A(X))&\longrightarrow&C(A(X))\\\
&f&\longmapsto&g\end{array}$

where $\displaystyle g(x)=\frac{f(x)+f(x^{\prime})}{2}$ for every $x$ in
$A(X)$. We will prove that $T$ is a conditional expectation on $C(A(X))$. In
this order, we can notice first that $T$ is strictly positive, order
continuous and satisfies $T1=1$.

Furthermore, $T$ is a projection since for all $f\in C(A(X))$ and for all $x$
in $A(X)$, we have

$\begin{array}[]{lrl}T\circ
T(f(x))&=&T\left(\frac{f(x)+f(x^{\prime})}{2}\right)\\\
&=&\frac{1}{2}(T(f(x))+T(f(x^{\prime})))\\\
&=&\frac{1}{2}\left(\frac{f(x)+f(x^{\prime})}{2}+\frac{f(x^{\prime})+f(x)}{2}\right)\\\
&=&T(f(x)).\end{array}$

It remains now to show that $R(T)$ is a Dedekind complete Riesz subspace.
There are two methods to prove it. The first one is a direct application of
Theorem 3.1 in [20]. The second possibility is the observation that

$R(T)=\\{g\in C(A(X))\text{ such that }g(x)=g(x^{\prime})\;\text{ for all
}x\in A(X)\\}.$

Indeed, let $g$ be in $R(T)$. By definition, there exists $f\in C(A(X))$ such
that $Tf=g$. Then
$g(x^{\prime})=\frac{f(x^{\prime})+f(x^{\prime\prime})}{2}=\frac{f(x^{\prime})+f(x)}{2}=g(x)$
for all $x\in A(X)$. Conversely, it is clear that $Tg=g$ for any $g\in
C(A(X))$ satisfying $g(x)=g(x^{\prime})$ for all $x$ in $A(X)$ which makes an
end to the proof,

∎

The important question that remains open is to characterize the spaces $X$ for
which there are conditional expectation operators on $C(X)$.

We express our thanks to Professors Marat Pliev, Youssef Azouzi and Vladimir
Troitsky for the fruitful discussions we had about this work.

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# Probing the Diversity of Type Ia Supernova Light Curves in the Open
Supernova Catalog

Chang Bi Department of Mathematics and Statistics, University of Victoria,
3800 Finnerty Road, V8P 5C2, Victoria, BC, Canada  National Research Council
of Canada, Herzberg Astronomy & Astrophysics Research Centre,
5071 West Saanich Road, Victoria, BC V9E 2E7, Canada  Tyrone E. Woods National
Research Council of Canada, Herzberg Astronomy & Astrophysics Research Centre,
5071 West Saanich Road, Victoria, BC V9E 2E7, Canada  Department of Physics
and Astronomy, Allen Building, 30A Sifton Rd, University of Manitoba, Winnipeg
MB R3T 2N2, Canada  Sébastien Fabbro National Research Council of Canada,
Herzberg Astronomy & Astrophysics Research Centre,
5071 West Saanich Road, Victoria, BC V9E 2E7, Canada

(Received 12 Oct 2023; Revised 22 Dec 2023; Accepted 03 Jan 2024)

###### Abstract

The ever-growing sample of observed supernovae enhances our capacity for
comprehensive supernova population studies, providing a richer dataset for
understanding the diverse characteristics of Type Ia supernovae and possibly
that of their progenitors. Here, we present a data-driven analysis of observed
Type Ia supernova photometric light curves collected in the Open Supernova
Catalog. Where available, we add the environmental information from the host
galaxy. We focus on identifying sub-classes of Type Ia supernovae without
imposing the pre-defined sub-classes found in the literature to date. To do
so, we employ an implicit-rank minimizing autoencoder neural network for
developing low-dimensional data representations, providing a compact
representation of the supernova light curve diversity. When we analyze light
curves alone, we find that one of our resulting latent variables is strongly
correlated with redshift, allowing us to approximately “de-redshift” the other
latent variables describing each event. After doing so, we find that three of
our latent variables account for $\sim$95% of the variance in our sample, and
provide a natural separation between 91T and 91bg thermonuclear supernovae. Of
note, the 02cx subclass is not unambiguously delineated from the 91bg sample
in our results, nor do either the over-luminous 91T or the under-luminous
91bg/02cx samples form a clearly distinct population from the broader sample
of “other” SN Ia events. We identify the physical characteristics of supernova
light curves which best distinguish SNe 91T from SNe 91bg & 02cx, and discuss
prospects for future refinements and applications to other classes of
supernovae as well as other transients.

supernova — machine learning — catalogs — surveys

††journal: ApJ

##  1 Introduction

Supernovae are easily among the most optically-luminous astrophysical
transients (LSST Science Collaboration et al., 2009). These explosive
endpoints of stellar evolution shape their surrounding interstellar gas,
enriching it as a primary source of heavy elements in the Universe (Johnson,
2019), and carving out the “hot” phase of the interstellar medium (Chevalier,
1977). Their expanding shock waves are also powerful cosmic ray accelerators
(Baade & Zwicky, 1934; Blandford & Eichler, 1987), and may trigger the
formation of new stars (e.g., Kothes et al., 2001).

Table 1: Stastistics for the Open Supernova Catalogue light curve data considered for this analysis Condition | Measurements | Supernovae
---|---|---
initial sample | 1,273,312 | 22,528
contains photometry data | 1,273,312 | 14,468
pass quality cuts | 1,273,185 | 14,460
has host information | 954,824 | 9,456
with considered filters | 877,630 | 8,617
with at least 25 measurements | 855,613 | 5,142

Early efforts in the 20th century to understand the mechanism and origin of
supernovae focused on classifying these events based on their spectroscopic
properties (Minkowski, 1941; Turatto, 2003). Type Ia supernovae (SNe Ia) were
defined by their strong singly-ionized silicon absorption features and the
absence of hydrogen lines. These explosions have since been shown to be
consistent with the runaway thermonuclear burning and disruption of carbon-
oxygen white dwarfs, based on their observed energetic and nucleosynthetic
yields (Hillebrandt & Niemeyer, 2000). The ability to calibrate SN Ia
luminosities empirically, based on their decline times (Rust, 1974; Pskovskii,
1977; Phillips, 1993), or on a timescale stretch (Goldhaber et al., 2001),
permitted the use of SNe Ia as luminosity distance indicators, rendering them
an integral part of the “cosmic distance ladder” and revealing the
accelerating expansion of the Universe (Riess et al., 1998; Perlmutter et al.,
1999). The evolutionary channel which causes these white dwarfs to explode,
however, remains mainly unknown (Maoz et al., 2014). Multiple distinct sub-
classes of Type Ia supernovae, varying in inferred total luminosity, decline
rate, and chemical abundance yields, suggest multiple channels play a role,
but connecting these observed classes to theoretical models is a key, ongoing
theoretical challenge (e.g., Ruiter, 2020).

The advent of dedicated, wide, high-cadence transient surveys, culminating
soon with the Vera Rubin Observatory’s Legacy Survey of Space and Time (LSST,
LSST Science Collaboration et al., 2009), easily outstrips the community’s
existing capacity to follow-up event triggers with time-intensive
spectroscopy. There is now a pressing need to be able to classify transients
based on time-series photometric data (“light curves”) alone. The concurrent
exponential growth of machine learning presents a natural approach, and has
become an essential tool in the transient community (Malz et al., 2019),
including the classification of supernovae (Sasdelli et al., 2016; Pasquet et
al., 2019; Brunel et al., 2019; Villar et al., 2020).

In the following, we present a framework for exploring Type Ia supernovae
based on their light curves and environmental properties alone, using a
dimensionality reduction method known as an implicit rank minimizing
autoencoder (Jing et al., 2020) to guide the interpretation.

In $\S$2, we discuss the Open Supernova Catalog (Guillochon et al., 2017) and
our processing and feature engineering of the available photometric data
therein. In $\S$3, we briefly put our work in the context of other efforts to
automate the classification of supernovae, before describing the
implementation and training of our implicit-rank minimizing autoencoder in
detail, and include further discussion of our use of the concept of mutual
information in $\S$4. In $\S$5, we identify one of our resulting latent
parameters as being strongly correlated with the redshift, and use principal
component analysis (PCA) to “de-redshift” our other latent variables. We find
that three such variables alone describe $\sim$95% of the variance in our
sample, and provide a clear delineation between over-luminous SN 91T events
and under-luminous 91bg events.

Finally, we summarize our results in $\S$7 and discuss prospects for future
improvements and further searches for structure within existing supernova
classes.

##  2 Data Preparation

The Open Supernova Catalog is a freely-available, online repository of
photometric and spectroscopic supernova data, spanning wavelengths from radio
to X-rays, as well as related metadata where available (e.g., host offset,
redshift) (Guillochon et al., 2017). At the time of writing, it was comprised
of 82,863 individual entries for both supernovae as well as related transients
and candidate events. We began our analysis by downloading the entire Open
Supernova Catalog via its standard
API111https://github.com/astrocatalogs/OACAPI. We note while the Open
Supernova Catalog effort has been discontinued, the data is still present on
the repository..

This provided us with 22,528 claimed supernova events. Of these, 14,460 events
had photometric data and reasonable values for the event time (MJD),
magnitudes, and photometric errors; 9,456 events had complete host information
available (drawn from GHOST, Gagliano et al., 2021, and other sources
therein). The data is heterogeneous, and required selection cuts to ease the
analysis, removing events with:

  1. 1.

No photometric data points available in the following 16 bands: $B,V,R,I$,
$u,g,r,i,z$, $u^{\prime},g^{\prime},r^{\prime},i^{\prime},z^{\prime}$.

  2. 2.

SNR $\leq 5$ in all photometric datapoints.

  3. 3.

$\leq 25$ measurements for a given event.

After applying the selection cuts, 5,142 events with 855,613 measurements
remained. Details of the effect of each cut are shown in table 1. We further
select all events that were exclusively identified as Ia (as some events have
multiple, different claimed types from different sources), often from
spectroscopy. This leaves us with 3,640 unambiguous Ia events which have a
combined 681,462 measurements. These Ia events are then labelled as 91T, 91bg,
02cx depending on their claimed type if known, and as others if unknown.222The
dataset is publicly available at
https://www.canfar.net/citation/landing?doi=23.0034

As we will further show, the supernova subtype labels are not used in all
steps of the analysis. Following this downselect and re-labelling of the
dataset, we then perform the following operations in order to prepare the
sample for further analysis:

Convert magnitude to flux: The magnitudes $\mathbf{m}$ and uncertainties
$\mathbf{\sigma_{m}}$ are then converted to flux $\mathbf{f}$ and its
uncertainty $\mathbf{\sigma_{f}}$. The missing uncertainties are manually set
to a constant value, $0.2$ mag (Carroll & Ostlie, 2017):

$\mathbf{f}=10^{\frac{22.5-\mathbf{m}}{2.5}}\,\,,\mathbf{\sigma_{f}}=\frac{\ln
10}{2.5}\times\mathbf{f}\times\mathbf{\sigma_{m}}$

Data splitting: Once the data is transformed into flux, we split it into train
and test sets. We experiment with two split ratios: a balanced 50% training to
50% testing, and a skewed 70% training to 30% testing. Initially, we adopted a
50-50 split, aiming to increase the test set sample size, given the limited
availability of samples with labeled subtypes. Subsequently, to align more
closely with prevalent community standards, we adjusted our approach to the
70-30 split.

Data Augmentation: We augment our dataset artificially with a procedure
similar to ParSNIP (Boone, 2021). This is intended to make the trained model
more robust against artifacts in the data such as noise, missing values, the
uncertainty in determining the time at maximum, etc. The data is augmented by
multiple degenerated copies of the same event, while ensuring to preserve the
same physical properties. It helps the training process to focus on learning
the intrinsic property of the events instead of focusing on the observational
or instrumental effects. The following transformations are used to augment the
training set, applied randomly in no particular order:

  1. 1.

shift the date of each event, by sampling a time shift $s\sim\cal{N}$(0, 20)
days. This transformation intends to simulate an uncertainty in predicting the
reference date.

  2. 2.

drop a fractional $p$ percent of light curve points, where $p\sim\cal{U}$(0,
0.5), in order to mimic the randomness observations of observation sampling.

  3. 3.

scale the amplitude of each event by log-normal $\mathbf{Lognormal}$(0, 0.5).
This simulates the multiplicative noise.

  4. 4.

add noise to each observation according to $\cal{N}$(0, a), where
$a\sim\mathbf{Lognormal}(\log(b),1)$, $b\sim\mathbf{Lognormal}(-4,1)$ This
simulates the heteroscedacity nature of the data.

Pre-processing with Gaussian Process Regression: light curve data are
irregularly sampled in both time and wavelength dimensions and with a wide
range of measurement uncertainties. With a procedure very similar to Boone
(2019), we model the light curves with Gaussian Processes (GP) regression with
a Matérn 3/2 kernel. The smooth modelling enables re-interpolation of the
irregular light curves into a grid that is evenly spaced in time and
wavelength, together with an estimated uncertainty. We choose to sample the
modelled GP light curves onto a grid of 300 sidereal days with 13 bands. The
resulting grid can be thought of as a light curve image (LCI), similar to
previous supernova studies (Pasquet et al., 2019; Villar et al., 2020). The
width of the LCI represents a time span from the -149th day relative to the
date of maximum brightness up to the 150th day; and the height represents 13
equally spaced wavelengths ranging from 3542.19Å to 10506.54Å. Each LCI
contains two layers, the first layer represents the converted flux value, and
the second contains its corresponding the predicted uncertainty from the GP
interpolation. Each LCI is then normalized into the range of [0, 1] by
dividing through by its maximum value. A lower bound of 0.01 is also set for
the uncertainty layer to compensate for the underestimated predicted
uncertainty.

Figure 1: Examples of light curve images (LCI). The red dots represent the
location of original data points. The color of the background indicates the
normalized flux at a given time and wavelength, which is not necessarily
aligned with the density of the original data points. Figure 2: Distribution
of SNe Ia event classes in the training and test sets.  Figure 3: Distribution
of event redshifts in the training and test set.  Figure 4: histogram for host
galaxy offset.

##  3 Machine-learning classification of type Ia supernovae

###  3.1 Related Studies

Previous studies have investigated the application of machine learning and
deep learning architecture to photometric classification. In PELICAN(Pasquet
et al., 2019), the team presents a contrastive deep learning architecture with
a supervised module that infers class label directly from a LCI as well. The
LCI is a single layer image without the uncertainty. The team performed
photometric classification using an architecture that is similar to the ones
used in image classification. To deal with the missing values presented in the
LCI, multiple modules and regularizers are part of the architecture such that
the network is capable of inferring class label from the incomplete images.
The architecture was tested on simulated light curves of the LSST Deep Fields,
and was able to detect 87.4% of Type Ia supernovae with a precision higher
than 98%.

SuperRAENN (Villar et al., 2020) presents a semi-supervised learning method
that involves three major steps in the pipeline that are similar to the ones
presented here: 1) Gaussian Process (GP) interpolation on the multi-band
lightcurves; 2) a recurrent autoencoder to encode a common representation 3)
classification on the encoded vectors with a supervised random forest
(Breiman, 2001). The completed pipeline has an accuracy of 87% across five SN
classes (Type Ia, Ibc, II, IIn, SLSN-I) on PS1-MDS.

Avocado (Boone, 2019) also shows a similar traditional approach, consisting
of: 1) Gaussian Process (GP) interpolation on the multi-band light curves; 2)
data augmentation of the light curves and different redshift; 3) extraction of
physical features from the light curve GP fit 4) classification on extracted
features with a gradient-boosted decision tree. This algorithm was the best-
performing among the 1094 models considered in the blinded phase of the
Photometric LSST Astronomical Time-Series Classification Challenge (Malz et
al., 2019), scoring 0.468 on the organizers’ logarithmic-loss metric with flat
weights for all object classes in the training set, and achieving an AUC of
0.957 for the classification of SNe Ia.

A particularly powerful approach is to combine photometric light curves with
the enormous complementary information in supernova spectra, where available.
Adopting a machine learning approach to spectral classification of supernovae
and their sub-types was pioneered by Módolo et al. (2015), using artificial
neural networks, and Sasdelli et al. (2015), using principal component
analysis. Sasdelli et al. (2017) demonstrated that even with a very small
sample of events it was possible to delineate different explosion models
within their principal component space. Subsequent efforts have expanded a
hybrid approach combining LCI with spectra and other metadata such as host
galaxy characteristics (e.g., Sasdelli et al., 2016; Brunel et al., 2019;
Muthukrishna et al., 2019; Santos et al., 2020; Chen et al., 2020; Arantes
Filho et al., 2021). Taken together, these studies have shown the great
promise for a hybrid approach to refine the characterization of Type Ia
supernova sub-types.

###  3.2 Using autoencoders

Our dataset is composed of multi-band supernova light curves, and where
available their contextual data including the host galaxy environment,
redshift, and Milky Way line-of-sight extinction. Our goal is to be able to
explore the diversity of supernovae without initially imposing a pre-defined
classification scheme, to eventually associate physical models to identified
similar data. For this exploration, we need to homogenize and reduce the
dimensionality of the data set. Reduction to two dimensions for visualization
enables better interpretation. Dimensionality reduction is not required for a
classification objective but having a single consistent and flexible encoding
of the data renders the analysis and exploration more tractable. The light
curves data in this case are also quite heterogeneous, with multiple bands and
time gaps between observations. Here, we learn an encoded version of all the
data of one supernova event, with the encoding framed within the neural
network autoencoder.

In their traditional form (Bank et al., 2023), autoencoders provide a
nonlinear mapping of input data into an encoded representation, capturing
enough information to be able to reconstruct the input data with minimal loss
in the least-square sense. This initial form of self-supervision has been
shown to be flexible and has been utilized for a wide variety of applications.
Several variants of autoencoders exist, and have previously been explored for
supernova classification and analysis (see $\S$3.1 above). For our analysis,
we explored several variants of autoencoders suited for a minimally
constrained encoding, capable of capturing the diversity of the current
supernova dataset. A popular variant is the Variational Autoencoder (VAE)
(Kingma & Welling, 2019), that enables resampling from a Gaussian distribution
on the latent space, thereby becoming more compact for generative modelling.
Here we explicitly keep the latent space more flexible, and focus mode on the
encoder part. We regularize the encoder with an implicit rank minimizing
autoencoder which we detail in the following section.

###  3.3 Implicit Rank Minimizing Weighted Autoencoders

Autoencoders are represented by neural networks, a composition of linear
functions each activated by non-linear activations. By keeping the last
composed functions (layers) before encoding the data, Jing et al. (2020) have
shown an autoencoder implicitly minimizing the rank of the covariance matrix
for latent variables. The latent variables of such a network summarize the
variation of the input data with a minimal number of significant uncorrelated
axes.

Here we exploit this simple version of the autoencoder motivated by the fact
that rank-minimizing decomposition typically extracts more informative and
less redundant features. Let $\mathbf{x}_{i}$ represents all pixels of a LCI;
the optimal weights of the network $f_{\theta}$ are then found by minimizing
the cost function:

$\hat{\theta}=\operatorname*{arg\,min}_{\theta}\frac{1}{N}\sum_{i=1}^{N}\left(\frac{\mathbf{x}_{i}-f_{\theta}(\mathbf{x}_{i},\mathbf{m}_{i})}{\sigma_{\mathbf{x}_{i}}}\right)^{2}$
(1)

where $\theta$ denotes the weights of the autoencoder network; $N$ denotes the
total number of LCI samples; $\sigma_{\mathbf{x}}$ denotes the uncertainty of
$\mathbf{x}$, and $\mathbf{m}$ denotes the associated metadata. The full
network is then composed of an encoder and decoder $f=e\circ d$, where the
encoder can be thought of a succession of non-linear and rank-minimizing
linear layers $e=e_{\mathrm{NL}}\circ e_{\mathrm{L}}$. The losses are weighted
according to their uncertainties similar to a Gaussian log-likelihood, to
encourage the network to reconstruct a light curve that well approximates the
observations with low uncertainty while giving the network flexibility on
observations that have high uncertainty. We note that when the metadata
encoding is enabled, the architecture is not strictly an autoencoder as there
is no decoder to reconstruct the metadata.

When the LCI training set is fed into the network, it is converted into a low-
dimensional vector representation by the encoder. The vectors then pass
through a series of linear layers (rank-minimizing block), further reducing
the rank of their covariance matrix. During training, the decoder uses a
vector $\mathbf{z}$, concatenated with available metadata (see above), to
reconstruct the input LCI. After training, the encoder, along with the rank-
minimizing block, can be used to perform the dimensional reduction.

Figure 5: Overview of IRMAE structure. $\mathbf{z}$ is the vector
representation of the LCI, denoted latent variables. The metadata can be
composed of environmental factors e.g. redshift, colors of the host galaxy and
angular separation.

##  4 Mutual Information Analysis

Before proceeding to our analysis, we introduce one more concept which will
allow us to compare our derived latent variables with observed quantities.
Mutual Information is a non-parametric quantity to measure the dependency of
two random variables. Let $(X,Y)$ be a pair of random variables, $P_{X}(x)$
and $P_{Y}(y)$ be the marginal probability of $X$ and $Y$, and $P_{XY}(x,y)$
be their joint probability. The mutual information(Cover, 1999) of them is
given as:

$I(X;Y)=D_{KL}(P_{XY}||P_{X}P_{Y})$ (2)

where $D_{KL}$ is the Kullback–Leibler divergence(Joyce, 2011), which measures
the similarity of two distribution, in this case, the joint distribution and
the product of two marginal probabilities. The value of mutual information is
non-negative. The mutual information is zero when $X$ and $Y$ are independent
and the value increase along with dependency. The procedure to calculate
mutual information depends on whether the variables are continuous or
discrete. (Ross, 2014)

##  5 Results

###  5.1 Parameter dependence on redshift

After reducing the dimensionality of the LCI, we obtain 30 latent variables,
of which the covariance has a low rank, meaning there are only a small number
of linearly independent components in the space of latent variables
contributing significantly to the overall variability. However, the latent
variables are not necessarily disentangled. The significant linearly
independent components can be a linear combination of the latent variables.
This is reflected in the correlation matrix in figure 6. Moreover, the latent
variables are also entangled with redshift, with most showing a moderate
degree of correlation with redshift, seen in Fig. 7.

Figure 6: Correlation matrix of latent variables of the autoencoder. Most
latent variables are highly correlated. This means that there are redundant
information among the latent variables. Figure 7: Mutual information between
latent variables and redshift. Most of the latent variables also has moderate
dependency with redshift.

###  5.2 De-redshifting using PCA

The two effects mentioned above make analyzing the variability of latent
variables difficult. To de-correlate the latent variables, while
removing/mitigating the effect of redshift, we concatenate the latent variable
with the redshift value and apply a principal component analysis (PCA) on the
concatenated variable. In this way, we transfer the concatenated variable to a
new space $S$, whose axes are not correlated.

We again check the correlation between PCA transformed latent variable and the
value of redshift. We find that only a small number of transformed latent
variables are correlated to redshift, and only one of them shows a strong
correlation (see figure 8). We identify this axis as the one corresponding to
the redshift effect and remove it from the space $S$.

Figure 8: Mutual information between principal components and redshift. Two
principal components have noticeable dependence with redshift, while the other
components have less dependency. This suggests the information about redshift
is concentrated mostly in this two axes.

###  5.3 Independent Component Analysis

By applying the PCA, latent variables are transformed to a new space where all
axes are not correlated to each other. However, no correlation implies
independence only when the variables are Gaussian, which is not always the
case. Independent component analysis (ICA) is a method that finds independent
axes by pushing the assumption of non-normality. It assumes the observed
signal is a mixture of independent non-Gaussian source signal. Suppose we have
an observation $x$ that is a mixture of $n$ source signal $s_{i}$ weighted by
$a_{i}$.

$x=\sum_{i=1}^{n}a_{i}s_{i}$ (3)

The central limit theorem establishes that, in many situations, the summation
of independent random variables are asymptotically normal. This means that the
observation $x$ should be more similar to the Gaussian distribution than the
individual source signal $s_{i}$. ICA uses this principle to find the source
signal by maximizing the non-normality of the source signal under the
constraint that all source signals are uncorrelated. In our case, we consider
the entangled latent variable as the observed variable $x$ and the
disentangled variables as the source signal $s$. We find applying ICA on top
of the previous procedure further mitigates the effect of redshift on the
variables, as shown in Fig. 9.

Figure 9: Mutual information between principal components and redshift. The
overall dependency with redshift is much lower in the new independent
variables comparing to the old ones in Figure 7.

###  5.4 Separating SN Ia subclasses

We manually inspect the rest of the transformed latent variables. We find that
two variables are particularly useful for separating some Type Ia supernova
sub-classes, in particular delineating luminous 91T-like events and
underluminous 91bg-like and 02cx-like events. In Fig. 10, we observe that
91T-like SNe are mostly located in the upper right part of the plot, well
separated from 02cx-like. We also find similar behaviour in the ICA map (see
Fig. 11), where 91T-like are accumulated in lower left of the map while 91bg-
like and 02cx-like are located in the upper right. Further, we train a Random
Forest classifier to verify these observations (Fig.12 and Table 2).
Interestingly, we do not find that either the over-luminous 91T or the under-
luminous 91bg populations are separated from the main body of “other”
supernovae, including all “normal” SNe Ia in the sample.

Note however that we do find that there are three 91T-like supernovae located
in the 91bg-like and 02cx-like region in our parameter space. Upon further
scrutiny, we found that these outliers are in fact the same supernovae in both
maps, namely SN2005eq, SN2016isl, and SNF20080522-000. To assess what this
means for our analysis, we next turn to the physical interpretation of the
mean light curves within each class and contrasting these with the outlier
events.

Figure 10: PC1 vs PC5. The transparent gray points are unlabeled data. In these hand-picked principal axes, we can observe the separation between 91T-like and the other two. Figure 11: UMAP for ICA components. The transparent gray points are unlabeled data. This shows the visualization of all independent components. We got the same observation as Figure 10. Figure 12: Confusion matrix for the classification on test and combined sets. A random forest classifier is trained on the encoded train set and applied on both test and the combined sets. Table 2: Classification on parameter space | Test Set | Combined Set
---|---|---
Accuracy | 0.928571 | 0.999846
Precision | 0.935714 | 0.999847
Recall | 0.928571 | 0.999846
F1 Score | 0.923308 | 0.999846

###  5.5 Mean light curves and physical interpretation

One possible explanation for the outlier events in our sample may be missing
observations in I band, resulting in over-smoothing of the light curves. To
investigate this, we turn to checking the raw and imputed light curves for
these three events.

Among them, SN2016isl and SNF20080522-000 are missing observations in the $I$
band. However, instead of an over smoothed light curve, we found an
interpolation artifact, such that the light curve in the higher wavelength
band has distorted shape in longer wavelength, see Fig. 13.

Figure 13: This, along with, Figure 14 and 15 shows the GP regression curves
for each subtype in selected bands of LCI. These plots intended to show the
pattern of LCI for each subtype and try to explain the three ”misplaced”
91T-like events. The events with solid line are those in question. As shown in
this plot, two of the events has artifacts in the higher wavelength band (8765
Angstroms) of the LCI.

Unlike the other two misplaced events, SN2005eq has observations in the I
band, and the imputed curves also exhibit the double-peaks. This implies that
the autoencoder may not discriminate the light curves based on the signature
double peaks, but other factors.

In our samples, the imputed light curves for 91T-like supernovae typically
have higher relative brightness in shorter wavelength band, comparing to 91bg-
like supernovae (Figs. 14 and 15). Although the imputed light curve of
SN2005eq peaks at the band with the longest wavelength, this is only due to
the extrapolation by the GP; we found that the original light curve does not.

Contrary to physical modelling, data-driven modelling techniques struggle with
extrapolation and GP is no exception. GP regression can be considered as the
average of random walks with specified covariance function conditioned in the
context points. Without the constraint of the context points, the prediction
results of GP regression are usually incorrect. As a result, due to the
missing observations in longer wavelength bands, the imputed light curves for
SN2005eq are peaked at much longer wavelengths than the actual light curves.
This may explain why SN2005eq is misplaced in the 91bg-like region.

Figure 14: This, along with Figure 15 shows the problem of SN2005eq. The plots
here show the bands where SN2005eq is extrapolated, which is similar to 91bg-
like in relative magnitude. Figure 15: GP regression curves in band 11.

##  6 Discussion

The dataset includes information about the supernova host galaxies, such as
their color, redshift, and angular separation from the supernova. However, in
our final model, we use only the host galaxy redshift as metadata, since we
found that including the color and angular separation of the host galaxies did
not significantly improve our results. The current method suffers from
extrapolation problems during the Gaussian process imputation. Due to the
heterogeneous nature of the dataset, it is challenging to avoid cutting large
amounts of data while maintaining quality. The present goal is to develop a
method to address this challenge such that we can use one model to describe
all data from difference sources (with different bands, time range, and
cadence). Ideally, by weighting the loss function of the network with
uncertainty, the autoencoder can focus on the parts of LCI where data exist
and summarize the physical information of the data into an abstract
representation.

However, the extrapolation problem in LCI reveals that the predictive
uncertainty from Gaussian process regression may not properly reflect the
error. The predictive uncertainty can still be small in extrapolated areas of
the LCI. This causes the autoencoder to include the imputation artifacts in
its learning and inferring process, polluting the abstract representation.

In order to mitigate the imputation issue, we propose two solutions. The first
is to apply stricter conditions in the quality cut, such that the data become
easier to impute; the second one is to adjust the setting of GP regression
that is tailored to our dataset but it will not guarantee that generalizes to
other datasets. Unfortunately, either approach would hinder our ability to
construct a universal model that describe compose data with multiple sources,
however this is an area for future work.

##  7 Conclusions

Our approach involved an implicit rank-minimizing autoencoder, which
effectively compressed the LCI parameter space from the Open Supernova
Catalog, achieving a reduction to 30 latent variables. While this number might
not seem remarkable, the true strength of the method lies in its ability to
condense the rank of the latent space, rather than just its dimensionality.
Using PCA, we were able to further identify one principal component as
strongly correlated with the redshift (confirmed using ICA). This allowed us
to de-redshift our parameter space, and identify three latent variables which
otherwise accounted for $\sim$95% of the variance of our sample. Two such
components provided an interesting delineation of Type Ia supernova
subclasses, with confirmed 91T events lying to one side of the resulting
mapping and 91bg and 02cx events lying at the other extreme, and with only
three outlier events for which we were able to identify to causes. Notably,
while the overluminous 91T events and the underluminous 91bg and 02cx events
are well-separated in this space, neither is particularly distinguishable from
the overall population of “other” Type Ia supernova events in the sample,
including all ‘normal’ events. In the future, directly combining this
photometric data with spectroscopic observations, where available, may provide
a powerful means to investigate the physics underlying different Type Ia
supernovae subclasses and their possibly heteraogenous origins.

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