The need for modelling causal knowledge at different levels of granularity arises in several settings. Causal Abstraction provides a framework for formalizing this problem by relating two Structural Causal Models at different levels of detail. Despite increasing interest in applying causal abstraction, e.g. in the interpretability of large machine learning models, the graphical and parametrical conditions under which a causal model can abstract another are not known. Furthermore, learning causal abstractions from data is still an open problem. In this work, we tackle both issues for linear causal models with linear abstraction functions. First, we characterize how the low-level coefficients and the abstraction function determine the high-level coefficients and how the high-level model constrains the causal ordering of low-level variables. Then, we apply our theoretical results to learn high-level and low-level causal models and their abstraction function from observational data. In particular, we introduce Abs-LiNGAM, a method that leverages the constraints induced by the learned high-level model and the abstraction function to speedup the recovery of the larger low-level model, under the assumption of non-Gaussian noise terms. In simulated settings, we show the effectiveness of learning causal abstractions from data and the potential of our method in improving scalability of causal discovery.

An overview of our contributions:
(a.) A linear SCM $\mathcal{H}$, representing the abstract causal model,
is a $\mat{T}$-abstraction
of a linear SCM $\mathcal{L}$, representing the concrete causal model,
whenever the
linear transformation $\mat{T}$
to abstract variables
is interventionally consistent, i.e., whenever it relates
both values and
interventions on the abstract model and the concrete model.
We prove
for each abstract variable $Y$,
the transformation $\mat{T}$
a block $\Pi(Y)$ of concrete causal variables
that necessarily follows
the causal ordering
of the abstract model
and whose parameters
are constrained
by the abstract coefficients.
For each block,
the abstraction function
depends on a possibly smaller
subset of relevant variables,
which we portray as dashed.
(b.) We
a method
to speedup the causal discovery
of the concrete model $\mathcal{L}$
given an additional dataset $\dset_{J}$
sampled from the joint distribution
of the abstract and the concrete model.
In order,
(i.) reconstructs the transformation $\mat{T}$,
(ii.) fits the abstract model by abstracting the concrete dataset $\dset_{\scm{L}}$,
(iii.) infers a set of constraints $\mat{K}$ on which paths cannot exist in the concrete graph,
and finally (iv.) discovers the concrete model in a search space reduced by the constraints.

§ INTRODUCTION

Causal Abstraction formalizes the property of distinct causal models to describe the same phenomenon with different levels of detail <cit.>. Despite having different variables and mechanisms, whenever two Structural Causal Models (SCMs) are in an abstraction relation, there must always exist at least one implementation on the low-level concrete model of any property of the high-level abstract one — such as values, interventions, mechanisms, and endogenous or exogenous distributions.

Abstract causal models allow the interpretation of causal models with large number of variables, such as in climate phenomena <cit.> or brain activation patterns <cit.>. Causal Abstraction has also found wide interest in explainable AI to align machine representations with human-interpretable concepts in feedforward neural networks <cit.>, concept-based neural networks <cit.>, and Large Language Models <cit.>.

Previous works on the definition of Causal Abstraction do not focus on the graphical and parametrical conditions for two models to be in an abstraction relation. Furthermore, the problem of learning abstractions from data, when the high-level model is not known, is still open. In this context, <cit.> and <cit.> propose methods to learn an abstraction function assuming to know both the low-level and the abstract model, while <cit.> and <cit.> assume to have at least the graphical structure of the abstract model.

In this paper, we tackle these issues by focusing on the scenario where two linear SCMs are abstracted by a linear transformation, as shown in <Ref>.
In particular,
we study necessary and sufficient conditions
for abstraction
in terms of the edges
and the coefficients of the models.
We then propose Abs-LiNGAM,
a strategy
to learn from data
the abstract model,
the concrete model,
and their abstraction function
under the further assumption
of non-Gaussian exogenous noise.
We summarize our contributions as follows:

We first prove that abstract edges
necessarily require
edges in the low-level model
to connect relevant variables,
i.e., variables on which
the abstraction function
directly depends (<Ref>).
we show that the abstraction necessarily arranges concrete variables in adjacent and disjoint blocks that must follow the abstract causal ordering (<Ref>).

We then prove a necessary and sufficient condition
for causal abstraction
that relates
the coefficients
of the linear models and
the abstraction function (<Ref>).
In this way,
we can characterize
the set of all concrete models
that are abstracted by a given abstract SCM
and design a complete and correct algorithm to sample any model from this set (<Ref>).

We introduce Abs-LiNGAM,
a method to speedup the causal discovery
of large linear non-Gaussian models
given an additional and small dataset sampled
from the observational joint distribution
of the model
and one of its abstractions.
Abs-LiNGAM recovers the abstraction function,
learns the abstract model
using low-level data,
and finally
constrains the recovery
of the concrete model
by ensuring that the necessary conditions
we introduced
are satisfied (<Ref>).

As we report in <Ref>,
experiments in simulated settings
show that Abs-LiNGAM substantially reduces the search space, and thus the execution time, compared to directly solving
the problem on the low-level dataset
with DirectLiNGAM <cit.>.

We also publicly release online
the code
of Abs-LiNGAM and the experimental settings[<https://github.com/rmassidda/causabs>].

§ BACKGROUND

Given a set of variables $\set{X}$,
we denote the domain
of each variable ${X\in\set{X}}$
as $\dom{X}$
and of any subset ${\set{V}\subseteq\set{X}}$
as ${\dom{\set{V}}}$.
a Structural Causal Model <cit.>
as a tuple ${\scm{M} =
\left(
\set{X},
\set{E},
\dist{P}_{\set{E}}\right)}$,

$\set{X}$ is a set containing $d$ distinct endogenous variables,

$\set{E}$ is a set containing $d$ distinct exogenous variables,

is a causal mechanism, i.e. a function
that determines the value
of the variable $X\in\set{X}$
given its parents $\pa{X}$
and the exogenous noise term $E_X\in\set{E}$,

$\dist{P}_{\set{E}}$ is the joint distribution over $\set{E}$.

We assume
that parental relations
a directed acyclic graph $\scm{G_M}$
and, consequently,
that the reduced form of the model
always has a unique solution <cit.>.
By slightly abusing the notation,
we denote as $\scm{M}$
both the SCM
and its reduced form $\scm{M}\colon\dom{\set{E}}\to\dom{\set{X}}$
mapping exogenous to endogenous values.
A hard intervention
is an assignment $i = (\set{V}\gets\vec{v})$
on a subset of variables $\set{V}\subseteq\set{X}$
that replaces
each mechanism of the variables $\set{V}$
with a constant value $\vec{v}\in\dom{\set{V}}$.
We denote as $\set{I}^\ast$
the set of all hard interventions
on an SCM, containg all possible assignments
to any subset of endogenous variables,
also including the empty intervention.
an intervention $i$
results in a different SCM $\scm{M}^i$
defined by
the tuple
\dist{P}_{\set{E}}\right)}$,
${f_X^i = f_X}$
if $X\notin\set{V}$
$f_X^i(\cdot) = v_X$
We then define
the restriction of an intervened causal model
as the set of values that the model can take
after the intervention,
which we denote as

\begin{align}
\rst{\scm{M}^{\set{V}\gets\vec{v}}}
= \setdef{\vec{x}\in\dom{\set{X}}\mid\vec{x}_{\set{V}} = \vec{v}},
\end{align}

where $\rst{\scm{M}}=\dom{\set{X}}$
for a non-intervened SCM.

We assume
and causal sufficiency,
i.e., the absence of hidden confounding and selection bias <cit.>.
In particular,
implies the absence of
canceling paths across variables,
while causal sufficiency
implies mutual independence
of exogenous terms, as in
$E_1 \indep E_2$ for any $E_1,E_2\in\set{E}$.

A linear SCM ${\scm{M}=\left(\set{X}, \set{E}, \mat{W}, \dist{P}_{\set{E}}\right)}$,
also known as a linear Additive Noise Model (ANM) <cit.>,
is an SCM
whose structural equations
are linear
and represented
by an upper-triangular adjacency matrix ${\mat{W}\in\real^{d \times d}}$,
as in

\begin{align}
\mat{X} = \mat{W}^\tr \mat{X} + \mat{E}.
\end{align}

We can compute the reduced form of the model
in closed form

\begin{align}
\scm{M}(\vec{e})
\mat{F}^\tr \vec{e},
\end{align}

where $\mat{F} = {(\mat{I}-\mat{W})}^\inv$ is a $d\times d$ linear transformation.

Causal Abstraction theory
relates variables
different SCMs
to determine
whether they
in a consistent way
the same system
at different levels of detail <cit.>.
Overall, we refer to
concrete, or low-level, causal models
as $\scm{L}=(\set{X},\set{E},\set{f},\dist{P}_{\set{E}})$
and to
abstract, or high-level, causal models
as $\scm{H}=(\set{Y},\set{U},\set{g},\dist{P}_{\set{U}})$,
where ${|\set{X}| \geq |\set{Y}|}$.
An abstraction
requires to consider
two subsets of
on the concrete
and the abstract model.

In this work,
we focus on strong abstractions,
any concrete
or abstract intervention
is allowed, i.e.,
given a surjective function
is a $\tau$-abstraction
of $\scm{L}$
if and only if
there exists a surjective function $\gamma\colon\dom{\set{E}}\to\dom{\set{U}}$
on the exogenous variables
such that,
for any low-level intervention $i\in\set{I}$
any exogenous configuration
it holds

\begin{align}\label{eq:intconst}
\tau(\scm{L}^i(\vec{e}))
\scm{H}^{\omega(i)}(\gamma(\vec{e})),
\end{align}

the intervention map $\omega\colon\set{I}\to\set{J}$
is uniquely induced by the value abstraction function $\tau$ <cit.>.
Formally, $\omega(i)=j$
if and only if
$\rst{\scm{H}^j} = \tau(\rst{\scm{L}^i})$
and $j\in\set{J}$,
otherwise $\omega(i)$ is undefined.
We refer to <Ref> as the interventional consistency property and to the function $\gamma$ as the exogenous abstraction function.
we refer to the equation

\begin{align}
\tau(\scm{L}(\vec{e}))
\scm{H}(\gamma(\vec{e}))
\end{align}

on non-intervened models
as observational consistency.
the empty intervention
is necessarily a fixed point
of the intervention map <cit.>,
interventional consistency
observational consistency.

as for the causal mechanisms,
we assume that the abstraction function
does not yield cancelling paths
towards abstract variables.
the composition
of the abstraction function $\tau$,
with the concrete model $\scm{L}$
must not cancel the effect
of concrete variables $\set{X}$
on abstract variables $\set{Y}$.

§ THEORY OF LINEAR CAUSAL ABSTRACTION

In this section,
we study
and structural
of linear causal models
in a linear abstraction relation.
we prove that
the set of concrete relevant variables
on which each abstract variable
are necessarily disjoint
we prove
necessary and sufficient conditions
on the existence of an abstract edge
in terms of the
directed paths
between relevant variables
in the concrete graph
We then show
that the abstraction function
constrains the causal ordering
of concrete blocks
composed of both
relevant and
non-relevant variables
given the notion of concrete block,
we provide an equivalent
formulation of abstraction
based on the model parameters,
which also
the set of possible
concretizations of an abstract model

§ ABSTRACT INFORMATION FOR NON-GAUSSIAN LINEAR DISCOVERY

In this section, we introduce Abs-LiNGAM (<Ref>),
a strategy to exploit our results
on $\mat{T}$-abstraction
to speedup observational causal discovery
of linear non-Gaussian models, e.g. LiNGAM <cit.>.
The intuition is that
whenever we have a $\mat{T}$-abstraction
of an unknown model to learn,
we can exclude all the candidate solutions
not satisfying the graphical conditions we presented in the previous sections.
in Abs-LiNGAM,
we demonstrate
how to infer
prior knowledge for the concrete model
from a small number
of paired concrete-abstract samples,
even when the abstract model
and the abstraction function
are unknown, and an abstract dataset is not directly available.
In the following,
we formalize the data-generation process
and the steps of Abs-LiNGAM.

§.§ Data-Generation Process

As in many real-world applications,
where observations are produced
by sensors or other data-collecting devices,
we assume that samples
from the low-level concrete model
have a significantly larger availability
than high-level abstract samples.
We formalize this
by defining two datasets

\begin{align}
\dset_{\scm{L}} &\sim \dist{P}_{\scm{L}}\\
\dset_J &\sim \dist{P}_{\scm{L}, \scm{H}},
\end{align}

where the former contains
concrete samples only
and the latter
paired observations
from the joint observational distribution
of both models,
such that $|\dset_{J}| \ll |\dset_{\scm{L}}|$.
we define the following
data-generating process,
where we produce
a significantly lower number
of abstract samples.

\begin{align}
\vec{e}^{(i)} &\sim \operatorname{Exponential}
&\text{for } i=1,\dots,|\dset_{\scm{L}}|,\\
\vec{x}^{(i)} &= \scm{L}(\vec{e}^{(i)})
&\text{for } i=1,\dots,|\dset_{\scm{L}}|,\\
\vec{y}^{(i)} &= \scm{H}(\gamma(\vec{e}^{(i)}))
&\text{for } i=1,\dots,|\dset_{J}|.
\end{align}

Since we assume linear and non-Gaussian data,
the models are identifiable
in the limit of infinite data <cit.>.
In <Ref>,
we discuss preliminary results
to tackle an additional
scenario where
we consider
abstract observations
to be perturbed
by random noise.

§.§ Abs-LiNGAM

Since we assume a linear transformation,
we can fit the abstraction function
from the joint dataset $\dset_J$
by solving a least-squares problem <cit.>.
for each abstract variable $Y_i\in\set{Y}$,
we can identify
its set of relevant variables $\hat{\Pi}_R(Y)$,

\begin{align}
\hat{\Pi}_R(Y_i) = \{X_k \mid {[\hat{\vec{t}}_i]}_k \neq 0\}.
\end{align}

In practice,
we mask the coefficients
of the fitted abstraction
transformation $\hat{\mat{T}}$
with a small threshold
to handle numerical instability,
which, whenever a sufficient number
of joint samples $|\mathcal{D_J}|$ is available,
ensures that each relevant block pertains to a single abstract variable.

Abstract Causal Discovery.
Then, we focus on learning
the abstract causal structure
from data.
we assume
abstract samples
to be scarce,
even in our simplified setting of linear and non-Gaussian models,
the abstract model
might not be discoverable
by the high-level samples
in the joint dataset $\dset_J$ alone.
after having identified the abstraction function,
we can use it on the concrete dataset
to abstract each sample as in

\begin{align}
\dset_{\hat{\scm{H}}}= \{ \hat{\mat{T}}^\tr \vec{x} \mid \vec{x} \in \dset_{\scm{L}} \}.
\end{align}

In fact,
the target model
is a $\mat{T}$-abstraction,
the observational consistency property
ensures that
abstracting concrete samples
is equivalent to directly sampling
from the abstract distribution,
as in the data-generating process.
we can use the newly generated
abstract samples
with any causal discovery algorithm
for linear non-Gaussian models.

Concrete Causal Discovery
we can use the constraints induced by the abstract model to speedup discovery
of the concrete causal model.
As an immediate consequence of <Ref>,
the existence
of an abstract directed path
$Y_i \anc Y_j$
entails the existence
of at least
a concrete directed path
between variables
in the corresponding relevant blocks $\Pi_R(Y_i)$ and $\Pi_R(Y_j)$.
We cannot,
infer which
of the possibly many
ancestral relations
the concrete model contains.
On the other hand,
whenever an abstract path does not exist,
we can infer that
any variable
in the source block
does not cause,
neither directly or indirectly,
any variable in the target block.
We can therefore
restrict the search space
of the concrete causal discovery problem
by excluding
all solutions
that do not satisfy
the following set of constraints

\begin{equation}
\begin{aligned}
\set{K} = \{
&X_k \centernot\anc X_h
\mid
\,X_k \in \Pi_R(Y_i)\\
\land
&\,X_h \in \Pi_R(Y_j)
\land
\,Y_i \centernot\anc Y_j
\}.
\end{aligned}
\end{equation}

We use the DirectLiNGAM algorithm <cit.>
to solve the concrete causal discovery problem,
as it can integrate
prior knowledge
in the form of
forbidden direct paths
and thus restrict the set of candidate solutions.

§ EXPERIMENTAL RESULTS

In this section,
we discuss
our analysis
on the performance
of Abs-LiNGAM (<Ref>)
on simulated data.
In particular,
we validate
a small amount
of paired concrete-abstract
can reduce
the search space,
and thus the execution time,
of DirectLiNGAM <cit.>,
without compromising
the quality of the retrieved
concrete causal structure.
As baseline,
we compare
against applying DirectLiNGAM
to the concrete dataset
without any abstract-induced
prior knowledge.

For each run,
we sample
the parameters
of an abstraction function
of an abstract linear SCM.
We then generate
a concrete causal model
by sampling
one of the
of the abstract model with <Ref>.
We provide
details on our experimental setup
and additional results
in <Ref> and in

We study
the performance
of Abs-LiNGAM
for an increasing number
of paired samples (<Ref>).
Since Abs-LiNGAM is a multi-step algorithm,
the quality of the retrieved
concrete causal model
strictly depends on the correctness
of the abstraction function, the consequent generated abstract data and abstract causal discovery.
As expected,
the size of the paired dataset $|\dset_J|$
is too small,
wrongly identifies
concrete paths
as forbidden
and, compared to the baseline,
fails to retrieve the correct
concrete causal model.
whenever the number
of paired samples
approaches the number
of concrete nodes $|\set{X}|$,
performs similarly
to the baseline
and correctly retrieves
the concrete causal model.
We observe
the same trend
for concrete graphs
of increasing size (<Ref>),
prior knowledge induced from the abstract model
significantly reduces
the execution time
compared to the baseline.

we found that
bootstrapping abstract causal discovery, i.e.,
aggregating several iterations
on randomly extracted sub-datasets,
the performance
on the downstream
concrete discovery task
without noticeably affecting
the execution time,
which is still dominated
by the final concrete causal discovery run.

§ RELATED WORKS

Several works addressed the problem
of clustering together variables
to reduce dimensionality
and maintain the
identifiability of causal effect.
Both <cit.> and <cit.>
deal with the problem of partitioning
a causal graph into clusters
where causal relations at the micro-level
are translated
as causal edges at the macro-level.
study instead a particular class of groups,
which they define as transit clusters,
where only part of the variables
are allowed to have
ingoing or outgoing edgs
from the cluster.

Differently from previous works,
our work
focuses instead
on the necessary conditions
for causal abstraction
and results in different
definitions for the grouping of micro-variables.
It is however
an interesting direction
to assess
whether different assumptions,
for instance on the intervention map,
might lead to comparable definitions.

In parallel, several recent papers explored the problem of fitting an abstraction function from data by focusing on either discrete <cit.> or linear <cit.> SCMs. Notably, apart from interventional samples, all these works assume to have at least partial knowledge of the graphs, the intervention map, or the set of concrete relevant variables corresponding to each abstract one.

Based on our theoretical results on the graphical and parametric conditions of linear abstraction for linear causal models,
we instead propose
to learn both the abstract and the concrete model,
and their abstraction function directly from observational data and
without any prior knowledge or any constraint on the graphical structure of the two models.

§ CONCLUSION

In this paper,
we studied
the necessary and sufficient conditions
on the causal ordering
and the parameters
for two linear Structural Causal Models
to be in a linear abstraction.
we introduced
the first procedure
to sample from
the set of all possible
concretizations of an abstract SCM, which can be used in other abstraction applications.

We also proposed Abs-LiNGAM,
a strategy
to speedup causal discovery
of a linear non-Gaussian concrete causal model
given an additional dataset
of paired observations
on concrete and abstract variables.
we empirically
highlighted how Abs-LiNGAM
abstract information
to reduce the search space
and improve
execution time
without sacrificing
on the
of the discovered

An interesting direction for future work
is to extend our results
to non-linear models and non-linear abstraction functions,
and to tackle the causal sufficiency assumption,
which requires full-observability of the concrete realizations.

This work has been supported by EU-EIC EMERGE (Grant
No. 101070918), by H2020 TAILOR (Grant
No. 952215) and by the EU NextGenerationEU programme
under the funding schemes PNRR-PE-AI (PE00000013)
FAIR — Future Artificial Intelligence Research.

§ APPENDIX

We organize the Appendix as follows.
In <Ref>,
we report further information
on our notation
by summarizing it in a glossary.
Then, in <Ref>,
we report all the proof
for the theoretical results
discussed in the main body.
Finally, in <Ref>,
we present further details
on the generative process
of the synthetic datasets
used for our empirical study,
of which we report additional results
in <Ref>.

§ GLOSSARY

Notation     Definition

$\set{X}$     Set of endogenous concrete variables

$\set{E}$     Set of exogenous concrete variables

$\set{Y}$     Set of endogenous abstract variables

$\set{U}$     Set of exogenous abstract variables

$d$     Number of concrete variables

$b$     Number of abstract variables

$\scm{L}$     Concrete Causal Model

$\scm{H}$     Abstract Causal Model

$\mat{W}$     Weighted Adjacencies of $\scm{L}$

$\mat{M}$     Weighted Adjacencies of $\scm{H}$

$\mat{F}$     Reduced Form of $\scm{L}$

$\mat{G}$     Reduced Form of $\scm{H}$

$\tau$     Endogenous Abstraction Function

$\gamma$     Exogenous Abstraction Function

$\mat{T}$     Linear Endog. Abstraction Transformation

$\mat{S}$     Linear Exog. Abstraction Transformation

$\vec{t}_j$     Vector in $\mat{T}$ abstracting $Y_j$ from $\Pi_R(Y_j)$

$\vec{s}_j$     Vector in $\mat{S}$ abstracting $U_j$ from $\vec{e}_{\Pi(Y_j)}$

$\Pi_R(Y)$     Set of relevant variables for $Y$

$\Pi(Y)$     Block of $Y$

$N_i$     Number of variables in $\Pi(Y_i)$

$\mat{W}_{ij}$     Submatrix of weights from $\Pi(Y_i)$ to $\Pi(Y_j)$

$\mat{F}_{ij}$     Submatrix of sub-model from $\Pi(Y_i)$ to $\Pi(Y_j)$

$\dset_{\scm{L}}$     Dataset sampled from $\dist{P}_{\set{X}}$

$\dset_J$     Dataset sampled from the joint $\dist{P}_{\set{X},\set{Y}}$

§ PROOFS

§.§ Lemma <ref>

<Ref> (Disjoint Relevant).
Let $\scm{H}$ be a $\mat{T}$-abstraction of $\scm{L}$,
where $\scm{H}$ and $\scm{L}$
are two linear SCMs
respectively on variables $\set{Y}$ and $\set{X}$.
for any pair of distinct abstract variables $Y_1, Y_2\in\set{Y}$,
it holds that $\Pi_R(Y_1)\cap\Pi_R(Y_2) = \emptyset$,
where $\Pi_R(Y_1) \neq \emptyset$ and $\Pi_R(Y_2) \neq \emptyset$.

we show that
given an abstract intervention ${j = (Y_1 \gets k)}$
on $Y_1$,
any concrete intervention $i$
such that $\omega(i) = j$
must fix all relevant variables $\rel{Y_1}$.
if we assume the existence
of a non-intervened variable $X_s\in\rel{Y_1}$
the function $\scm{L}^i_{\rel{Y_1}}$
would be non-constant.
$\tau_{Y_1}$ depends on $X_s$
by definition of relevant variable,
interventional consistency would not hold,
as in

\begin{align}
\tau_{Y_1} \circ \scm{L}^i_{\Pi_R(Y_1)}
\neq
\scm{H}^j_{Y_1} \circ \gamma
= k.
\end{align}

for any abstract intervention $j=(Y_1 \gets k)$,
the corresponding concrete interventions
must have form

\begin{align}
i = (\Pi_R(Y_1) \gets \vec{v}),
\end{align}

for any vector $\vec{v}$ such that $\tau_{Y_1}(\vec{v}) = k$,
without intervening on further relevant variables.

We firstly
prove <Ref>
whenever $Y_1 \centernot\anc Y_2$.
we assume the existence
of a non-empty subset $\set{V} = \Pi_R(Y_1) \cap \Pi_R(Y_2)$
of shared variables.
Since $Y_1$ has no causal effect on $Y_2$,
given an high-level intervention
${j=(Y_1 \gets k)}$,
it must hold that

\begin{align}
\scm{H}^j_{Y_2} = \scm{H}_{Y_2}.
\end{align}

However, by intervening on $Y_1$,
any concretization must also fix $\set{V}$.
Therefore, we prove the property by contradiction, as

\begin{align}
\scm{H}^j_{Y_2} \circ \gamma &=
\tau_{Y_2} \circ \scm{L}^i_{\Pi_R(Y_2)} \\
& \neq \tau_{Y_2} \circ \scm{L}_{\Pi_R(Y_2)} \\
&= \scm{H}_{Y_2} \circ \gamma\\
\scm{H}^j_{Y_2} \neq \scm{H}_{Y_2},
\end{align}

given the surjectivity of $\gamma$
and the lack of cancelling paths.

Finally, we can tackle the last scenario,
where $Y_1 \anc Y_2$,
by showing that $\Pi_R(Y) \cap \Pi_R(\an{Y}) = \emptyset$,
where $\an{Y}$ is the set of ancestors of $Y$.
Given the model acyclicity,
for any abstract intervention $j=(Y\gets k)$,
it must hold

\begin{align}
\scm{H}^j_{\an{Y}} = \scm{H}_{\an{Y}}.
\end{align}

if the relevant variables of $Y$
were to overlap
with the relevant variables of its ancestors,
we could show that

\begin{align}
\scm{H}^j_{\an{Y}} \circ \gamma
&= \tau_{\an{Y}} \circ \scm{L}^i_{\an{Y}} \\
&\neq \tau_{\an{Y}} \circ \scm{L}_{\an{Y}} \\
&= \scm{H}_{\an{Y}} \circ \gamma\\
\scm{H}^j_{\an{Y}} \neq \scm{H}_{\an{Y}}.
\end{align}

since interventional consistency does not hold,
$\scm{H}$ is not a $\mat{T}$-abstraction of $\scm{L}$,
which contradicts the hypothesis
and concludes the proof.

§.§ Corollary <ref>

<Ref> (Constructive Abstraction).
Let $\scm{H}$
be a strong $\tau$-abstraction
of $\scm{L}$ where
$\scm{H}$ and $\scm{L}$ are linear SCMs
$\tau$ is a linear transformation.
$\scm{H}$ is a constructive $\tau$-abstraction
of $\scm{L}$.

By definition of linear transformation,
the set of low-level variables
on which an abstract variable $Y\in\set{Y}$
depends through the linear abstraction function $\tau$
coincides with its
set of relevant variables $\rel{Y}\subseteq\set{X}$.
by showing
that the relevant sets are disjoint
whenever the SCMs $\scm{H}$ and $\scm{L}$
are linear,
a $\mat{T}$-abstraction on linear SCMs
is also a constructive abstraction.
By definition of $\mat{T}$-abstraction,
this is equivalent
to state that
a linear $\tau$-abstraction
on linear SCMs
is a constructive abstraction
under our assumption
on the absence of cancelling paths.

§.§ Lemma <ref>

<Ref> (Sufficient Directed Paths)
Let $\scm{H}$ be a $\mat{T}$-abstraction of $\scm{L}$,
where $\scm{H}$ and $\scm{L}$
are two linear SCMs
respectively on variables $\set{Y}$ and $\set{X}$ with graphs $\mathcal{G}_\scm{H}$ and $\mathcal{G}_\scm{L}$.
for any pair
of relevant variables $X_1,X_2\in\Pi_R(\set{Y})$,
such that
with $Y_1 \neq Y_2 \in \set{Y}$,
it holds

\begin{align}
X_1 \taudirect X_2 \ \ \mathrm{in} \ \mathcal{G}_\scm{L}
\implies
Y_1 \to Y_2 \ \ \mathrm{in} \ \mathcal{G}_\scm{H}.
\end{align}

Let $Y_1, Y_2$ be two distinct abstract variables
let $i,i'$
be two concrete interventions
that fix any relevant variable
except for those in the relevant set $\Pi_R(Y_2)$,
whose assignments differ only in $X_1\in\Pi_R(Y_1)$.

\begin{align}
i &= (\set{V} \gets \vec{v},\,\Pi_R(Y_1) \gets \vec{c})\\
i' &= (\set{V} \gets \vec{v},\,\Pi_R(Y_1) \gets \vec{c}'),
\end{align}

\begin{align}
\set{V}
\bigcup_{Y \in \set{Y} \setminus \{Y_1, Y_2\}} \Pi_R(Y).
\end{align}

Given $X_1 \taudirect X_2$,
there exists at least a directed path
composed only of non-relevant variables,
that are therefore non-intervened.
due to the faithfulness assumption,
the concrete model does not have cancelling paths
and, therefore,
an intervention on a variable
always has an effect on its descendants.
In particular,
since $i,i'$ constrain $X_1$
to two different values,
it holds that

\begin{align}
\scm{L}^i_{X_2} &\neq \scm{L}^{i'}_{X_2}\\
\tau_{Y_2}\circ\scm{L}^i &\neq \tau_{Y_2}\circ\scm{L}^{i'}\\
\scm{H}^j_{Y_2} \circ \gamma &\neq \scm{H}^{j'}_{Y_2} \circ \gamma,
\end{align}

where, given the intervention map,
the concrete interventions
to the following
abstract interventions

\begin{align}
j &=
( \set{Y} \setminus \{Y_1, Y_2\} \gets \tau(\vec{v}), Y_1 \gets \tau_{Y_1}(\vec{c}))\\
j' &=
( \set{Y} \setminus \{Y_1, Y_2\} \gets \tau(\vec{v}), Y_1 \gets \tau_{Y_1}(\vec{c}')).
\end{align}

due to
the surjectivity of $\gamma$,
it also holds

\begin{align}
\scm{H}^j_{Y_2} \neq \scm{H}^{j'}_{Y_2}.
\end{align}

since $j$ and $j'$
differ only in $Y_1$
and fix everything but $Y_2$,
$Y_1$ has a direct effect on $Y_2$,
i.e., $Y_1 \to Y_2$.

§.§ Corollary <ref>

<Ref> (Sufficient Directed Paths)
Let $\scm{H}$ be a $\mat{T}$-abstraction of $\scm{L}$,
where $\scm{H}$ and $\scm{L}$
are two linear SCMs
respectively on variables $\set{Y}$ and $\set{X}$ with graphs $\mathcal{G}_\scm{H}$ and $\mathcal{G}_\scm{L}$.
for any pair
of relevant variables $X_1,X_2\in\Pi_R(\set{Y})$,
such that
with $Y_1 \neq Y_2 \in \set{Y}$,
it holds that

\begin{align}
X_1 \anc X_2 \ \ \mathrm{in} \ \mathcal{G}_\scm{L}
\implies
Y_1 \anc Y_2 \ \ \mathrm{in} \ \mathcal{G}_\scm{H}.
\end{align}

Given <Ref>,
whenever there exists a $\mat{T}$-direct
path between relevant variables $X_1\in\rel{Y_1}$
and $X_2\in\rel{Y_2}$
there must exist an abstract
edge $Y_1 \to Y_2$.
However, if the path is not $\mat{T}$-direct,
then there must exists
some relevant variable $X_3\in\rel{Y_3}$
for another abstract variable $Y_3$ along the path.
We firstly
consider the case where $Y_3 \neq Y_1$ and $Y_3 \neq Y_2$.
there must exist an edge $Y_1 \to Y_3$ and,
by applying the same argument on the path $X_3 \anc X_2$,
the corollary holds for $Y_1 \anc Y_2$.
Due to the acyclicity of the abstract graph,
the case where $Y_3 = Y_1$ or $Y_3 = Y_2$
can arise only at the beginning (resp. the end) of the path.
In this case,
we could consider the successive variable
until we get one different from $Y_1, Y_2$, if any.
If there is none, then there exists a $\mat{T}$-direct path between the relevant variables of $Y_1, Y_2$ and we fallback to the scenario of <Ref>,
which directly entails $Y_1 \anc Y_2$.

§.§ Example <ref>

<Ref> (Unfaithful Concrete Model)

To prove $\mat{T}$-abstraction
of the example, we anticipate
the parametrical characterization
of linear abstraction
which we introduce in <Ref>.
In particular,
given the adjacencies
of the model,

\begin{align}
\mat{W} &= \begin{bmatrix}
0 & 1 & -1 & 0 & 1\\
0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 0
\end{bmatrix}\\
\mat{M} &= \begin{bmatrix}
0 & 0 & 1\\
0 & 0 & 1\\
0 & 0 & 0
\end{bmatrix}
\end{align}

the necessary form for
the exogenous abstraction function,
which we will introduce in <Ref>,

\begin{align}
\mat{S} = \begin{bmatrix}
1 & 0& 0\\
0 & 1& 0\\
0 & 1& 0\\
0 & 1& 0\\
0 & 0& 1
\end{bmatrix}.
\end{align}

we can prove abstraction by showing that
for any $Y_i, Y_j$
it holds that

\begin{align}
\mat{W}_{ij}\vec{s}_j = m_{ij} \vec{t}_i.
\end{align}

For this example,
of particular interest is the case $Y_1 \to Y_2$,
where it holds that

\begin{align}
\mat{W}_{1,2}\vec{s}_2 &= m_{1,2} \vec{t}_1\\
\begin{bmatrix}
1 & -1 & 0
\end{bmatrix}
\begin{bmatrix}
\end{bmatrix}
0 \cdot \begin{bmatrix}1\end{bmatrix}\\
0 &= 0,
\end{align}

and thus $\scm{H}$ $\mat{T}$-abstracts $\scm{L}$.

§.§ Theorem <ref>

<Ref> (Abstract Connectivity)
Let $\scm{H}$ be a $\mat{T}$-abstraction of $\scm{L}$,
where $\scm{H}$ and $\scm{L}$
are two linear SCMs
respectively on variables $\set{Y}$ and $\set{X}$ with graphs $\mathcal{G}_\scm{H}$ and $\mathcal{G}_\scm{L}$.
there exists an edge ${Y_1 \to Y_2}$ in $\mathcal{G}_\scm{H}$
if and only if
for each $X_1\in\Pi_R(Y_1)$
there exists $X_2\in\Pi_R(Y_2)$
such that $X_1 \taudirect X_2$ in $\mathcal{G}_\scm{L}$.

The sufficient condition follows immediately from <Ref>, where we already proved that any $\mat{T}$-direct path between relevant variables entails an abstract edge.

To prove the necessary condition,
we consider instead two abstract interventions $j,j'$
which differ only in $Y_1$
and fix everything but $Y_2$.

\begin{align}
j  &= (Y_1 \gets k, \set{V} \gets \set{v})\\
j' &= (Y_1 \gets k', \set{V} \gets \set{v}),
\end{align}

where $\set{V} = \set{Y} \setminus \{Y_1, Y_2\}$.
since $Y_1$ has a direct linear effect on $Y_2$,
it holds that

\begin{align}\label{eq:proofnecessary}
\scm{H}^j_{Y_2} &\neq \scm{H}^{j'}_{Y_2}\\
\scm{H}^j_{Y_2} \circ \gamma &\neq \scm{H}^{j'}_{Y_2} \circ \gamma\\
\tau_{Y_2} \circ \scm{L}^i &\neq \tau_{Y_2} \circ \scm{L}^{i'},
\end{align}

for any intervention $i, i'$ such that $\omega(i)=j$ and $\omega(i')=j'$.

Let now $X_1 \in \Pi_R(Y_1)$
be a relevant concrete variable
for $Y_1$, and $t_{11}$ be the non-zero coefficient
from $X_1$ to $Y_1$
in the linear abstraction transformation $\mat{T}$.
We can then build two concrete interventions $i,i'$
by setting all relevant variables of $Y_1$
to zero, except for $X_1$.
the interventions have the following form

\begin{align}
i &= (X_1 \gets \frac{k}{t_{11}},\, \Pi(Y_1) \setminus \{X_1\} \gets \vec{0},\, \ldots)\\
i' &= (X_1 \gets \frac{k'}{t_{11}},\, \Pi(Y_1) \setminus \{X_1\} \gets \vec{0},\, \ldots).
\end{align}

If we suppose that
it does not exist a variable $X_2\in\Pi_R(Y_2)$
such that $X_1 \taudirect X_2$,
all directed paths $X_1 \anc X_2$, if any,
are mediated by a relevant variable
of any abstract variable $Y\in \set{Y}\setminus \{Y_2\}$.
Consequently, given our construction of $j,j'$
and consequently $i,i'$,
any path is mediated by an intervened variable
and, therefore, it holds

\begin{align}
\tau_{Y_2} \circ \scm{L}^i = \tau_{Y_2} \circ \scm{L}^{i'},
\end{align}

which however breaks interventional consistency
and implies that $\scm{H}$ is not
a $\mat{T}$-abstraction of $\scm{L}$,
proving the necessary condition by contradiction.

§.§ Corollary <ref>

<Ref> (Connectivity Violation)
Let $\scm{H}$ and $\scm{L}$
be two linear SCMs
respectively on variables $\set{Y}$ and $\set{X}$ with graphs $\mathcal{G}_\scm{H}$ and $\mathcal{G}_\scm{L}$. Consider a linear transformation $\mat{T}$ between them leading to the sets of relevant variables $\Pi_R(\set{Y})$.
If there exists three variables
$X_2\in\Pi_R(Y_2)$, and
such that
both conditions hold

$X_1 \taudirect X_2$
in $\mathcal{G}_\scm{L}$, and

for any $X_4 \in \Pi_R(Y_2)$,
$X_3 \centernot\taudirect X_4$
is not in $\mathcal{G}_\scm{L}$,

then $\scm{H}$ is not a $\mat{T}$-abstraction of $\scm{L}$.

Follows directly from <Ref> which applied to the first item implies that $Y_1 \to Y_2$, and from <Ref>, which applied to the second item implies that $Y_1 \not \to Y_2$, hence providing a contradiction to the assumption that $\scm{H}$ is a $\mat{T}$-abstraction of $\scm{L}$.

§.§ Corollary <ref>

<Ref> (Exogenous Abstraction)
Let $\scm{H}=(\set{Y},\set{U},\set{g},\dist{P}_{\set{U}})$ be a $\mat{T}$-abstraction of
where $\scm{H}$ and $\scm{L}$
are two linear SCMs.
Then, the exogenous abstraction function
has form

\begin{align}
\gamma(\vec{e}) = \mat{S}^\tr \vec{e},
\end{align}

where $\mat{S} = \mat{F} \mat{T}\mat{G}^\inv$
and $\mat{F}, \mat{G}$
the linear
the reduced forms of $\scm{L}$ and $\scm{H}$, i.e., $\scm{L}(\vec{e}) = \mat{F}^T \vec{e}$ and $\scm{H}(\vec{u}) = \mat{G}^T \vec{u}$.

Since $\scm{H}$ $\mat{T}$-abstracts $\scm{L}$,
it must hold $\tau\circ\scm{L}=\scm{H}\circ\gamma$.
Consequently, due to the invertibility
of the reduced form $\scm{H}$
of linear SCMs, it holds that

\begin{align}
\gamma = \scm{H}^\inv\circ\tau\circ\scm{L}.
\end{align}

Since, $\scm{L}$, $\tau$, and $\scm{H}^\inv$
are linear transformations,
their composition coincides
with a linear transformation

§.§ Lemma <ref>

<Ref> (Block Composition)
Let $\scm{H}$ be a $\mat{T}$-abstraction of $\scm{L}$,
where $\scm{H}$ and $\scm{L}$
are two linear SCMs
respectively on variables $\set{Y}$ and $\set{X}$.
for any abstract variable $Y\in\set{Y}$,
it holds
$X\in\Pi(Y)$ if and only if

* $X\in\Pi_R(Y)$, or

* $X \not \in \Pi_R(\set{Y})$, i.e., $X$ is irrelevant, and there exists $X'\in\Pi_R(Y)$ s.t. $X \taudirect X'$.

Let $Y$ be an abstract variable
and ${j = (\pa{Y} \gets \vec{k})}$
be a hard intervention
fixing all of its endogenous parents.
the value of the abstract variable,
$\scm{H}^j_Y(\vec{u})$ depends only
on its exogenous term $U_Y$.
given the definition of concrete block,
the formulation

\begin{align}
\scm{H}^j_Y(\gamma(\vec{e})) &=
\scm{H}^j_Y(\mat{S}^\tr \vec{e}),
\end{align}

depends only on the exogenous
terms $\vec{e}_{\Pi(Y)}$.
Therefore, given the interventional consistency property

\begin{align}
\scm{H}^j_Y(\gamma(\vec{e})) &= \tau_Y(\scm{L}^i_{\Pi_R(Y)}(\vec{e})),
\end{align}

and the lack of cancelling paths,
also depends only on the exogenous terms $\vec{e}_{\Pi(Y)}$,
for any concrete intervention

\begin{align}
i = (\Pi_R(\pa{Y}) \gets \vec{c}),
\end{align}

where $\tau_{\pa{Y}}(\vec{c}) = \vec{k}$.
given the intervention $i$,
the structural mechanisms of $\Pi_R(Y)$
depend only on the exogenous noise
of the relevant variables
and on those variables
whose direct path
is non-mediated by another relevant variable.
Given <Ref>,
any of such relevant variables
must be in the relevant set of a parent,
and thus be constrained by the intervention $i$.
depends only on its relevant variables
and the irrelevant variables
with a $\mat{T}$-direct path
towards the former.

§.§ Lemma <ref>

<Ref> (Disjoint Block)
Let $\scm{H}$ be a $\mat{T}$-abstraction of $\scm{L}$,
where $\scm{H}$ and $\scm{L}$
are two linear SCMs
respectively on variables $\set{Y}$ and $\set{X}$.
If for any two distinct endogenous variables $Y_1, Y_2$
it holds that $\Pi(Y_1) \cap \Pi(Y_2) \neq \emptyset$, then
the abstract model is not causally sufficient.

By definition of concrete block (<Ref>),
each abstract exogenous term $U_Y$
is a function $\gamma$
of the noise terms of the block $\Pi(Y)$.
given two variables $Y_1, Y_2\in\set{Y}$,
we can write

\begin{align}
U_1 &= \gamma_{1} ( E_{\Pi(Y_1)})\\
U_2 &= \gamma_{2} ( E_{\Pi(Y_2)}).
\end{align}

whenever the blocks share
a subset of variables ${\set{S}} = \Pi(Y_1) \cap \Pi(Y_2)$,
both $U_1$ and $U_2$
are a function of the exogenous terms

\begin{align}
\set{V} =
\{ E_X \in \set{E}
\mid X \in \set{S}\}.
\end{align}

the exogenous terms $U_1, U_2$
are not independent and
the variables $Y_1, Y_2$
are then confounded.

§.§ Theorem <ref>

<Ref> (Block Ordering)
Let $\scm{H}$ be a $\mat{T}$-abstraction of $\scm{L}$,
where $\scm{H}$ and $\scm{L}$
are two linear SCMs
respectively on variables $\set{Y}$ and $\set{X}$ with graphs $\mathcal{G}_\scm{H}$ and $\mathcal{G}_\scm{L}$.
for any valid topological ordering $\prec_{\scm{H}}$ of $\mathcal{G}_\scm{H}$
there exists a valid ordering $\prec_{\scm{L}}$ of $\mathcal{G}_\scm{L}$
such that
for any $Y_1, Y_2, Y \in \set{Y}$:

$Y_1 \prec_{\scm{H}} Y_2 \iff
\Pi(Y_1) \prec_{\scm{L}} \Pi(Y_2)$, and

$\Pi(Y) \prec_{\scm{L}} \big( \set{X} \setminus \Pi (\set{Y}) \big). %

Firstly, we recall
that in a valid topological order,
a variable precedes another
only if
there is a directed path
from the former to the latter <cit.>.

\begin{align}
X_1 \anc X_2 &\implies X_1 \prec X_2
\end{align}

Since we always compare abstract variables
with abstract variables
and concrete variables with concrete variables,
in the following
we ease the notation
by avoiding the subscript on the precedence operator $\prec$.

We show the existence
of a valid topological ordering
on the concrete model by construction.
Given the topological ordering on the abstract model,
we assign to each abstract node $Y\in\set{Y}$
an integer $\rho_{\set{Y}}(Y) \in \{1, \ldots, |\set{Y}|\}$
such that

\begin{align}
Y_1 \prec Y_2 \iff \rho_{\set{Y}}(Y_1) < \rho_{\set{Y}}(Y_2).
\end{align}

Then, we can take any valid topological
ordering within any concrete block $\Pi(Y)$
and assign in the same way $\rho_{\Pi(Y)}(X)$
for any
We do the same
for the set $\set{Q}$
of concrete variables
outside of any block,
which we formally define as follows

\begin{align}
\set{Q} = \set{X} \setminus \bigcup_{Y\in\set{Y}} \Pi(Y).
\end{align}

We then assign
the “position”
of each
concrete variable $X\in\set{X}$
through a further integer
defined as follows,

\begin{align}
\rho_{\set{X}} = \begin{cases}
\sum_{Y' \prec Y} |\Pi(Y')| + \rho_{\Pi(Y)}(X)
& \exists Y .\, X \in \Pi(X)\\
\sum_{Y \in\set{Y}} |\Pi(Y)| + \rho_{\set{Q}}(X)
& X \in \set{Q}.
\end{cases}
\end{align}

Notably, since the blocks do not overlap (<Ref>),
the assignment is unique.
We finally define the concrete topological ordering
for any $X_1, X_2 \in\set{X}$ as

\begin{align}
X_1 \prec X_2 \iff \rho_{\set{X}}(X_1) < \rho_{\set{X}}(X_2).
\end{align}

Given this ordering, it holds by construction that

\begin{align}
\forall Y_1, Y_2 \in \set{Y}.\,
Y_1 \prec_{\scm{H}} Y_2 \iff
\Pi(Y_1) \prec_{\scm{L}} \Pi(Y_2)\\
\forall Y \in \set{Y}.\,
\Pi(Y) \prec_{\scm{L}} \{X\in\set{X} \mid X \notin \bigcup_{Y\in\set{Y}} \Pi(Y)\}.
\end{align}

to finally prove the Theorem we have to show
that the ordering we defined is valid for the concrete graph.
we have to show that, for any $X_1, X_2\in\set{X}$,

\begin{align}
X_1 \to X_2 &\implies X_1 \prec X_2\\
&\implies \rho_{\set{X}}(X_1) < \rho_{\set{X}}(X_2).
\end{align}

Case $\{X_1, X_2\} \subset\Pi(Y) \lor \{X_1, X_2\}\subset\set{Q}$.
Whenever $X_1 \to X_2$ and $X_1, X_2$ are in the same block $\Pi(Y)$
for some $Y\in\set{Y}$ or are both in $\set{Q}$,
then $\rho_{\set{X}}(X_1) < \rho_{\set{X}}(X_2)$
by definition.

Case $X_1\in\Pi(Y_1), X_2\in\set{Q}$.
Also holds by definition.

Case $X_1\in\set{Q}, X_2\in\Pi(Y)$.
By definition of block,
this case never occurs,
since otherwise $X_1$ would be in $\Pi$ (<Ref>).
Case $X_1\in\Pi(Y_1), X_2\in\Pi(Y_2)$.
whenever $X_1 \to X_2$ such that $X_1\in\Pi(Y_1)$
for some $Y_1$ and $X_2\in\Pi(Y_2)$ for some $Y_2$,
then $X_1$ is relevant, otherwise it would have also been
in the block $\Pi(Y_2)$, which are necessarily disjoint (<Ref>).
Therefore, given the sufficient condition
on the existence of an abstract edge (<Ref>),
it must hold

\begin{align}
Y_1 &\to Y_2 \\
\implies Y_1 &\prec Y_2 \\
\implies \Pi(Y_1) &\prec \Pi(Y_2) \\
\implies X_1 &\prec X_2.
\end{align}

§.§ Lemma <ref>

<Ref> (Submodel Abstraction)
Let $\scm{H}$ and $\scm{L}$
be two linear SCMs
respectively on variables $\set{Y}$ and $\set{X}$.
$\scm{H}$ is a $\mat{T}$-abstraction of $\scm{L}$
if and only if
$\scm{H}$ is a $\mat{T}$-abstraction of $\scm{L}^\prime$,
where $\scm{L}^\prime$ is a submodel of $\scm{L}$
defined on the subset of variables
$\set{X}^\prime = \Pi(\set{Y})$, i.e., all of the variables in the concrete blocks.

The Lemma directly follows from <Ref>,
where the variables not in any block
always follow in the topological ordering
the remaining. Therefore, by removing them,
for any intervention $i$
the interventional consistency
$\tau \circ \scm{L}'^i  = \tau \circ \scm{L}^i$
still holds
since they do not influence any relevant variable, hence the abstraction function $\tau$, nor any block, hence the exogenous abstraction function $\gamma$.
Similarly, we could add as many variables
and mechanism not influencing the blocks
and interventional consistency would still hold.

§.§ Lemma <ref>

<Ref> (Exogenous Abstraction)
Let $\scm{H}=(\set{Y}, \set{U}, \mat{M},  \dist{P}_{\set{U}})$
and $\scm{L}=(\set{X},  \set{E}, \mat{W}, \dist{P}_{\set{E}})$
be two linear SCMs
such that $\scm{H}$
is a $\mat{T}$-abstraction
of $\scm{L}$, such that $\mat{W}$ follows permutation $\pi_{\scm{H}}$.
Then, the exogenous abstraction function $\gamma\colon\dom{\set{E}}\to\dom{\set{U}}$
is unique and
has form ${\gamma(\vec{e}) = \mat{S}^\tr \vec{e}}$
for a linear transformation ${\mat{S}\in\real^{d \times b}}$
defined as the upper-diagonal block matrix

\begin{align}
\mat{S} = \begin{bmatrix}
\vec{s}_1 & \mat{0} & \cdots & \mat{0} \\
\mat{0} & \vec{s}_2 & \cdots & \mat{0} \\
\vdots & \vdots & \ddots & \vdots \\
\mat{0} & \mat{0} & \cdots & \vec{s}_b,
\end{bmatrix}
\end{align}

$\vec{s}_k = \mat{F}_{kk} \vec{t}_k = {(\mat{I} - \mat{W}_{kk})}^\inv\vec{t}_k$
for any $Y_k\in\set{Y}$.

Given the definition of $\mat{T}$-abstraction,
we can rephrase observational consistency as

\begin{align}
\tau \circ \scm{L} &= \scm{H} \circ \gamma\\
\mat{FT} &= \mat{SG}
\end{align}

where $\mat{F}$ and $\mat{G}$
are respectively the reduced
forms of the concrete and the abstract SCM.
by exploiting the block-definition of $\mat{T}$,
we can reformulate the left side of the equation as

\begin{align}
\begin{bmatrix}
\mat{F}_{11} & \mat{F}_{12} & \cdots & \mat{F}_{1b} \\
\mat{0} & \mat{F}_{22} & \cdots & \mat{F}_{2b} \\
\vdots & \vdots & \ddots & \vdots \\
\mat{0} & \mat{0} & \cdots & \mat{F}_{bb}
\end{bmatrix}
\begin{bmatrix}
\vec{t}_1 & \vec{0} & \cdots & \vec{0} \\
\vec{0} & \vec{t}_2 & \cdots & \vec{0} \\
\vdots & \vdots & \ddots & \vdots \\
\vec{0} & \vec{0} & \cdots & \vec{t}_b
\end{bmatrix}\\
\mat{F}_{11}\vec{t}_1 & \mat{F}_{12}\vec{t}_2 & \cdots & \mat{F}_{1b}\vec{t}_b \\
\mat{0} & \mat{F}_{22}\vec{t}_2 & \cdots & \mat{F}_{2b}\vec{t}_b \\
\vdots & \vdots & \ddots & \vdots \\
\mat{0} & \mat{0} & \cdots & \mat{F}_{bb}\vec{t}_b
\end{bmatrix}
\end{align}

Given that
block variables are not shared (<Ref>)
and follow the same topological order of $\mat{T}$,
the exogenous transformation must also have form

\begin{align}
\mat{S} = \begin{bmatrix}
\vec{s}_1 & \mat{0} & \cdots & \mat{0} \\
\mat{0} & \vec{s}_2 & \cdots & \mat{0} \\
\vdots & \vdots & \ddots & \vdots \\
\mat{0} & \mat{0} & \cdots & \vec{s}_b,
\end{bmatrix}.
\end{align}

We can therefore reformulate the right
side $\mat{SG}$
of the observational consistency
equation as

\begin{align}
\begin{bmatrix}
\vec{s}_1 & \mat{0} & \cdots & \mat{0} \\
\mat{0} & \vec{s}_2 & \cdots & \mat{0} \\
\vdots & \vdots & \ddots & \vdots \\
\mat{0} & \mat{0} & \cdots & \vec{s}_b,
\end{bmatrix}
\begin{bmatrix}
1 & g_{12} & \cdots & g_{1b} \\
0 & 1 & \cdots & g_{2b} \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 1
\end{bmatrix}\\
\vec{s}_1 & g_{12}\vec{s}_1 & \cdots & g_{1b}\vec{s}_1 \\
\mat{0} & \vec{s}_2 & \cdots & g_{2b}\vec{s}_2 \\
\vdots & \vdots & \ddots & \vdots \\
\mat{0} & \mat{0} & \cdots & \vec{s}_b.
\end{bmatrix}
\end{align}

Consequently, for any $Y_i\in\set{Y}$,
it holds $\vec{s}_i = \mat{F}_{ii}\vec{t}_i$.

§.§ Theorem <ref>

<Ref> (Block Abstraction)
Let $\scm{H}=(\set{Y}, \set{U}, \mat{M}, \dist{P}_{\set{U}})$
and $\scm{L}=(\set{X}, \set{E}, \mat{W}, \dist{P}_{\set{E}})$
be two linear SCMs with graphs $\mathcal{G}_{\scm{H}}$ and $\mathcal{G}_{\scm{L}}$ respectively.
Then $\scm{H}$ is a linear $\mat{T}$-abstraction of $\scm{L}$
if and only if
for any valid topological ordering $\prec_{\scm{H}}$ of $\mathcal{G}_{\scm{H}}$
there exists a valid ordering $\prec_{\scm{L}}$ of $\mathcal{G}_{\scm{L}}$
such that,
for any $Y_i,Y_j\in\set{Y}$ it holds

\begin{align}\label{eq:weightconsistency}
Y_i \prec_{\scm{H}} Y_j &\iff \Pi(Y_i) \prec_{\scm{L}} \Pi(Y_j), \ \mathrm{and}\\
\mat{W}_{ij}\vec{s}_j &= m_{ij} \vec{t}_i,
\end{align}

where $\mat{W}_{ij}$ is the $i$-th element on the $j$-th column of $\mat{W}$, and $m_{ij}$ is the $i$-th element on the $j$-th column of $\mat{M}$.

Firstly, we introduce
the following decomposition
of the reduced forms
of the concrete and the abstract model,
which we separately prove in <Ref>.

\begin{align}
\mat{F}_{ij} &= \begin{cases}
{(\mat{I} - \mat{W}_{ii})}^\inv & \text{if } i=j\\
\mat{F}_{ii} (\mat{W}_{ij} + \mat{R}_{ij}) \mat{F}_{jj} & \text{if } i<j\\
\mat{0} & \text{otherwise},\\
\end{cases}\\
\mat{R}_{ij} &= \sum_{i < k < j} \mat{W}_{ik} \mat{F}_{kk} (\mat{W}_{kj} + \mat{R}_{kj})\\
g_{ij} &= \begin{cases}
1 & \text{if } i=j\\
m_{ij} + \rho_{ij} & \text{if } i<j\\
0 & \text{otherwise},\\
\end{cases}\\
\rho_{ij} &= \sum_{i < k < j} m_{ik} (m_{kj} + \rho_{kj})
\end{align}

Necessary Condition.
We show that $\mat{T}$-abstraction
implies both conditions.
For the existence of a valid concrete ordering,
we invite the reader to consult the proof
of <Ref>.
Therefore, we focus on proving that $\mat{T}$-abstraction
entails $\mat{W}_{ij} \vec{s}_j = m_{ij} \vec{t}_i$
for any $Y_i, Y_j \in \set{Y}$.
Given the decomposition
consistency condition $\mat{FT}=\mat{SG}$
from the proof of <Ref>,
for each $i<j$, it must hold that

\begin{align}
\mat{F}_{ij} \vec{t}_j
&= \vec{s}_{i} g_{ij}\\
\mat{F}_{ii} (\mat{W}_{ij} + \mat{R}_{ij}) \mat{F}_{jj} \vec{t}_j
&= \mat{F}_{ii} \vec{t}_i (m_{ij} + \rho_{ij})\\
(\mat{W}_{ij} + \mat{R}_{ij}) \vec{s}_j
&= \vec{t}_i (m_{ij} + \rho_{ij})\\
\mat{W}_{ij} \vec{s}_j
&= m_{ij} \vec{t}_i,
\end{align}

where the first step comes from
the previously introduced decomposition,
proved in <Ref>.
To prove the last step
we firstly notice that

\begin{align}
\mat{R}_{ij} \vec{s}_j = \rho_{ij}\vec{t}_i
\iff
\mat{W}_{ij} \vec{s}_j = m_{ij} \vec{t}_i.
\end{align}

We then prove the statement for each row
by induction on the columns.
We take $j=i+1$ as base case,
where it holds

\begin{align}
\mat{R}_{ij}\vec{s}_j &= \rho_{ij} \vec{t}_{i}\\
\mat{0}\vec{s}_j &= 0 \cdot \vec{t}_{i}\\
\vec{0} &= \vec{0}\\
\implies
\mat{W}_{ij} \vec{s}_j &= m_{ij} \vec{t}_i.
\end{align}

we can show that

\begin{align}
&\mat{R}_{ij} \vec{s}_j =
\sum_{i < k < j}
\mat{W}_{ik} \mat{F}_{kk} (\mat{W}_{kj} + \mat{R}_{kj})
\vec{s}_j\\
&=\sum_{i < k < j}
\mat{W}_{ik} \mat{F}_{kk} \mat{W}_{kj}
\vec{s}_j +
\mat{W}_{ik} \mat{F}_{kk} \mat{R}_{kj}
\vec{s}_j\\
&=\sum_{i < k < j}
\mat{W}_{ik} \mat{F}_{kk} \mat{W}_{kj}
\vec{s}_j +
\mat{W}_{ik} \mat{F}_{kk} \rho_{kj}
\vec{t}_k\\
&=\sum_{i < k < j}
\mat{W}_{ik} \mat{F}_{kk} m_{kj}
\vec{t}_k +
\mat{W}_{ik} \mat{F}_{kk} \rho_{kj}
\vec{t}_k\\
&=\sum_{i < k < j}
\mat{W}_{ik} \mat{F}_{kk}
\vec{t}_k
m_{kj} +
\mat{W}_{ik} \mat{F}_{kk}
\vec{t}_k
\rho_{kj}\\
&=\sum_{i < k < j}
m_{ik}\vec{t}_i m_{kj} +
m_{ik}\vec{t}_i \rho_{kj}\\
&=\sum_{i < k < j}
m_{ik} (m_{kj} + \rho_{kj}) \vec{t}_i\\
&= \rho_{ij} \vec{t}_i.
\end{align}

Sufficient Condition.
We now show that the conditions imply
interventional consistency of the abstraction.
That is, we want to prove that

\begin{align}
\tau_Y \circ \scm{L}^\iota_{\Pi(Y)}
&= \scm{H}^{\omega(\iota)}_Y \circ \gamma,
\end{align}

for any concrete intervention $\iota$
on the relevant sets
defined by the linear abstraction transformation $\mat{T}$.
Firstly, we notice that the equation
is immediately true for any abstract variable $Y\in\set{Y}$
whenever the intervention targets its relevant set.
we focus on the case
where the abstract intervention $\omega(\iota)$
does not affect $Y$.
given that we assume that the
topological ordering of the blocks
coincides with that of the abstract variables,
we can decompose the concrete model as

\begin{align}
\scm{L}^\iota_{\Pi(Y_j)}(\vec{e})
\sum_{Y_i\in\pa{Y_j}}
\left(
\scm{L}^\iota_{\Pi(Y_i)}(\vec{e})
\right]}^\tr
\mat{W}_{ij} + \vec{e}^\tr_{\Pi(Y_j)}
\right) \mat{F}_{jj},
\end{align}

where we
(i.) compute the linear contribution of the parents,
(ii.) sum the exogenos noise of the block,
(iii.) and apply the submodel composed of the internal connections in the block.
Similarly, we can decompose the abstract model as

\begin{align}
\scm{H}_{Y_j}^{\omega(\iota)}(\vec{u})
\sum_{Y_i\in\pa{Y_j}}
\scm{H}_{Y_i}^{\omega(\iota)}(\vec{u})
\cdot
+ u_j.
\end{align}

Abstraction holds whenever
interventional consistency is satisfied
by at least an exogenous transformation $\gamma$.
To continue the proof, we then define it as the linear transformation
from <Ref>, where $\vec{s}_j = \mat{F}_{jj}\vec{t}_j$ for any $Y_j\in\set{Y}$.
we can reformulate interventional consistency as

\begin{align}
\begin{split}
\sum_{Y_i\in\pa{Y_j}}
\left(
\scm{L}^\iota_{\Pi(Y_i)}(\vec{e})
\right]}^\tr
\mat{W}_{ij} + \vec{e}^\tr_{\Pi(Y_j)}
\right) \mat{F}_{jj} \vec{t}_j
\\=
\sum_{Y_i\in\pa{Y_j}}
\scm{H}_{Y_i}^{\omega(\iota)}(\mat{S}^\tr \vec{e})
\cdot
+ \vec{e}_{\Pi(Y_j)}^\tr \vec{s}_j,
\end{split}
\end{align}

which further simplifies to

\begin{align}
\begin{split}
\scm{L}^\iota_{\Pi(Y_i)}(\vec{e})
\right]}^\tr
\mat{W}_{ij} \mat{F}_{jj} \vec{t}_j
\\=
\scm{H}_{Y_i}^{\omega(\iota)}(\mat{S}^\tr \vec{e})
\cdot
\end{split}
\end{align}

given our choice of the exogenous transformation $\mat{S}$.
We prove this last equation by induction
on the topological ordering
of the abstract graph.
In fact, as a base case,
for any root of the graph
the equation holds
given that the parent set is the empty set.
we can finally show
$\mat{W}_{ij}\vec{s}_j = m_{ij} \vec{t}_i$
implies abstraction
as follows

\begin{align}
\scm{H}_{Y_i}^{\omega(\iota)}(\mat{S}^\tr \vec{e})
\cdot
\scm{L}^\iota_{\Pi(Y_i)}(\vec{e})
\right]}^\tr \vec{t}_i
\cdot
\scm{L}^\iota_{\Pi(Y_i)}(\vec{e})
\right]}^\tr
\mat{W}_{ij}
\vec{s}_{j}\\
\scm{L}^\iota_{\Pi(Y_i)}(\vec{e})
\right]}^\tr
\mat{W}_{ij}
\mat{F}_{jj}
\vec{t}_{j}.
\end{align}

§.§ Model Reduction Decomposition

In the following,
we prove the decomposition
of the model reduction matrix $\mat{F}$
from the proof in <Ref>.
To simplifiy the notation, we define the matrix $\mat{A}=(\mat{I}-\mat{W})$.

to solve $\mat{FA}=\mat{I}$
leads to

\begin{align}
{\mat{F}}_{ij} = \begin{cases}
\mat{A}_{ii}^\inv & i =j\\
- \sum_{i<k\leq j} \mat{F}_{ii} \mat{A}_{ik} {\mat{F}}_{kj} & i <j\\
0 & i > j
\end{cases}.
\end{align}

Therefore, we want to prove that whenever $i<j$,
it holds

\begin{align}
- \sum_{i<k\leq j} \mat{F}_{ii} \mat{A}_{ik} {\mat{F}}_{kj}
\mat{F}_{ii}(\mat{W}_{ij} + \mat{R}_{ij})\mat{F}_{jj},
\end{align}

\begin{align}
\mat{R}_{ij} =
\sum_{i < k < j} \mat{W}_{ik} \mat{F}_{kk} (\mat{W}_{kj} + \mat{R}_{kj}).
\end{align}

we simplify the thesis
as follows

\begin{align}
\mat{F}_{ii}(\mat{W}_{ij} + \mat{R}_{ij})\mat{F}_{jj}
- \sum_{i<k\leq j} \mat{F}_{ii} \mat{A}_{ik} {\mat{F}}_{kj}
\\
\mat{F}_{ii}(\mat{W}_{ij} + \mat{R}_{ij})\mat{F}_{jj}
\sum_{i<k\leq j} \mat{F}_{ii} \mat{W}_{ik} {\mat{F}}_{kj}
\\
\mat{F}_{ii}\mat{R}_{ij}\mat{F}_{jj}
\sum_{i<k<j} \mat{F}_{ii} \mat{W}_{ik} {\mat{F}}_{kj}
\\
\mat{R}_{ij}\mat{F}_{jj}
\sum_{i<k<j} \mat{W}_{ik} {\mat{F}}_{kj}.
\end{align}

We finally prove our thesis
by induction
on the decreasing row component $i$,
from $i=j-1$.
In the base case,
both sides of the equation
reduce to zero
and thus the statement holds.
We then prove the inductive case
by showing that if the statement
holds for any $k>i$, then it also
holds for $i$.

\begin{align}
&\sum_{i<k<j} \mat{W}_{ik} {\mat{F}}_{kj}\\
\sum_{i<k<j} \mat{W}_{ik} \mat{F}_{kk} (\mat{W}_{kj} + \mat{R}_{kj}) \mat{F}_{jj}\\
&=\mat{R}_{ij} \mat{F}_{jj}.
\end{align}

§ DATASET

In the following,
we report further details
on the simulation procedure
used to generate the dataset
for the experiments,
which we also visualize in <Ref>.

Abstract Model.
Given a number of desired nodes
and edges,
we sample the abstract model
by randomly sampling
an Erdős-Rényi graph
for the given parameters.
Then, we sample the weights
of the edges
from the uniform distribution
in the interval $[-2, -0.5] \cup [0.5, 2]$.

Abstraction Function.
Given the abstract model,
we sample the abstraction function
by firstly assigning a block size to each node
from the uniform distribution,
whose minimum and maximum values
are given as input.
Then, within each block we randomly choose
at least half of the nodes to be relevant
and randomly assign the remaining
as relevant or not.
We also sample a further block to contain the ignored variables,
for which the abstraction function maps to zero.
We finally sample the abstraction coefficients
from the uniform distribution
in the interval $[-2, -0.5] \cup [0.5, 2]$.

Concrete Model.
Given an abstract model and an abstraction function,
we sample the concrete model
using the algorithm in <Ref>.
Firtsly, we sample
the causal relations within each block
by randomly sampling
an upper triangular matrix
with non-zero entries
from the standard normal distribution.
we employ the Dirichlet distribution
to sample each vector $\vec{v}$
with sum one as requested by the algorithm
to explore the right-inverses
of the exogenous abstraction function.
we randomly sample
from the standard normal distribution.
the weights
to connect ignored variables.

Data Generation.
As we detailed in the main body,
we sample the data from the concrete model
by first sampling the non-Gaussian noise
and then by abstracting the noise
to sample from the abstract model.
In all experiments, we use the Exponential distribution.
We then normalize the data to have zero mean and unit variance
and permute all the variables
in both the concrete and abstract samples.

§ ADDITIVE NOISE ON ABSTRACT OBSERVATIONS

Results of Abs-LiNGAM
over pairs of abstract ($b=5$ nodes)
and concrete ($d\in[25,50]$ nodes)
linear SCMs
after perturbing the abstract observations
with normal noise
of increasing variance $\sigma^2$.
We denote as “Top-1” the strategy
where we force the selection
of at most a single abstract variable
per concrete one and as “Top-1-Refit”
the one where we then refit each abstraction vector.
All results are averaged over 30 independent runs
with $|\dset_{\scm{L}}| = 20000$ concrete samples
and $|\dset_{\scm{J}}| = 150$ paired samples.

In this section, we discuss strategies to handle a further scenario where we consider abstract observations to be further perturbed by random noise. We consider the following generative model for the abstract observations:

\begin{align}
\vec{e}^{(i)} &\sim \operatorname{Exponential}
&\text{for } i=1,\dots,|\dset_{\scm{L}}|,\\
\vec{x}^{(i)} &= \scm{L}(\vec{e}^{(i)})
&\text{for } i=1,\dots,|\dset_{\scm{L}}|,\\
\vec{y}^{(i)} &= \scm{H}(\gamma(\vec{e}^{(i)})) + \vec{\epsilon}^{(i)}
&\text{for } i=1,\dots,|\dset_{J}|,
\end{align}

where $\vec{\epsilon} \sim \mathcal{N}(0, \sigma^2)$
is a Gaussian noise term
and the data-generating process
is the same of <Ref>.
Due to the presence of noise,
minimizing the least-squares error
does not ensure to recover
the true abstraction function.
We thus propose two strategies
to identify the concrete blocks
of each abstract variable.
By exploiting the fact
that each concrete variable
pertains to a single abstract variable,
we can filter the resulting matrix $\hat{\mat{T}}$
to select only the largest component per row
if it is above the threshold.
We find then beneficial
to refit the model once we have identified
the block in this way,
as in

\begin{align}
\vec{t}_i = \argmin_{\vec{t}_i} \left\|
\vec{x}_{\Pi_R{Y_i}} - \vec{t}_i^\tr \vec{y}_i
\right\|^2_2.
\end{align}

In <Ref>,
we report results
for the reconstruction
of the blocks from the paired samples
for increasing
variance $\sigma^2$
of the noise term
for these strategies.

§ ADDITIONAL RESULTS

In this section,
we report additional results
on our experiments
on Abs-LiNGAM (<Ref>).
We mostly consider three settings:
small, where the number of nodes in the abstract model is $b=5$ and the number of nodes in the concrete model is $d\in[25,50]$; medium, where the number of nodes in the abstract model is $b=10$ and the number of nodes in the concrete model is $d\in[50,100]$; and large, where the number of nodes in the abstract model is $b=10$ and the number of nodes in the concrete model is $d\in[100,150]$.
We then report
results on the sensitivity of Abs-LiNGAM
to the number of paired samples $\dset_J$ (<Ref>),
the number of concrete samples $\dset_{\scm{L}}$ (<Ref>),
and the number of nodes in the concrete model $d$ (<Ref>).
Further, we report
results on the quality of the retrieved prior knowledge
given the threshold
used to mask the learned abstraction function $\hat{\mat{T}}$ (<Ref>)
and the threshold used to mask the learned abstract model $\hat{\scm{H}}$ (<Ref>).
we study the retrieval of the prior knowledge
for different number of bootstrap samples
to identify the abstract model $\hat{\scm{H}}$ (<Ref>).
To provide further insights
on the performance of Abs-LiNGAM,
we also report precision and recall
on the three settings
We finally report additional results on the reconstruction
of the abstraction function $\hat{\mat{T}}$
in the small (<Ref>),
medium (<Ref>),
and large (<Ref>) settings.

Visualization of a pair of concrete-abstract models
and their abstraction function.
The abstract graph has 5 nodes and 8 edges
while the concrete has 5 blocks of random size from $[5, 10]$,
with an additional block for the ignored variables.

Results of Abs-LiNGAM
over pairs of abstract ($b=5$ nodes)
and concrete ($d\in[25,50]$ nodes)
linear SCMs.
In all subfigures
we plot the results
for an increasing number of paired samples $\dset_J$
we report the average
size of the concrete graphs
as a vertical dashed line.
Abs-LiNGAM-GT denotes
a ground truth oracle
where the abstraction function
and the abstract model
are given.
The first plot (top left) shows the ROC-AUC
of the retrieved concrete causal model $\hat{\scm{L}}$.
The second plot (top right)
shows the execution time
required to retrieve the concrete causal model.
The third and fourth plots (bottom) show the precision and recall of the prior knowledge
inferred by the learned abstraction function $\hat{\mat{T}}$
and the consequent abstract model $\hat{\scm{H}}$.
All results are averaged over 30 independent runs
with $|\dset_{\scm{L}}| = 15000$ concrete samples.

Results of Abs-LiNGAM
over pairs of abstract ($b=10$ nodes)
and concrete ($d\in[50,100]$ nodes)
linear SCMs.
In all subfigures
we plot the results
for an increasing number of paired samples $\dset_J$
we report the average
size of the concrete graphs
as a vertical dashed line.
Abs-LiNGAM-GT denotes
a ground truth oracle
where the abstraction function
and the abstract model
are given.
The first plot (top left) shows the ROC-AUC
of the retrieved concrete causal model $\hat{\scm{L}}$.
The second plot (top right)
shows the execution time
required to retrieve the concrete causal model.
The third and fourth plots (bottom) show the precision and recall of the prior knowledge
inferred by the learned abstraction function $\hat{\mat{T}}$
and the consequent abstract model $\hat{\scm{H}}$.
All results are averaged over 30 independent runs
with $|\dset_{\scm{L}}| = 15000$ concrete samples.

Results of Abs-LiNGAM
over pairs of abstract ($b=10$ nodes)
and concrete ($d\in[100,150]$ nodes)
linear SCMs.
In all subfigures
we plot the results
for an increasing number of paired samples $\dset_J$
we report the average
size of the concrete graphs
as a vertical dashed line.
Abs-LiNGAM-GT denotes
a ground truth oracle
where the abstraction function
and the abstract model
are given.
The first plot (top left) shows the ROC-AUC
of the retrieved concrete causal model $\hat{\scm{L}}$.
The second plot (top right)
shows the execution time
required to retrieve the concrete causal model.
The third and fourth plots (bottom) show the precision and recall of the prior knowledge
inferred by the learned abstraction function $\hat{\mat{T}}$
and the consequent abstract model $\hat{\scm{H}}$.
All results are averaged over 30 independent runs
with $|\dset_{\scm{L}}| = 15000$ concrete samples.

Results of Abs-LiNGAM
over pairs of abstract ($b=5$ nodes)
and concrete ($d\in[25,50]$ nodes)
linear SCMs.
In all subfigures
we plot the results
for an increasing number
of concrete samples $|\dset_{\scm{L}}|$.
Abs-LiNGAM-GT denotes
a ground truth oracle
where the abstraction function
and the abstract model
are given.
The first plot (top left) shows the ROC-AUC
of the retrieved concrete causal model $\hat{\scm{L}}$.
The second plot (top right)
shows the execution time
required to retrieve the concrete causal model.
The third and fourth plots (bottom) show the precision and recall of the prior knowledge
inferred by the learned abstraction function $\hat{\mat{T}}$
and the consequent abstract model $\hat{\scm{H}}$.
All results are averaged over 30 independent runs
with $|\dset_J| = 100$ paired samples.

Results of Abs-LiNGAM
over pairs of abstract ($b=10$ nodes)
and concrete ($d\in[50,100]$ nodes)
linear SCMs.
In all subfigures
we plot the results
for an increasing number
of concrete samples $|\dset_{\scm{L}}|$.
Abs-LiNGAM-GT denotes
a ground truth oracle
where the abstraction function
and the abstract model
are given.
The first plot (top left) shows the ROC-AUC
of the retrieved concrete causal model $\hat{\scm{L}}$.
The second plot (top right)
shows the execution time
required to retrieve the concrete causal model.
The third and fourth plots (bottom) show the precision and recall of the prior knowledge
inferred by the learned abstraction function $\hat{\mat{T}}$
and the consequent abstract model $\hat{\scm{H}}$.
All results are averaged over 30 independent runs
with $|\dset_J| = 200$ paired samples.

Results of Abs-LiNGAM
over pairs of abstract ($b=10$ nodes)
and concrete ($d\in[100,150]$ nodes)
linear SCMs.
In all subfigures
we plot the results
for an increasing number
of concrete samples $|\dset_{\scm{L}}|$.
Abs-LiNGAM-GT denotes
a ground truth oracle
where the abstraction function
and the abstract model
are given.
The first plot (top left) shows the ROC-AUC
of the retrieved concrete causal model $\hat{\scm{L}}$.
The second plot (top right)
shows the execution time
required to retrieve the concrete causal model.
The third and fourth plots (bottom) show the precision and recall of the prior knowledge
inferred by the learned abstraction function $\hat{\mat{T}}$
and the consequent abstract model $\hat{\scm{H}}$.
All results are averaged over 30 independent runs
with $|\dset_J| = 300$ paired samples.

Results of Abs-LiNGAM
over pairs of abstract ($b=5$ nodes)
and concrete models with increasing size $d\in[5,60]$.
Abs-LiNGAM-GT denotes
a ground truth oracle
where the abstraction function
and the abstract model
are given.
The first plot (top left) shows the ROC-AUC
of the retrieved concrete causal model $\hat{\scm{L}}$.
The second plot (top right)
shows the execution time
required to retrieve the concrete causal model.
The third and fourth plots (bottom) show the precision and recall of the prior knowledge
inferred by the learned abstraction function $\hat{\mat{T}}$
and the consequent abstract model $\hat{\scm{H}}$.
All results are averaged over 30 independent runs
with $|\dset_{\scm{L}}| = 1500$ concrete samples
and $|dset_J| = 2 \cdot |\set{X}|$ paired samples.

Results of Abs-LiNGAM
over pairs of abstract ($b=10$ nodes)
and concrete models with increasing size $d\in[10,120]$.
Abs-LiNGAM-GT denotes
a ground truth oracle
where the abstraction function
and the abstract model
are given.
The first plot (top left) shows the ROC-AUC
of the retrieved concrete causal model $\hat{\scm{L}}$.
The second plot (top right)
shows the execution time
required to retrieve the concrete causal model.
The third and fourth plots (bottom) show the precision and recall of the prior knowledge
inferred by the learned abstraction function $\hat{\mat{T}}$
and the consequent abstract model $\hat{\scm{H}}$.
All results are averaged over 30 independent runs
with $|\dset_{\scm{L}}| = 1500$ concrete samples
and $|dset_J| = 2 \cdot |\set{X}|$ paired samples.

Results of Abs-LiNGAM
over pairs of abstract ($b=10$ nodes)
and concrete models with increasing size $d\in[20,180]$.
Abs-LiNGAM-GT denotes
a ground truth oracle
where the abstraction function
and the abstract model
are given.
The first plot (top left) shows the ROC-AUC
of the retrieved concrete causal model $\hat{\scm{L}}$.
The second plot (top right)
shows the execution time
required to retrieve the concrete causal model.
The third and fourth plots (bottom) show the precision and recall of the prior knowledge
inferred by the learned abstraction function $\hat{\mat{T}}$
and the consequent abstract model $\hat{\scm{H}}$.
All results are averaged over 30 independent runs
with $|\dset_{\scm{L}}| = 1500$ concrete samples
and $|dset_J| = 2 \cdot |\set{X}|$ paired samples.

\begin{align*}
b &= 5 \\
d &\in [25,50]
\end{align*}

\begin{align*}
b &= 10 \\
d &\in [50,100]
\end{align*}

\begin{align*}
b &= 10 \\
d &\in [100,150]
\end{align*}

Analysis of the prior knowledge
inferred by the learned abstraction function $\hat{\mat{T}}$
and the consequent abstract model $\hat{\scm{H}}$
on a concrete model ($d\in[25,50]$ nodes).
We report precision (left) and recall (right)
of the prior knowledge
for different thresholds
to mask the learned abstraction function $\hat{\mat{T}}$.

\begin{align*}
b &= 5 \\
d &\in [25,50]
\end{align*}

\begin{align*}
b &= 10 \\
d &\in [50,100]
\end{align*}

\begin{align*}
b &= 10 \\
d &\in [100,150]
\end{align*}

Analysis of the prior knowledge
inferred by the learned abstraction function $\hat{\mat{T}}$
and the consequent abstract model $\hat{\scm{H}}$
on a concrete model.
We report precision (left) and recall (right)
of the prior knowledge
for different thresholds
to mask the learned abstract model $\hat{\scm{H}}$.

\begin{align*}
b &= 5 \\
d &\in [25,50]
\end{align*}

\begin{align*}
b &= 5 \\
d &\in [25,50]
\end{align*}

\begin{align*}
b &= 5 \\
d &\in [25,50]
\end{align*}

Analysis of the prior knowledge
inferred by the learned abstraction function $\hat{\mat{T}}$
and the consequent abstract model $\hat{\scm{H}}$,
with $b$ nodes,
on a concrete model with $d$ nodes.
We report precision (left) and recall (right)
of the prior knowledge
for different number
of bootstrapped samples
to fit the abstract model $\hat{\scm{H}}$.

Method                        ROCAUC        Precision     Recall        Time

Abs-Fit (Bootstrap=0)        0.965$\pm$0.066   0.957$\pm$0.123   0.940$\pm$0.074   42$\pm$13

Abs-Fit (Bootstrap=1)        0.977$\pm$0.012   0.980$\pm$0.015   0.953$\pm$0.026   42$\pm$14

Abs-Fit (Bootstrap=2)        0.977$\pm$0.012   0.980$\pm$0.015   0.952$\pm$0.027   43$\pm$11

Abs-Fit (Bootstrap=5)        0.977$\pm$0.012   0.980$\pm$0.015   0.952$\pm$0.027   45$\pm$13

Abs-Fit (Bootstrap=10)       0.977$\pm$0.012   0.980$\pm$0.015   0.952$\pm$0.027   46$\pm$12

Abs-LiNGAM-GT                0.977$\pm$0.011   0.982$\pm$0.013   0.953$\pm$0.026   45$\pm$15

DirectLiNGAM                 0.977$\pm$0.011   0.980$\pm$0.013   0.953$\pm$0.026   61$\pm$12

Results of Abs-LiNGAM
over pairs of abstract ($b=5$ nodes)
and concrete ($d\in[25,50]$ nodes)
linear SCMs.
Abs-LiNGAM-GT denotes
a ground truth oracle
where the abstraction function
and the abstract model
are given.
All results are averaged over 30 independent runs
with $|\mathcal{D}_{\mathcal{L}}| = 15000$
$|\mathcal{D}_J| = 150$
paired samples.

Method                        ROCAUC        Precision     Recall        Time

Abs-LiNGAM (Bootstrap=0)     0.963$\pm$0.043   0.939$\pm$0.119   0.926$\pm$0.067   179$\pm$53

Abs-LiNGAM (Bootstrap=1)     0.952$\pm$0.066   0.914$\pm$0.169   0.914$\pm$0.085   181$\pm$53

Abs-LiNGAM (Bootstrap=2)     0.968$\pm$0.027   0.956$\pm$0.041   0.930$\pm$0.061   182$\pm$50

Abs-LiNGAM (Bootstrap=5)     0.968$\pm$0.027   0.955$\pm$0.041   0.930$\pm$0.061   189$\pm$51

Abs-LiNGAM (Bootstrap=10)    0.968$\pm$0.027   0.954$\pm$0.040   0.930$\pm$0.061   194$\pm$51

Abs-LiNGAM-GT                0.969$\pm$0.026   0.965$\pm$0.022   0.931$\pm$0.060   186$\pm$54

DirectLiNGAM                 0.968$\pm$0.025   0.958$\pm$0.020   0.930$\pm$0.061   394$\pm$94

Results of Abs-LiNGAM
over pairs of abstract ($b=10$ nodes)
and concrete ($d\in[50,100]$ nodes)
linear SCMs.
Abs-LiNGAM-GT denotes
a ground truth oracle
where the abstraction function
and the abstract model
are given.
All results are averaged over 30 independent runs
with $|\mathcal{D}_{\mathcal{L}}| = 15000$
$|\mathcal{D}_J| = 270$
paired samples.

Method                        ROCAUC        Precision     Recall        Time

Abs-LiNGAM                   0.927$\pm$0.070   0.919$\pm$0.119   0.845$\pm$0.132   748$\pm$121

Abs-LiNGAM (Bootstrap=1)     0.913$\pm$0.083   0.877$\pm$0.187   0.834$\pm$0.136   731$\pm$116

Abs-LiNGAM (Bootstrap=2)     0.925$\pm$0.072   0.912$\pm$0.130   0.844$\pm$0.132   738$\pm$123

Abs-LiNGAM (Bootstrap=5)     0.926$\pm$0.067   0.913$\pm$0.109   0.844$\pm$0.130   755$\pm$140

Abs-LiNGAM (Bootstrap=10)    0.927$\pm$0.065   0.918$\pm$0.090   0.844$\pm$0.130   775$\pm$183

Abs-LiNGAM-GT                0.927$\pm$0.069   0.920$\pm$0.117   0.845$\pm$0.131   763$\pm$116

DirectLiNGAM                 0.928$\pm$0.061   0.925$\pm$0.047   0.844$\pm$0.128   1608$\pm$212

Results of Abs-LiNGAM
over pairs of abstract ($b=10$ nodes)
and concrete ($d\in[100,150]$ nodes)
linear SCMs.
Abs-LiNGAM-GT denotes
a ground truth oracle
where the abstraction function
and the abstract model
are given.
All results are averaged over 30 independent runs
with $|\mathcal{D}_{\mathcal{L}}| = 15000$
$|\mathcal{D}_J| = 270$
paired samples.

Abstract per Concrete

Reconstruction metrics
of the linear abstraction function $\mathbf{T}$
over pairs of abstract ($b=5$ nodes)
and concrete ($d\in[25,50]$ nodes)
linear SCMs
for an increasing number
of paired samples $|\mathcal{D_J}|$.
For different thresholds,
we report the normalized Hamming Distance (left),
the F1 score (center),
and the average number of abstract variables
assigned to each concrete variable (right).
All results are averaged over 30 independent runs.

Abstract per Concrete

Reconstruction metrics
of the linear abstraction function $\mathbf{T}$
over pairs of abstract ($b=10$ nodes)
and concrete ($d\in[50,100]$ nodes)
linear SCMs
for an increasing number
of paired samples $|\mathcal{D_J}|$.
For different thresholds,
we report the normalized Hamming Distance (left),
the F1 score (center),
and the average number of abstract variables
assigned to each concrete variable (right).
All results are averaged over 30 independent runs.

Abstract per Concrete

Reconstruction metrics
of the linear abstraction function $\mathbf{T}$
over pairs of abstract ($b=10$ nodes)
and concrete ($d\in[100,150]$ nodes)
linear SCMs
for an increasing number
of paired samples $|\mathcal{D_J}|$.
For different thresholds,
we report the normalized Hamming Distance (left),
the F1 score (center),
and the average number of abstract variables
assigned to each concrete variable (right).
All results are averaged over 30 independent runs.