# Exploiting Preference Elicitation in Interactive and User-centered
Algorithmic Recourse: An Initial Exploration

Seyedehdelaram Esfahani  ,  Giovanni De Toni  ,  Bruno Lepri  ,  Andrea
Passerini  ,  Katya Tentori  and  Massimo Zancanaro

(2018)

###### Abstract.

Algorithmic Recourse aims to provide actionable explanations, or recourse
plans, to overturn potentially unfavourable decisions taken by automated
machine learning models. In this paper, we propose an interaction paradigm
based on a guided interaction pattern aimed at both eliciting the users’
preferences and heading them toward effective recourse interventions. In a
fictional task of money lending, we compare this approach with an exploratory
interaction pattern based on a combination of alternative plans and the
possibility of freely changing the configurations by the users themselves. Our
results suggest that users may recognize that the guided interaction paradigm
improves efficiency. However, they also feel less freedom to experiment with
”what-if” scenarios. Nevertheless, the time spent on the purely exploratory
interface tends to be perceived as a lack of efficiency, which reduces
attractiveness, perspicuity, and dependability. Conversely, for the guided
interface, more time on the interface seems to increase its attractiveness,
perspicuity, and dependability while not impacting the perceived efficiency.
That might suggest that this type of interfaces should combine these two
approaches by trying to support exploratory behavior while gently pushing
toward a guided effective solution.

Human-centred AI, Algorithmic Recourse, Counterfactual Examples

††copyright: acmlicensed††journalyear: 2018††doi: XXXXXXX.XXXXXXX††conference:
User Modelling and User-Adapted Interaction; July, 1-4, 2024; Cagliari,
Italy††isbn: 978-1-4503-XXXX-X/18/06††ccs: Human-centered computing HCI design
and evaluation methods††ccs: Computing methodologies Machine learning

##  1\. Introduction

Alice is applying for a loan at a respected bank in her hometown that uses an
automated machine learning system to decide whether to accept or reject loan
applicants. Unfortunately, Alice gets denied the loan and the bank cannot
provide her with any explanation about their decision, since their system is a
”black-box”: it performs well, but it is too complicated for humans to
understand. Alice’s story is fictional, but there are already examples of
machine learning systems applied to decision-making tasks with a high impact
on human lives, such as credit scoring (Bhatore et al., 2020) and recidivism
prediction (Wang et al., 2023a).

Indeed, more than just an explanation about the reasons for the rejection of
her request, Alice would benefit from practical suggestions on how to reverse
the algorithm decision. Algorithmic recourse (AR) (Wachter et al., 2017;
Karimi et al., 2020a) is a particular case of explanations based on
counterfactual analysis (Guidotti, 2022) that can provide a sequence of
actions capable of overturning an unfavorable machine-driven decision (Ustun
et al., 2019). For example, a recourse suggestion would provide Alice with the
actions she needs to take to obtain a loan (e.g., ”increase your monthly
salary by 100$”).

In this respect, algorithmic recourse can better respond to the need for
transparency and fairness required for AI-based systems by several national
authorities in a context where AI-based tools can significantly impact
people’s lives (Wachter et al., 2017). This aspect is gaining increasing
importance as one effective way to implement the rights sought by the European
GDPR (Voigt and Von Dem Bussche, 2017).

Although there is a growing body of research on this topic (Karimi et al.,
2020a; Verma et al., 2020), the assessment of the state-of-the-art recourse
algorithms is usually based on quantitative benchmarking with metrics such as
validity, minimality or similarity (Bodria et al., 2023). While helpful in
appreciating the potential of algorithmic recourse strategies, these
approaches fall short of assessing the effectiveness of recourse strategies
for real users.

While an algorithm must be efficient and effective, it is also of paramount
importance that the interaction paradigm sustains and fosters empowerment by
providing a creative space of opportunities to explore while limiting
confusion and false expectations. In this direction, the recent work by Wang
and colleagues (Wang et al., 2023b) proposes an algorithm based on integer
linear programming to generate counterfactual explanations, together with a
graphical user interface called GAM Coach, to enable end users to develop
recourse plans iteratively. The authors refer to their approach as an
exploratory interface by citing the seminal work by Shneiderman on Human-
centered AI Systems (Shneiderman, 2020, 2022) since it allows users to freely
manipulate the values of the features involved in the decision process.

An evaluation study of GAM Coach provided helpful insights on designing user
interfaces to support users in leveraging these algorithms effectively.
Nevertheless, the authors noted how the exploration might become complicated
and require several interactions to learn how it works. As a design lesson,
they suggest that “developers can use the log data […] to train a new model to
predict users’ preference configurations” (Wang et al., 2023b). Another
interesting result of that study is that, although participants appreciate
being able to control recourse generation with what-if questions,
“interactivity and transparency could occasionally confuse users with
counterintuitive recourse plans”. (Wang et al., 2023b).

Indeed, a less explored but critical desideratum is the degree of
personalization of the recourse plans to match the users’ needs. Tailoring
recourse to users requires an interactive strategy to include humans in the
generation process since some users’ preferences cannot be inferred by their
features alone (Toni et al., 2024).

Our work investigates a different interaction paradigm with respect to the
purely exploratory approach of GAM Coach, while still aimed at implementing
the proactive principle proposed for Human-centered AI Systems: ”to probe the
algorithm boundaries with different inputs” (Shneiderman, 2020). Specifically,
we use an algorithmic recourse method enriched with preference elicitation
capabilities which realizes the personalization of a recourse plan. We propose
an approach based on a guided-interaction pattern aimed at directly eliciting
the user’s preferences needed by the algorithm to head the user toward a
satisfying recourse intervention quickly.

We believe that limiting the exploratory opportunities within an optimized
step-by-step process may help relieve the user of the possible overwhelming
amount of choices in a typical recourse negotiation while still preserving the
gist of the prospective strategy by providing users with ”a better
understanding of each step in the process so that they can prevent mistakes”
(Shneiderman, 2022). In this regard, our primary research question concerns
the user experience of a novel interaction strategy that we refer to as guided
interaction. This strategy maintains an exploratory interaction style while
establishing boundaries via an optimized preference elicitation process.

> RQ: To what extent do users recognize and appreciate a guided-interaction
> strategy as measured by the pragmatic and hedonic dimensions of their
> overall experience?

To provide a first answer to our research question, we present the design of a
graphical user interface based on the guided-interaction pattern as the
foreground of a novel preference elicitation algorithm for recourse. Then, we
compare it with a pilot study in a typical scenario for recourse problems (a
fictional task of money lending) to a purely exploratory interface inspired by
the GAM Coach one (Wang et al., 2023b).

##  2\. Related Work

Algorithmic Recourse is concerned with providing actionable suggestions, or
interventions, that show how to overturn an unfavorable prediction of a
machine learning model via counterfactual scenarios (Wachter et al., 2017;
Byrne, 2019). Formally, let a user sampled from a distribution $P(X)$ be
${\bm{\mathrm{x}}}=\\{x_{1},\ldots,x_{d}\\}\in\mathcal{X}$, where $x_{i}$ is a
single feature (e.g., age, education level, etc.). Each user $x$ is assigned a
binary outcome $y\in\\{0,1\\}$ following $P(Y\mid X)$. Given a dataset
$\\{(x,y)_{i}\\}_{i=1}^{N}$ sampled from $P(X,Y)$, we can learn a classifier
$h:\mathcal{X}\rightarrow\\{0,1\\}$ which predicts the class $y$ given a user
$x$. Following our previous example, $y=1$ could mean ”$x$ will repay the
loan”. We also define the (possibly unknown) user preferences as a vector
${\bm{\mathrm{w}}}\in\mathcal{W}$ sampled from a distribution $P(W)$. An
intervention is a sequence of actions $I=\\{a^{(1)},\ldots,a^{(N)}\\}$ where
each action $a^{(i)}\in\mathcal{A}$ suggests a modification to a single
actionable feature e.g., ”reduce your monthly spending by 100$”. Given the
classifier $h:\mathcal{X}\rightarrow\\{0,1\\}$, in algorithmic recourse we aim
at finding the best sequence of actions $I$ optimizing the following
objective:

(1)
$I^{*}=\operatorname*{argmin}_{I}\;c(I,{\bm{\mathrm{x}}},{\bm{\mathrm{w}}})\quad\mathrm{s.t.}\quad
h(I({\bm{\mathrm{x}}}))\neq h({\bm{\mathrm{x}}})$

where
$c:\mathcal{I}\times\mathcal{X}\times\mathcal{W}\rightarrow\mathbb{R}^{+}$ is
a cost function. In practice, the idea is to find the cheapest intervention
following the user preferences that, if applied to the initial profile, will
induce a different outcome by the classifier. Optimizing Eq. 1 means solving
both a NP-hard combinatorial problem together with estimating the user
preferences. There is already a plethora of methods for optimizing Eq. 1
(Karimi et al., 2020a; Verma et al., 2020), which however discard user
preferences or make some unrealistic assumptions. Many algorithmic recourse
approaches assume preferences are fully specified beforehand (Wachter et al.,
2017; Ramakrishnan et al., 2020; Karimi et al., 2020b; De Toni et al., 2023)
or they assume users can define them via numerical constraints (Poyiadzi et
al., 2020) or by quantifying the cost of each action (Mahajan et al., 2019).
However, it is well known that it is hard for users to specify preferences
quantitatively (Keeney and Raiffa, 1993).

For what concerns user interaction and assessment of user experience, recent
works have explored ways to revive users’ agency when building counterfactual
examples to understand machine learning models, such as ViCE (Gomez et al.,
2020) and DECE (Cheng et al., 2021). However, to the best of our knowledge,
there are no other proposals, beyond GAM Coach (Wang et al., 2023b),
considering both algorithmic aspects and users’ interaction aspects.

##  3\. Structuring Interaction with Recourse based on Preference Elicitation

On the side of the algorithmic recourse, our approach is inspired by the
algorithm proposed by De Toni and colleagues (Toni et al., 2024) which
integrates recourse with Preference Elicitation (Boutilier, 2002) to
interactively estimate user preferences via pairwise comparison queries. While
a detailed description and a formal evaluation of the algorithm are beyond the
scope of this short paper, we present the gist of the algorithm to facilitate
the understanding of the proposed guided-interaction paradigm.

Our approach is based on the idea of eliciting user preferences via an
interaction protocol that asks the user to choose the preferred intervention
from a small set $k$ of alternatives. The options are selected and proposed to
the user at each algorithm interaction. We use a greedy approach to select the
alternatives maximizing the information gain (Viappiani and Boutilier, 2010).
The user choice is then used to update a distribution over cost weights P(W),
progressively converging towards the true user preferences
${\bm{\mathrm{w}}}^{*}$ used by Eq. 1. Lastly, this preference elicitation
strategy is integrated into a reinforcement learning agent coupled with a
discrete search procedure (Monte Carlo Tree Search (Coulom, 2006)) to
efficiently discover recourse solutions. This approach realizes an instance of
the coactive learning strategy as proposed by (Shivaswamy and Joachims, 2015).

The guided-interaction paradigm has been designed to take advantage of this
algorithmic approach (see Fig. 1).

  1. (1)

First, the user is provided an intervention that can overturn the decision
given by the classifier;

  2. (2)

Second, the user has to rate the proposed intervention with a 5-point Likert
scale (”Terrible”, ”Bad”, ”Neutral”, ”Good” and ”Great”);

  3. (3)

Then, for each feature modified by the intervention, the user can specify how
achievable such change is on a 5-point Likert scale ranging from ”Very
difficult” to ”Very easy”. Users can also specify a preferred set of options
for categorical features or an acceptable range for numerical features. See 1,
left, for an example of the preference panel;

  4. (4)

Lastly, the user can decide to accept the proposed intervention or ask to
generate a new one.

Figure 1. The guided interface that uses the algorithmic recourse method
enriched with preference elicitation: the user can only express preferences on
the proposed features since they have been selected by the algorithm as those
that optimize the learning of a personalized cost function.

On the left, the main screenshot of the interface that implements the guided
method. On the right is the detail for the specification of the acceptable
range.

###  3.1. The control interface

The graphical interface designed as a control condition of the study has been
inspired by GAM Coach (Wang et al., 2023b) but with a look-and-feel closer to
our interface discussed above. This has been done to reduce possible biases
due to non-relevant aspects. The control interface implements a purely
exploratory perspective: one or more plans are generated, possibly considering
previous constraints on features set by the user, but without learning from
the user preferences. On the other hand, the interface allows the user to
progressively disclose all the features and specify constraints on features
not present in the suggested plan(s). Figure 2 shows the main screen of the
control interface.

Figure 2. Control interface used in the study; inspired by GAM Coach (Wang et
al., 2023b): the system proposes one or more plans, and the users can express
their attitude and constraints on any available feature. On the left, two
plans each one offering the possibility of accepting it or modifying it; on
the right, one of the plans while being modified, all the actionable features
can be accessed.

Two screenshots of the interface used as the control condition in the study.

##  4\. The Pilot Study

For the pilot study, a loan application task has been chosen as a typical
scenario for recourse assessment. In order to increase the number of features
on which to base the decisions, two datasets: the ”Adult Census Income”
(Becker and Kohavi, 1996) and the ”Lending
Club”111https://www.kaggle.com/datasets/wordsforthewise/lending-club have been
combined. Each dataset contains both continuous and categorical features for a
total of 104 attributes. We selected a subset of 31 attributes that we believe
the user would be reasonably capable of acting upon (such as, Job Category and
Education Level). Then, we defined an action set $\mathcal{A}$ comprising all
the features, those 31 features were marked as actionable, and all the others
as non-actionable but still used in the machine learning process.

As the black-box classifier $h$, we trained an ensemble composed of two multi-
layered perceptron architectures by minimizing the empirical risk over the
training data via stochastic gradient descent (Ruder, 2016). We trained the
classifier to be as realistic as possible by using both the actionable and
unactionable user attributes. In our loan scenario, we assume a loan is
granted to a user only if both ensemble components give a positive
classification.

Study design. The study employed a within-subjects design. Each participant
was asked to solve the same task (overturn an unfavorable outcome) with the
guided-style interface and with the purely exploratory-style interface as a
control condition. The order of the interfaces was randomized.

Procedure. To make sure that the first interaction with the systems resulted
in a rejection of the loan request (which was needed to start the recourse
process), we used an approach based on personas: fictional characters for
which the specific characteristics were prepared in advanced (Matthews et al.,
2012). While personas are commonly used by designers to describe and empathize
with users, we adopted this approach in user testing by asking our users to
play the role of a specific persona rather than responding on the interface
for themselves. We also believe that this approach had the advantage of
minimizing the potential influence of previous experience, personal knowledge,
and attitudes toward lending. We defined two personas, one female and one
male. Each participant was asked to role-play one persona while using one
interface and the other persona while using the other one. The association
between interfaces and personas was also randomized.

Measures. We adopted a mixed methods approach employing quantitative and
qualitative measures. We employed the short version of the User-Experience
Questionnaire (UEQ) (Schankin et al., 2022; Schrepp et al., 2017). This
questionnaire consists of 26 semantic differential items on six scales:
Attractiveness, which measures the overall impression; Perspicuity, which
measures the ease of use and ease of learning, as well as, perceived
familiarity; Efficiency, how efficient and fast is perceived the product;
Dependability, how much the user feel in control of and safe during the
interaction; Stimulation, how exciting and motivating is to use the product;
and finally Novelty, that measures the degree to which the product is
considered innovative and creative, and how much it captures users’ attention.
Since the questionnaire was administered by one experimenter, we selected one
time for each scale, using the following anchors: Attractiveness: unattractive
vs attractive; Perspicuity: complicated vs easy; Efficiency: inefficient vs
efficient; Stimulation: boring vs exciting; Novelty: conventional vs
inventive; Dependability: obstructive vs supportive. The experimenter
administered the questionnaire at the end of the task with each interface.
Besides scoring the semantic differential of each item, the experimenter also
asked for a brief motivation. Those verbal reports were then transcribed and
analyzed with Thematic Analysis (Braun et al., 2019).

Participants. The call for participants was promoted among local university
students, and the sampling was organized through a snowball effect.
Ultimately, 12 participants, three for each combination of persona-interface,
were recruited. The participants had an average age of 25.6 years (standard
deviation 4.3; minimum 19 and max 33 years). Among them, there were ten
females and two males.

##  5\. Results and Discussion

Figure 3 presents the scores of the semantic differentials. The overall scores
were good with an average above 4, except Perspicuity and Dependability, whose
scores ranged between 3 and 4. None of the observed differences are
statistically significant, as confirmed by Friedman’s ranking tests (with
Bonferroni’s correction).

Figure 3. Scores of the UEQ questionnaires for the two interfaces: the value 1
corresponds to the negative anchor and the value 5 corresponds to the positive
anchor for the semantic differential items.

For the guided-style there is a positive correlation between the duration of
the task and the dimension of Attractiveness (Pearsons’ ro=2.6), Perspicuity
(ro=0.73) and Dependability (ro=0.15) while the same correlations are negative
for the exploratory-style interface (respectively, -0.23, -0.32, and -0.12);
for the Efficiency score, the correlation is negative for the exploratory-
style interface and not statistically different from 0 for the guided-style
interface. Stimulation does not seem to correlate with duration in either one
of the interfaces. While these results are preliminary due to the low number
of participants, it’s noteworthy that our users appreciate the freedom
provided by the explorative-style interface, unless it leads to time wastage.
Conversely, more time spent on the guided-style interface is perceived as
useful, possibly because it offers a more direct path to finding a solution.

From a Thematic Analysis of the verbal reports collected as motivations for
the scores of the scales, it emerges that our participants articulated their
experiences along two different dimensions. On one side, the freedom to
propose changes and manipulate the input (even though our participants never
engaged in such elaborate behaviors). On the other side, the need to avoid
confusion and quickly reach a solution.

For example, p1011 commenting on Perspicuity for the exploratory interface
said “I could solve them without effort” and in commenting Efficiency for the
guided “I barely put effort for this”. Yet, in commenting on Stimulation, she
said “I was more excited to use this one [the exploratory] because I had more
plans to check. I liked it more.” Nevertheless, for the guided interface, she
reflected that “Maybe for someone that wants loan approval in real life, it
would be more interesting.”. P1013 discussing Attractiveness said “I prefer
the first platform [the guided-style], in the second one [the exploratory] I
could express my preferences with more details but it took too much time and
confused me.” P1016 on commenting the Perspicuity of exploratory “it was
harder than the first one, but this one you had more options to change I
prefer this one.” She was also more explicit on commenting Stimulation for the
exploratory style “[this one is] more exciting than [the one] before because I
had more control over changing. The fact that you have more plans is more
exciting than having one plan.” On a similar line, P018 on Stimulation of the
exploratory ”this one is more exciting […] you could change the plan then you
have more plans again.” P1014 on commenting on the Dependability of the
exploratory summarized these terms: “I didn’t expect to receive many plans
[positive surprise], and negative surprised because of a long list of
information in the modifying page [the list of all the disclosed features].”.
In a similar venue, P1019 on the Efficiency of the exploratory “I think it was
5, but the other one was shorter, and people (may) find the other one more
efficient.”.

##  6\. Conclusion

Our work addressed a core aspect of the design of Human-Centred AI, namely,
how to balance user control with system autonomy (Shneiderman, 2022), in the
context of AI-based algorithm recourse. Our design was framed in the strategy
of prospective interaction. Still, it differentiates from a purely exploratory
approach by framing the exploration within the boundaries of an optimized
preference elicitation process.

The aim of our work was not just to compare our interface to GAM Coach (Wang
et al., 2023b) but rather to compare the two approaches of prospective methods
for AI-driven systems: a purely exploratory approach and a purely guided
approach. It is worth noting that they differ concerning (i) how the
algorithmic recourse is realized in the backend and (ii) how the interface
fosters either free interactions up to what-if explorations or rather an
efficient but constrained step-by-step process. However, in both cases, users
are engaged in proactively ”probing the algorithm boundaries with different
inputs” (Shneiderman, 2022). Our results suggest that users may recognize that
the guided interaction paradigm improves efficiency. However, they also feel
less freedom to experiment with ”what-if” scenarios. Nevertheless, the time
spent on the purely exploratory interface tends to be perceived as a lack of
efficiency, which reduces attractiveness, perspicuity, and dependability.
Conversely, for the guided-style interface, more time on the interface seems
to increase its attractiveness, perspicuity, and dependability while not
impacting the perceived efficiency. The evidence of this study might suggest
that this type of interfaces should combine these two approaches by trying to
support exploratory behavior while gently pushing toward a guided effective
solution.

We acknowledge that the main limitations of our study include the small sample
size and the likelihood that our participants may not have fully grasped the
task’s difficulty due to their lack of previous experience in money lending.
Nevertheless, albeit still preliminary, we believe our work may help shed new
light on the ongoing debate on Human-Centred AI and lay the basis for further
studies on this topic.

###### Acknowledgements.

The work was partially supported by the following projects: AI@Trento, FBK-
Unitn (GDT), MUR PNRR project FAIR - Future AI Research (PE00000013) funded by
the NextGenerationEU (AP, BL and MZ) Horizon Europe Programme, grant number
101120237 - ELIAS (AP and BL) and grant number 101120763 - TANGO (AP, BL and
KT). Funded by the European Union. Views and opinions expressed are however
those of the author(s) only and do not necessarily reflect those of the
European Union or the European Health and Digital Executive Agency (HaDEA).
Neither the European Union nor the granting authority can be held responsible
for them.

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arbitrarily small. The 3SUM’ subroutine on $r$-sized subsets that are stored
on the dynamic data structure (from Figure 5) can be reduced to finding a
solution in any of the $O(r/R)^{3}$ instances of Convolution-3SUM for which we
supposedly have a bounded-error sublinear quantum algorithm. As we know, given
a bounded-error algorithm, we can cheaply reduce this error to any
$\varepsilon$ by running this subroutine $O(\log(1/\varepsilon))$ many times.
With that, the error probability both for the Convolution-3SUM subroutine and
the Grover subroutine can be made inverse polynomial in $n$ (of any degree
$d$) by just repeating these subroutines $O(\log n)$ times. Therefore, one can
make the value of $t_{2}p$ arbitrarily small by just choosing the right value
of $d$, with only logarithmic overhead to the run-time of our algorithm.

  3. 3.

Finally, we mention the errors that stem from the failure161616The term
_failure_ is used for operations that required more than $c\log^{4}(n)$ steps
and hence were aborted. of the data structure operations, which could stem
from any of the following: (1) The random hash function, as mentioned in
Equation 18, or (2) the probabilistic skip-list data structure used to store
the elements in each bucket in a sorted order (Example 2 from Figure 5), both
of which we discuss in the subsequent paragraphs.

#### Failure of data structure operations.

Let us first revisit the hash function from Equation 18 and its properties.
The hash function on any element $a$ is,

$h(a)=(za\bmod 2^{w})\div 2^{w-s},$ (21)

where $z$ is a random odd integer of $w$ bits and $s=\Theta(\log w)$. This
hash function is probabilistic and has the possible sources of errors:

  1. 1.

This hash function is _always_ (almost) linear, i.e. for any two numbers $a$
and $b$, $h(a)+h(b)+\\{0,1\\}=h(a+b)(\bmod 2^{s})$. Hence, this is not a
source of error, in fact similar to the classical case, we only have to create
another set of Convolution-3SUM instances with only slight modification to the
way the first set of instances are created.

  2. 2.

However, with probability $O(1/2^{s})$, the hash function creates false
positive cases, i.e. $h(a)+h(b)+\\{0,1\\}=h(c)(\bmod 2^{s})$ even when
$a+b\neq c$. For our algorithm to be sublinear in time, the value of $R$ has
to be equal to $2^{s}$ and $r^{\alpha}$, which means $R=n^{\zeta}$ for some
$0<\zeta<1$, making the probability of false positive cases equal to
$O(1/n^{\zeta})$. We propose the following way to deal with this situation.

The primary goal is to use the Convolution-3SUM subroutine as a black box on
instances of size $O(R)$ repeatedly and check if any of these $O(r/R)^{3}$
instance has a positive solution, for which we use Grover’s search subroutine.
The result of the Grover’s subroutine gives out the details of which instance
of size $O(R)$ has the solution, whose validity can be checked in $O(R)$
additional time. This would worsen the complexity to

$\sqrt{rR}+(r/R)^{1.5}\cdot T_{\textit{c-3SUM}}(R)+R$ (22)

as opposed to what we had in in Equation 19. However, given that
$R=r^{\alpha}$, the calculations in the proof of Corollary 5 still goes
through.

  3. 3.

Lastly, the hash function has _good load balancing_ property which means
expected number of elements in the bad buckets is at most $O(R)$ [Die96].
Using Markov’s inequality we can see that the probability of the number of bad
elements exceeding $k$ times the expected number of bad elements is upper
bounded by $\frac{1}{k}$. Therefore, even for a $k$ as small as $\poly(\log
n)$ the probability of error (in the asymptotic sense) is arbitrarily close to
$0$.

#### Errors stemming from the use of probabilistic skip-list data structure.

The later source of errors emerge because of the probabilistic skip-list data
structure used to store elements of each bucket in sorted order. The analysis
almost directly follows from the ideas presented in Section 3.2, which leads
to the following observation: Similar to the technique used in proving Theorem
4, here also we modify our algorithm to abort any data structure operation
that takes more than $c\log^{4}(n+m)$ steps. We see that the distance between
the final state of the algorithm in the ideal situation and in the situation
where we abort the _lengthy_ data structure operations is small. There are $r$
elements hashed into $R$ buckets using the hash function from Equation 21.
Every such bucket is equipped with a skip-list data structure of size $O(r/R)$
to stores all its elements in a sorted order, refer to Figure 5. Probability
that a data structure operation takes too long on any bucket can be reduced to
$O(R^{k}/r^{k})$ for any constant $k$ with only constant factor worsening of
the runtime of the walk algorithm. Given that our prescribed algorithm runs in
$O(n^{1-O(1)})$ time, the number of data structure operations is at most
sublinear in $n$. Therefore, employing results of Lemma 1,2 and 3 we can see
that the distance between the final state of our algorithm and the final state
of the ideal algorithm is at most $O(n\frac{R^{k/2}}{r^{k/2}})$. With the
right choice of $r,R$ and $k$, which indeed is possible171717Recall that
$R=r^{\alpha}$ and $r=n^{\beta}$. The following choices of $r,R$ and $k$ make
the error arbitrarily small: Let the Convolution-3SUM problem have a bounded-
error quantum algorithm that runs in $O(N^{1-\delta})$ for a $\delta>0$ on
input size $N$. Our algorithm uses this Convolution-3SUM subroutine on sets of
size $R=r^{\alpha}$ for some $0<\alpha<1$. Choosing this $\alpha$ from a range
$(\frac{1}{1+2\delta},1)$ gives a $r^{1-O(1)}$ algorithm to solve 3SUM’ on $r$
sized subsets, let the exact exponent for the same be $r^{1-\delta^{\prime}}$
with $\delta^{\prime}>0$. Using results of Theorem 1 we can see that for a
$\beta\in(\max(\frac{1}{2},\frac{1}{2\delta^{\prime}+1}),1)$ we could get a
sublinear quantum time algorithm for 3SUM’ problem. An additional restriction
that needs to imposed here is that, the errors need to be arbitrarily small,
which means we would like $n\frac{R^{k/2}}{r^{k/2}}=1/\poly(n)$. For that we
need to choose $\beta>\frac{2}{k(1-\alpha)}$. Therefore, it suffices to chose
a
$\beta\in(\max(\frac{1}{2},\frac{1}{2\delta^{\prime}+1},\frac{2}{k(1-\alpha)}),1)$.
Note that, $\frac{2}{k(1-\alpha)}$ is not always less than $1$. However, given
that $\alpha$ is only dependent on the value of $\delta$ and has to be in the
range $(\frac{1}{1+2\delta},1)$, by choosing a $k$ strictly greater than
$\frac{2}{1-\alpha}$, we can ensure that there exists a $\beta$ in the range
$(\max(\frac{1}{2},\frac{1}{2\delta^{\prime}+1},\frac{2}{k(1-\alpha)}),1)$.,
this value $O(n\frac{R^{k/2}}{r^{k/2}})$ becomes $O(\frac{1}{\poly(n)})$.

###  5.2 Conditional Quantum $\Omega(n^{1.5})$ Time Bound for 0-Edge-Weight-
Triangle problem

The quantum reduction from Convolution-3SUM to 0-Edge-Weight-Triangle problem
is a straightforward adaptation of the classical local reduction by [VW09],
which is as follows: Given an input instance of Convolution-3SUM, an array $A$
of $n$ elements, the reduction, for every $i\in[\sqrt{n}]$ creates an instance
$G_{i}$ which is a tripartite graph with a weight function associated with the
edges of each graph. We will show that there exists a 0-Edge-Weight-Triangle
in any of these $\sqrt{n}$ graphs, if and only if there exists a solution to
the Convolution-3SUM.

For every $i\in[\sqrt{n}]$, create a complete tripartite graph $G_{i}$ of
three partitioned sets of nodes $L_{i},R_{i},S_{i}$ which contain $\sqrt{n}$
nodes each. Let $L_{i}[t],R_{i}[t],S_{i}[t]$ denote the $t^{\text{th}}$ node
of the partition $L_{i},R_{i},S_{i}$, respectively. We then set the weights as
follows:

  1. 1.

$w(L_{i}[s],R_{i}[t])=A[(s-1)\sqrt{n}+t]$,

  2. 2.

$w(R_{i}[t],S_{i}[q])=A[(i-1)\sqrt{n}+q-t]$,

  3. 3.

$w(L_{i}[s],S_{i}[q])=-A[(s+i-2)\sqrt{n}+q]$.

Clearly, if there is a triangle in a graph $G_{i}$ having zero total edge
weight, then the value of the weights are solution to the Convolution-3SUM
problem. The other direction also holds: Suppose there is a solution to the
Convolution-3SUM at index $i_{1},i_{2},i_{3}$ such that
$A[i_{1}]+A[i_{2}]=A[i_{3}]$ then there exists a tripartite graph $G_{i}$ with
a 0-Edge-Weight-Triangle made by the nodes
$L_{i}[s],R_{i}[t],S_{i}[i_{2}^{\prime}+t]$ where $i-1=i_{2}\div\sqrt{n}$ and
$i_{2}^{\prime}=i_{2}\mod\sqrt{n}$ are the quotient and rest of the integer
division of $i_{2}$ by $\sqrt{n}$ (which we are assuming is an integer,
without loss of generality), and $s-1=i_{1}\div\sqrt{n}$, $t=i1\mod\sqrt{n}$.
It then holds $i_{2}=(i-1)\sqrt{n}+i_{2}^{\prime}$ with
$i_{2}^{\prime}\in\\{0,...,\sqrt{n}-1\\}$ and $i_{1}=(s-1)\sqrt{n}+t$ for
$0\leq t<\sqrt{n}$.

As the reduction is completely local, given an index $i\in[\sqrt{n}]$ and any
three indices $s,t,q\in[\sqrt{n}]$ we can in constant time query the weights
$w(L_{i}[s],R_{i}[t]),w(R_{i}[t],S_{i}[q]),w(L_{i}[s],S_{i}[q])$ associated
with nodes $L_{i}[s],R_{i}[t],S_{i}[q]$.

The following now follows:

######  Theorem 22.

There is no quantum algorithm for the 0-Edge-Weight-Triangle problem, running
in time $O(n^{1.5-\epsilon})$ for an $\epsilon>0$, unless Quantum-3SUM-
Conjecture is false.

###### Proof.

Let $T(v)=v^{\beta}$, for some $\beta>0$, denote the time taken quantumly to
compute whether a graph $G=(V,E)$ with $|V|=v$ nodes contains a 0-Edge-Weight-
Triangle.

Using Grover’s subroutine over $\sqrt{n}$ indices, one can in $O(n^{1/4}\cdot
T(\sqrt{n}))$ quantum time check if there exists an index $i$ such that the
graph $G_{i}$ contains a 0-Edge-Weight-Triangle. As argued above, this is
equivalent to checking for a solution to Convolution-3SUM on $n$ elements.
Therefore, by Theorem 5, it is required that $\frac{1}{4}+\frac{\beta}{2}\geq
1$, which is to say, $\beta\geq\frac{3}{2}$, unless the Quantum-3SUM-
Conjecture is false. ∎

##  6 Acknowledgments

Subhasree Patro is supported by the Robert Bosch Stiftung. Harry Buhrman,
Subhasree Patro, and Florian Speelman are additionally supported by NWO
Gravitation grants NETWORKS and QSC, and EU grant QuantAlgo. Bruno Loff’s
research is supported by National Funds through the Portuguese funding agency,
FCT - Fundação para a Ciência e a Tecnologia, within project UIDB/50014/2020.

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MethodsAdditional References

# Chromospheric and Coronal heating in active region plage by dissipation of
currents from braiding

Souvik Bose  Lockheed Martin Solar & Astrophysics Laboratory, Palo Alto, CA
94304, USA  Bay Area Environmental Research Institute, NASA Research Park,
Moffett Field, CA 94035, USA  Institute of Theoretical Astrophysics,
University of Oslo, PO Box 1029, Blindern 0315, Oslo, Norway  Rosseland Centre
for Solar Physics, University of Oslo, PO Box 1029, Blindern 0315, Oslo,
Norway<EMAIL_ADDRESS>Bart De Pontieu  Lockheed Martin Solar & Astrophysics
Laboratory, Palo Alto, CA 94304, USA  Institute of Theoretical Astrophysics,
University of Oslo, PO Box 1029, Blindern 0315, Oslo, Norway  Rosseland Centre
for Solar Physics, University of Oslo, PO Box 1029, Blindern 0315, Oslo,
Norway  Viggo Hansteen  Lockheed Martin Solar & Astrophysics Laboratory, Palo
Alto, CA 94304, USA  Bay Area Environmental Research Institute, NASA Research
Park, Moffett Field, CA 94035, USA  Institute of Theoretical Astrophysics,
University of Oslo, PO Box 1029, Blindern 0315, Oslo, Norway  Rosseland Centre
for Solar Physics, University of Oslo, PO Box 1029, Blindern 0315, Oslo,
Norway  Alberto Sainz Dalda  Lockheed Martin Solar & Astrophysics Laboratory,
Palo Alto, CA 94304, USA  Bay Area Environmental Research Institute, NASA
Research Park, Moffett Field, CA 94035, USA  Sabrina Savage  NASA Marshall
Space Flight Center, Huntsville, AL 35812, USA  Amy Winebarger  NASA Marshall
Space Flight Center, Huntsville, AL 35812, USA

###### Abstract

It remains unclear which physical processes are responsible for the dramatic
increase with height of the temperature in stellar atmospheres, known as the
chromospheric ($\sim$10,000 K) and coronal (several million K) heating
problems. Statistical studies of sun-like stars reveal that chromospheric and
coronal emissions are correlated on a global scale, constraining, in
principle, theoretical models of potential heating mechanisms. However, so
far, spatially resolved observations of the Sun have surprisingly failed to
show a similar correlation on small spatial scales, leaving models poorly
constrained. Here we use unique coordinated high-resolution observations of
the chromosphere (from the Interface Region Imaging Spectrograph or IRIS
satellite) and low corona (from the Hi-C 2.1 sounding rocket), and machine-
learning based inversion techniques to show a strong correlation on spatial
scales of a few hundred km between heating in the chromosphere and low corona
for regions with strong magnetic field (“plage"). These results are compatible
with recent advanced 3D radiative magnetohydrodynamic simulations in which
dissipation of current sheets formed due to the braiding of the magnetic field
lines deep in the atmosphere is responsible for heating the plasma
simultaneously to chromospheric and coronal temperatures. Our results provide
deep insight into the nature of the heating mechanism in solar active regions.

## Main

It has long been known from statistical studies of a wide variety of stars
that the chromospheric and coronal emission are well correlated on a global,
stellar scale[1], which suggests that the heating mechanisms in the
chromosphere and corona are related. This intriguing correlation potentially
provides critical constraints on the heating mechanism responsible for non-
thermal energy dissipation in a stellar atmosphere. This is because the
chromosphere and corona are very different in terms of partial ionization,
non-local thermodynamic equilibrium (non-LTE), stratification, and magnetic
field to plasma pressure ratio. These differences have varying effects on the
efficiency and likelihood of different heating mechanisms.

The global correlations, however, have not been translated into a direct
physical connection at high spatial and temporal scales in the solar
atmosphere[2, 3, 4], leaving potential heating mechanisms poorly constrained.
Detailed correlation studies on small spatial scales between coronal and
chromospheric heating are very challenging, because thermal conduction in the
corona efficiently redistributes heat along a wide range of heights and
locations. De Pontieu et al.[2, 3] studied the correlation between heating
sites in “moss"– dynamic, bright regions of enhanced emission observed in the
upper transition region (TR) at the footpoints of high-pressure and high-
temperature loops above an active region (AR) plage[5], and the underlying
chromospheric emission (visible in Ca H 3968 Å filter images from ground-based
telescopes). They found no correlation on arcsecond ($\sim$ 700 km) scales, in
puzzling contrast with the global, stellar studies. These earlier studies were
flawed, however, because recent results indicate that the bright
“chromospheric” emission in broadband Ca H 3968 Å filter images is not a
likely signature of chromospheric heating, but rather emanates from deeper,
hotter layers in the convection zone below evacuated strong photospheric
magnetic flux tubes[6, 7]. Upper TR moss provides an excellent laboratory to
study how chromospheric and coronal heating mechanisms are spatio-temporally
correlated. Because moss is formed in a thin layer it is very sensitive to
changes in the local heating rate and avoids the confusion introduced by line-
of-sight superposition that affects optically thin coronal diagnostics[3, 8].
Most current observations lack the spatial resolution to resolve moss, which
is structured on spatial scales of a few hundred km. Recent observations from
High-Resolution Coronal Imager (Hi-C) flight 1[9], and later Interface Region
Imaging Spectrograph (IRIS[10]) satellite, showed evidence of highly-
localized, small-scale, heating events in active region moss with durations
between 15 and 50s[11, 12]. This relatively rare short-term, rapid variability
is thought to be caused by the impact of non-thermal electrons, generated in
the corona by reconnection in nanoflares[13], on the upper transition region
and lower-lying chromosphere. While these observations hint at a relationship
between chromospheric and coronal heating on small scales, these rare events
do not seem to be common enough to explain the relatively steady emission
patterns in the low atmosphere.

In this paper we use unique coordinated observations of active region moss at
unprecedented high spatial and temporal resolution, from IRIS and Hi-C
sounding rocket flight 2.1[14] to investigate the relationship between
chromospheric and coronal heating on small spatial scales. We exploit recent
advances in modeling of non-LTE radiative transfer of spectral lines formed in
the chromosphere (refs.[15, 4]) that indicate that the cores of the
chromospheric Mg ii h&k lines in moss are well-defined signatures of
chromospheric heating, in contrast to broadband Ca ii H filter images. The
combination of state-of-the-art spectral inversions of such lines, combined
with the high spatio-temporal resolution of the 5-min timeseries of Hi-C 2.1
images allow us to determine unequivocally whether chromospheric and coronal
heating are spatially and temporally correlated on small spatial scales,
thereby providing strict constraints for heating based models.

Figure 1: The chromospheric and coronal scenery of the moss observed with
HiC-2.1 and IRIS on 29 May 2018. (a) Hi-C 172 Å observation of AR 12712
centered at solar ($X$,$Y$)=(-145",280"). The black rectangular box outlines
the moss region studied in this paper. (b) A composite layering showing Fe
ix/x Hi-C 172 Å (red, 1 MK), Fe xvi 335 Å (green, 2.5 MK) and Fe xviii 94 Å
(blue, 6 MK) images from AIA. The white and yellow-dashed boxes indicate the
FOV recorded in the IRIS context and high-cadence rasters respectively. (c) Mg
ii k3 emission (Trad) from the context raster. (d) and (e) chromospheric
plasma temperature ($T$) and electron density (Ne) maps corresponding to the
context raster averaged between log$\tau$=[$-$4.6, $-$4.2]. The FOV of the
high-cadence raster is outlined in the three panels. (f) Observed (red) and
inverted (black) Mg ii h&k profiles corresponding to a moss pixel indicated in
panel (c). A typical quiet-Sun profile is shown (blue) for reference. Panels
(g), (h), and (i) show the stratification of $T$, microturbluent velocity, and
Ne of the moss (red) and quiet-Sun (blue) inferred from $IRIS^{2}$ inversion.

Figure 1 shows the coronal and the chromospheric emission of the AR target
12712. The moss region, identified as a reticulated bright emission patch in
panel (a), is clearly seen to lie at the footpoints of the hotter ($\sim$2–6
MK, see panel b and Materials and Methods) AR loops, typical of a moss[5, 2,
3, 16]. IRIS co-observed the AR target, and we utilize the spectral
information primarily in the Mg ii h&k lines to explore the chromosphere
underneath the moss which is characterized by regions of enhanced brightness
in the Mg ii k3 (h3) core. In panel (c), such a region is indicated by the
area bounded between $X$ $\sim$ 80–100" and $Y$ $\sim$ 40–100", and it
corresponds to a chromospheric plage. This observation is in agreement with
Carlsson et al.[4] who also reported that the Mg ii k spectral line in plages
show little-to-no reversals of the upper chromospheric k3 core. Based on a
semi-empirical radiative transfer approach, they argued that the enhanced core
brightness in the plage represents conditions of hot and dense chromosphere,
which causes the source function to remain strongly coupled with the local
Planck function, thereby indicating locally strong heating.

Instead of an ad-hoc forward modeling approach, for the first time, we employ
a state-of-the-art, simultaneous, multi-line inversion scheme based on
$IRIS^{2}$[17] with six different spectral lines (including C ii 1335 Å,
refer: Materials and Methods). Figures 1 (d) and (e) show the plasma
temperature ($T$) and electron density (Ne) maps derived from the inversion
scheme, which upon inspection, immediately reveals clear evidence of hot and
dense plasma in the regions associated with enhanced Mg ii k3 radiation
temperature (Trad, panel c). The radiation temperature is the temperature for
which the Planck function reproduces the observed intensity at any given
wavelength. Interestingly, the moss region (identified from panels a and b)
show stronger enhancement in the Mg ii k3 radiation temperature that appears
to be well correlated with stronger $T$ and Ne values, which clearly suggests
that the chromosphere underneath the moss has hotter and denser plasma
associated with it. Including multiple atomic species in the inversion allows
us to effectively constrain a wide range of solar atmosphere spanning the
photosphere through the upper chromosphere (see Sainz Dalda et al. 2022, in
prep.). This approach represents a major advancement since it enables to
directly examine and correlate the physical parameters of the plasma, instead
of focusing on the “proxies" of heating (as had been done in the past[8, 3]).

A representative example of a Mg ii h&k spectrum associated with a mossy plage
pixel is shown in Fig. 1 (f). The spectrum shows a clear lack of self-reversal
in the k3 (h3) wavelength intensities indicating a flat-topped (single-
peaked[4]) profile unlike the quiet-Sun (QS) profiles, which show a distinct
absorption in k3 (h3). Moreover, the overall Trad and the width of the plage
spectrum is significantly enhanced compared to the QS case, which is in
agreement with earlier findings[4]. The synthetic spectra derived from
inversion show good fits not only for the example shown in Fig. 1 (f), but
also for various other examples shown in Supp. Figs. 5, 6, and 7.
Additionally, the C ii lines, which were simultaneously inverted and fitted
with the Mg ii h&k, show good fits as well; this suggests that the derived
physical parameters are well-constrained at upper chromospheric heights. The
inferred $T$, microturbulence and Ne as a function log$\tau$ ($\tau$ being an
“optical" measure of depth in the solar atmosphere) corresponding to the plage
and the QS spectra in panel (f) is shown in panels (g)–(i) of Fig. 1.
Interestingly, all the three parameters show a clear enhancement centered
around log$\tau$=$-$4 compared to the QS case, reinforcing that the
chromosphere underneath the moss is hot and dense, and has significant non-
thermal velocities (of $\sim$6 kms-1, here named as microturbulence). This
pattern is consistently observed for a vast majority of the plage including
the examples shown in Supp. Figs. 5, 6, and 7. In addition, the temperature
stratification shows a characteristic “step" (around log$\tau$=$-$4.5), and
the temperature minimum is pushed deeper in the solar atmosphere, which is
critical (in combination with high density) in reproducing the flat-topped,
single-peaked spectra.

Figure 2:  Spatio-temporal correlation for the mossy plage region ROI-B. Left:
(a), (b), and (c) show the Mg ii k3, Si iv and Hi-C 172 Å intensity (counts)
corresponding to the high-cadence raster (#4). The FOV under investigation is
shown in red. (d) and (e) show the 2D density distribution functions between
Mg ii k3 and Hi-C 172 Å, and Si iv and Hi-C 172 Å intensities. Right: (a) and
(b) show the $T$ and Ne maps corresponding to the same time and FOV as left
panel averaged between log$\tau$=[$-$4.6, $-$4.2], while (c) shows the Hi-C
172 Å as a reference. (d) and (e) show the 2D density distribution functions
between $T$ and Hi-C 172 Å, and Ne and Hi-C 172 Å. The Pearson correlation
coefficients ($r$) are shown on top left. An animation of this figure is
available.

The left and the right panels of Fig. 2 show the spatio-temporal correlation
analysis between the IRIS and the Hi-C intensities, and between the physical
parameters derived from the $IRIS^{2}$ inversion and Hi-C intensity for a
small field-of-view (FOV, of 13$\times$7 arcsec2) of the moss region marked as
the region of interest (ROI)-B. The analysis is derived from the high-cadence
IRIS rasters (see Materials and Methods) that overlaps the moss region as
shown in Fig. 1 (b). Visual inspection reveals that the intensity in ROI-B is
well correlated among the Mg ii k3, Si iv and Hi-C channels. This is supported
quantitatively in the form of 2-dimensional (2D) density distribution
functions shown in panels (d) and (e), where we see a tight correlation not
only between Mg ii k3 and Hi-C 172 Å intensities, but also between Si iv and
Hi-C 172 Å. The high values of Pearson coefficient ($r$) in both cases is in
direct contrast with earlier results (such as refs.[2, 3]) where the authors
failed to obtain a good correlation on (sub)arcsecond scales. The correlation
($r$) between Si iv and Hi-C 172 Å intensity is slightly worse than the
chromospheric (Mg ii k3) counterpart (also see Supp. Figs. 8 and 9), possibly
because TR Si iv is often impacted by shocks[18] and spicules[3, 7, 19] that
affect its brightness variation more significantly compared to Mg ii k.
Nonetheless, the correlation coefficients obtained in this paper are
significantly better than past findings which implies a tight coupling in the
heating of the solar atmosphere from chromosphere to coronal heights.

The tight coupling in the moss intensity among the different channels directly
translated to a strong visual correlation between the inferred physical
parameters (Fig. 2, right panel), i.e. $T$ (panel a) and Ne (panel b) shown at
<log$\tau$>=$-$4.4,111¡¿ represents an average of the physical parameter
between log$\tau$=[$-$4.6,$-$4.2] with Hi-C 172 Å intensity (panel c). In
ROI-B, the enhanced Hi-C 172 Å intensity is clearly characterized by enhanced
$T$ and Ne. The range of the derived $T$ and Ne is consistent with values that
are typically observed in a chromospheric plage[4, 20]. The 2D density
distribution functions shown in panels (d) and (e) further highlight the rigid
coupling of the chromospheric and (lower) coronal plasma in a moss. Moreover,
the animation associated with Fig. 2 suggests that the correlation remains
stable (and high) for the entire timeseries of the Hi-C data. This suggests
that the heating observed in ROI-B (and also over the whole moss in general)
is roughly constant and non-impulsive (unlike electron beams which causes a
sudden, temporal variability of $\sim$15–60s[11, 12]). Analysis of the spatio-
temporal correlation analysis of other ROIs in the moss region (see Supp.
Figs. 8 and 9) reveal similar results ( with slightly different values of
$r$), providing support for the non-impulsive nature of chromospheric and
coronal heating.

Figure 3: Heatmaps showing the depth dependent correlation for ROI-B. Left:
Pearson correlation coefficient ($r$, encoded in a rainbow color scheme) as a
function of depth in the solar atmosphere (log$\tau$, y-axis) and time
(#raster, x-axis) for $T$ vs. Hi-C 172 Å and right: Ne vs. Hi-C 172 Å
intensity. The T and Ne are derived from the multi-line inversions of the
chromospheric IRIS data and the region between the dashed lines in the two
panels indicate the range of optical depths within which the inversions are
sensitive (refer to main text). The number of rasters correspond to the entire
Hi-C 2.1 duration of $\sim$5 min.  Figure 4: Overview of the 3D numerical
simulation of an active region plage. (a) Vertical magnetic field strength
(Bz) at $z$=0 Mm saturated between $\pm$ 2 kG. ROIs-A and B cover an area of
10$\times$10 Mm2, and 1.5$\times$1.5 Mm2 respectively, whereas the quiet-Sun
(QS) patch encloses an area of 15$\times$15 Mm2. (b) and (c) Synthetic Mg ii
k3 and Fe ix 171 Å maps of the simulated plage. Purple contours in (b)
indicate the regions of enhanced brightness corresponding to the moss emission
in (c). (d) Shows the synthetic median Mg ii k spectra over the mossy pixels
(red) within the purple contours in (b) and the QS patch (blue), while (e)
shows the corresponding temperature stratification. (g) and (h) shows
illustrative examples of a synthetic Mg ii k spectrum and its “step-like"
temperature stratification (highlighted in gray) for a single mossy plage
pixel in ROI-B. (f) and (i) show the 2D density distribution functions for
ROIs-A and B. (j) shows the visualization of the magnetic field lines (yellow)
and their associated current sheets ($|\vec{j}|/|\vec{B}|$, $\vec{j}$ being
the current density) from the photosphere to the corona in ROI-A.

Although we found a good correlation between temperature, density (at mid-
chromospheric heights), and Hi-C 172 Å intensity, an important question is how
deep in the solar atmosphere does the correlation still hold? The $IRIS^{2}$
database provides the derived thermodynamic parameters as a function of
optical depth $\tau$. The latter is correlated with height, with smaller
depths occurring at greater heights. We therefore studied the correlation over
a range of heights in the solar atmosphere between log$\tau$=[$-$7.6,0], i.e.,
from the top of the chromosphere down to the photosphere. This is represented
in the form of a 2D heatmap in Fig. 3 that shows the Pearson $r$ for ROI-B as
a function of time (#raster) and optical depth (log$\tau$) for $T$ (left
panel) and Ne (right panel) vs. Hi-C 172 Å intensity. The sensitivity of our
inversions is predominantly limited between $-$6 $\leq$ log$\tau$ $\leq$ $-$2
(due to the sensitivities of the different lines included in the inversion,
see Materials and Methods and Sainz Dalda et al. 2022 in prep). Interestingly,
we find that there is a range of optical depths (within log$\tau$=[$-$6,$-$2])
for which $r$ is rather high ($\geq$ 0.6) in both panels (Fig. 3). The
temperature and density of the chromospheric plasma appear to be well
correlated with the coronal intensity of Hi-C 172 Å down to the temperature
minimum (log$\tau$$\sim$$-$3) for all the 14 rasters. The coupling immediately
weakens deeper in the atmosphere (for log$\tau$ > $-$3 in the photosphere)
suggesting different physical conditions of the plasma. However, there is no
doubt that hot and dense plasma dominates the lower to upper chromosphere
underneath, and is well correlated with lower coronal moss brightness. For
heights beyond log$\tau$=$-$6, the apparent absence (or presence) of
correlation in Fig. 3 does not indicate a lack (or sufficiency) of coupling.
It merely implies that the inferred values of T and Ne at these heights are
ill constrained due to little or no sensitivity of the inversions. We find
similar results for the other ROIs (with different peak values of $r$, Supp.
Figs. 10 and 11) where the correlation extends reasonably well down to at
least log$\tau$$\sim$$-$4 where the temperature is between 5–6 kK.

The tight correlation between the derived physical parameters with the Hi-C
172 Å intensity down to the temperature minimum strongly suggests the
prevalence of a common heating mechanism. Thermal conduction, which obviously
plays a role in the hot corona and upper transition region,[8] is negligible
at low temperatures in the lower chromosphere ($\sim$ 6kK), and thus cannot
explain the observed spatio-temporal correlation between low chromospheric and
upper transition region heating. Accelerated electron beams generated in
coronal nanoflares can have an impact on heating the chromosphere[21],
however, due to their impulsive nature such heating would show a high temporal
variability[11, 12], which is not observed in this study.

We find that our observations show a striking correspondence with an advanced
3D magnetohydrodynamic (MHD) numerical simulation of active region plage (Fig.
4 a). The numerical domain encompasses the convection zone, photosphere,
chromosphere, TR, and corona (see Materials and Methods). Optically thin
modeling of the coronal channels reveal the presence of moss-like emission
patches in Fe ix 171 Å that lie at the footpoints of hot loops visible in Fe
xvi 335 Å (Supp. Fig. 12). Synthetic Mg ii k3 and 171 Å (panels b and c)
intensity bear a close resemblance with observations where the moss coincides
with enhanced k3 radiation, and the Mg ii k spectra show single peaks (panels
d and g). The temperature minimum is shifted deeper in the solar atmosphere
for the moss pixels (panel d) with many showing characteristic “steps" around
low-chromosphere at $z$=0.4 Mm (panel h), similar to Fig. 1 (f). 2D density
distribution functions between 171 Å and k3 channels show a tight correlation
with high values of Pearson $r$ at both large (ROI-A, panel f) and small
(sub)arcsecond scales (ROI-B, panel i). The simulated scenario bears a close
resemblance with observations, and a detailed analysis of the magnetic field
topology reveals the presence of braiding in the field lines[22] deep in the
atmosphere, which leads to the formation of current sheets (panel j). These
sheets extend from the chromosphere (where they are stronger) and corona,
traversing several mega-meters and cause (quasi)steady heating in the mossy
plage regions. Dissipation of the current sheets leads to strong heating of
the plasma in the chromosphere and the corona that explains the cause of the
observed correlation. These types of small-scale current sheets from braiding
are more ubiquitous and very different from large-scale current sheets that
form when newly emerging flux interacts with pre-existing fields and that have
been implicated in local heating[23]. The current sheets in our simulation
occur on very small-scales of tens of km that are at (and beyond) the limit of
current observational capabilities[24]. High-resolution magnetic field
measurements at multiple heights are needed to resolve some of these sheets.
Future observations, e.g. with 4-m DKIST[25], may aid in this direction.

Furthermore, the nature of heating observed in the moss appears to be
incompatible with current models of wave-based heating. Predicted heating
rates from Alfvén wave turbulence models[26, 27] are highly time varying
because of the presence of high frequency waves. This causes a lack of
correlation between chromospheric and upper TR (lower coronal) emission, in
contrast to what we observe in this study. Future observations of longer
duration with Solar Orbiter’s Extreme Ultraviolet Imager[28] instrument and
the upcoming NASA MUlti-slit Solar Explorer (MUSE) mission[29] will provide
more detailed imaging and spectroscopy of a much wider variety of regions, and
help determine how widespread this heating mechanism is in the solar
atmosphere.

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## Materials and Methods

### Observations

The Hi-C 2.1 sounding rocket mission was launched on 29 May 2018 at 18:54 UT
from the White Sands Missile Range in New Mexico, USA. The target was AR 12712
and the instrument successfully acquired high spatial resolution (0.4"),
2k$\times$2k images with a 2s exposure at a cadence of 4.4s. The Solar
Pointing and Aerobee Control System (SPARCS) maintained a constant target for
the entire duration of the flight lasting 335s. One of the major goals of this
mission was to study the mass and energy coupling between the chromosphere and
the corona in ARs. Consequently, the observations were recorded in the upper
transition/lower coronal 172 Å passband (Fe ix/Fe x) that has a peak
temperature sensitivity of about 1 MK. The passband is similar to AIA 171 Å
channel, except that Hi-C 2.1 recorded observations at roughly three times the
spatial and temporal resolution of AIA thereby allowing an unprecedented view
of the million degree corona. The details of the instrument and its
characterization can be found here[14].

The Hi-C 2.1 sounding rocket flight was also coordinated with IRIS that co-
observed the AR target with multiple, co-temporal, high-cadence rasters
centered around the moss brightenings (indicated by the yellow dashed lines in
Fig. 1 b). It ran in very large, sparse 8-step raster mode (OBS-3600104031)
with a step size of 1" perpendicular to the slit and a raster cadence of 25 s
covering a FOV of 175"$\times$7" in the solar $Y$ and $X$ directions. 256
rasters were obtained over a duration of 1hr and 50 min, including the Hi-C
flight duration of 5 mins (equivalent to 14 rasters). In addition, IRIS also
recorded several very large, dense 400-step rasters (OBS-3600010078) covering
a FOV of 141"$\times$174" co-spatially with Hi-C, but for roughly 1hr and 51
min (raster cadence) before and after the sounding rocket flight. The step
size in this case was 0.35" perpendicular to the slit, while the step cadence
was 16.7s. The FOV of one such context raster is indicated in Fig. 1 (b) and
the purpose of these rasters was to provide contextual information in and
around the AR. Furthermore, IRIS recorded slit-jaw images (SJIs) in C ii 1330
Å, Si iv 1400 Å, Mg ii 2796 Å, and Mg continuum 2832 Å channels sampling
plasma from 6,000 to $\sim$100,000 K. The cadence and the spatial sampling of
these images are 13s and 0.33", respectively.

Finally, we have also used observations in the
SDO2012SoPh..275….3P/AIA2012SoPh..275…17L 94 Å (Fe xviii,$\sim$6 MK) and 335 Å
(Fe xvi,$\sim$2.5 MK) passbands solely for the purpose of visualizing the
presence of hot AR loops above the moss brightenings seen in Hi-C 172 Å. This
relationship is shown in the form an RGB composite image in Fig. 1 (b).

### Data processing

The above sets of observations were precisely co-aligned using SolarSoftWare
(SSW1998SoPh..182..497F) IDL routines including compensating for the roll
angle (1.7 degrees) between the IRIS and Hi-C observations. In particular, the
IRIS SJIs and Hi-C datasets were co-aligned through first, a visual comparison
and cross-correlation of Hi-C 172 Å and AIA 171 Å channels, followed by
internally co-aligning the AIA 1600 Å, 171 Å, 94 Å, and 335 Å channels (this
includes the 3hr cadence limb fitting), and finally by co-aligning the AIA
1600 Å and IRIS 1400 Å SJIs. The above steps resulted in a precise co-
alignment between IRIS SJIs and the Hi-C 2.1 data, where the former was
expandedpark1983image, rotated and spatio-temporally aligned to the latter.
Extensive visualization with CRISPEX2012ApJ…750…22V, an IDL widget-based tool,
showed good correspondence among the coordinated datasets.

For the spatio-temporal correlation analysis presented in the paper, synthetic
sparse, 8-step Hi-C 172 Å rasters were generated that were spatially and
temporally co-aligned with each of the IRIS rasters. This was done to ensure
consistency between the IRIS and Hi-C observations, since the spectrograph
slit takes time to “build" the FOV whereas Hi-C observations over the whole
FOV are instantaneous. It is to be noted that the derived parameters (e.g.
intensity of Mg ii k3, Si iv, $T$, Ne) from the IRIS rasters at original
resolution were resampled to Hi-C 2.1 pixel scale (0.129") before generating
the synthetic Hi-C rasters, which was done to not compromise the high spatial
resolution of Hi-C. A standard single Gaussian fitting algorithm was applied
to derive the Si iv peak intensity, while a double Gaussian approach
(including one absorption and emission Gaussian like ref.2015ApJ…811..127S)
was employed to fit the Mg ii k spectra. This way single peaked profiles were
easily fitted with the emission Gaussian whereas the remaining spectra with
depressed cores were fitted with a superposition of a broad emission and
narrower absorption Gaussian.

###  $\mathbf{IRIS^{2}}$ inversions

The inversions presented in this article were made using the $IRIS^{2}$
inversion framework. The core of this framework is a database created from the
inversion of multi-line representative profiles (RPs) and their corresponding
representative model atmospheres (RMAs).

This approached was first introduced by Sainz Dalda et al.[17]. There, the
authors calculated the RPs of the Mg ii h&k lines (including the Mg ii UV
lines located between the former ones) for a selection of IRIS data sets. The
RPs are the averaged profile of those profiles belonging to a cluster. A
cluster is calculated by using the $k-means$
techniqueSteinhaus57,macqueen1967. The profiles within a cluster share a
similar shape, i.e. the distribution of the intensity (spectral radiance) with
respect to the wavelength is similar. Therefore, the physical conditions of
the atmosphere where the radiation comes from are similar in the profiles
within a cluster, and it is encoded in the RP of the cluster. The database was
created considering a large variety of RPs belonging to different solar
features with different observational setups, e.g., exposure time, location on
the solar disk, raster steps, and so on. These RPs were inverted by the state-
of-the-art STockholm Inversion Code (STiC2019A&A…623A..74D), which considers
non-LTE radiative transfer and partial frequency distribution (PRD) of the
scattered photons. This code is capable of inverting simultaneously several
spectral lines from different atomic species, both in non-LTE and LTE. STiC
solves the radiative transfer problem to synthesize the spectral lines in the
RP, and in an iterative process that slightly changes the physical parameters
of a initial guess model, looks for the best possible fit between the
synthetic profile and the observed RP. This process is repeated until
convergence under a given limit or computational constraint. As a result of
the inversion, STiC provides a synthetic RP that best fits the observed RP,
and the RMA that reproduces that synthetic RP. An RMA consists of
thermodynamic parameters (e.g., $T$, Ne, LOS velocity, microturbulence) as a
function of optical depth ($\tau$). For a given observed Mg ii h&k profile in
an IRIS observation, $IRIS^{2}$ looks for the closest synthetic RP in their
database that fits that observed profile, and associates the RMA to the
location in the data set where the observed profile was recorded. This method
speeds the inversion of a data set up by a factor 105–106.

In this paper, we use an extended version of the $IRIS^{2}$ method. Sainz
Dalda et al. 2022 (in prep) use the same method and calculate the RPs for
multi-line profiles containing the C ii lines, the Mg ii h&k lines, the three
lines of the Mg ii UV triplet, and 6 photospheric lines located around the Mg
ii h&k spectral range. Thus, a multi-line RP is simultaneously inverted by
STiC providing a RMA, for which the thermodynamic parameters are constrained
from the top of the chromosphere to the bottom of the photosphere (see Sainz
Dalda et al. 2022 in prep for more details). Note that because we are
considering simultaneously several lines that are sensitive to variations of
the thermodynamics in the same region of the solar atmosphere, the $T$ and the
non-thermal contribution to the profile (microturbulence) are better
constrained than in the case that only considers one spectral line, which is
the case for both the chromosphere and the low photosphere. Thus, this set of
RPs and RMAs are likely the best constrained, comprehensive database of
thermodynamics models of the low solar atmosphere.

For this paper, $IRIS^{2}$ looked for the closest RP in the database
considering only the lines available in our data sets, which are the C ii
lines, the Mg ii h&k lines, the three Mg ii UV triplet lines, and the
photospheric Ni i at 2815.18 Åline. Fig. 5 shows the observed lines in the
chromosphere (red-dotted line) and their fits (black). Thus, although the RMA
provided by $IRIS^{2}$ contains information that was obtained from inverting 6
photospheric lines, the information presented in this paper comes from fitting
only 1 photospheric line. Therefore, the values in the region log$\tau$=[-2,0]
must be considered with caution. We are however very confident with the values
obtained in the chromosphere, especially between log$\tau$=[$-$6, $-$2], since
that information comes from the unique simultaneous inversion of 6
chromospheric lines, which provide model atmospheres with excellent constraint
in these heights, and the photospheric Ni i at 2815.18 Åline.

### Numerical Simulation

The 3D simulation of the plage-like region was performed in a computational
box with dimensions $72\times 72\times 61$ Mm spanning from 8.5 Mm below the
photosphere to 52.5 Mm above it. The resolution of this box is moderate: 100
km in the horizontal directions and $[20,100]$ km in the vertical direction
depending on location in the atmosphere; in the photosphere, chromosphere and
transition region the vertical resolution is of order 20 km. The model
calculations are run with the Bifrost radiative-MHD code2011A&A…531A.154G,
which includes optically thick radiative transfer in the photosphere and lower
chromosphere, effectively thin radiative losses in the mid- and upper
chromosphere, and optically thin radiative losses in the corona. Spitzer
thermal conduction along the magnetic field is handled by a multi-grid method
(or alternately by the hyperbolic method developed by ref.2018ApJ…859..161R)
to maintain time steps at a reasonable level. To initialize the model, two
circular patches of strong but opposite polarity magnetic flux (at 8.5 Mm
below the photosphere) were used to generate a potential field covering the
entire computational box. This field geometry was inserted into a relaxed
convective atmosphere after which the field and atmosphere were allowed to
relax together. At the time of the snapshot the photosphere in the strong
field regions has become quite cold (and darker than the surrounding
atmosphere) while the chromosphere and corona are hot and bright (Fig. 4 and
Supp. Fig. 12).

In order to compute the Mg ii lines we have used the RH 1.5D
code2001ApJ…557..389U,2015A&A…574A…3P. This code includes detailed radiative
physics including polarized transfer and PRD, but at the cost of only being
used for 1.5D (column-by-column) computations, and thus ignores the 3D effects
of lateral transport of radiation. 3D effects can play an important role in
modeling the Mg ii lines[15], especially in the line cores. However, in
regions where the temperature and the density of the plasma is enhanced (such
as in the mossy plage), 3D effects can be minimal due to lower scattering of
photons. Moreover, the lower time complexity of the 1.5D approach far
outweighs its limitations. The calculations are carried out in non-LTE
assuming PRD.

The emission of optically thin transition region and coronal lines, including
the Fe ix 17.1 nm and Fe xvi 33.5 nm lines are calculated using contribution
functions $G(T,n_{\rm e})$ from the CHIANTI2015A&A…582A..56D database package.
These contribution functions are integrated along the line of sight using the
local values of the electron density $n_{\rm e}$, the temperature $T$, and the
velocity along the line of sight uz at several frequencies covering the
emitting ions line profile. These calculations are done on GPUs using a Cuda
code developed by Juan Martinez-Sykora.

naturemag-doi supplementary

## Acknowledgments

S.B., V.H., A.S.D., and B.D.P. gratefully acknowledge support from NASA
contract NNG09FA40C (IRIS). B.D.P. was also supported by grant NNM16AA10P
(Hi-C). We acknowledge the High-resolution Coronal Imager (Hi-C 2.1)
instrument team for making the second re-flight data available under NASA
proposal 17-HTIDS17_2-003. MSFC/NASA led the mission with partners including
the Smithsonian Astrophysical Observatory, the University of Central
Lancashire, and the Lockheed Martin Solar and Astrophysics Laboratory. Hi-C
2.1 was launched out of the White Sands Missile Range on 2018 May 29. IRIS is
a NASA small explorer mission developed and operated by LMSAL with mission
operations executed at NASA Ames Research Center and major contributions to
downlink communications funded by ESA and the Norwegian Space Centre. The 3D
visualization of the numerical simulation was carried out using VAPOR, a
product of the Computational Information Systems Laboratory at the National
Center for Atmospheric Research.

## Author contributions statement

S.B. was responsible for most of the analysis and writing of the manuscript.
B.D.P. designed the study, performed initial analysis, and final editing. V.H.
performed the numerical simulations, calculation of synthetic observables, and
assisted with analysis of the simulations. A.S.D. performed the inversions.
S.S. and A.W. designed, built, and operated the Hi-C sounding rocket
experiment. All authors reviewed the manuscript.

## Competing Interests

The authors declare no competing interests.

## Supplementary Section

Figure 5: Comparison of the observed and fitted Mg ii h&k (top row) and C ii
1335 Å (middle row) spectra for a mossy plage pixel from IRIS raster #4. The
derived stratification of T, microturbulent velocity (Vturb), and Ne and their
comparison with a typical QS atmosphere is shown in the bottom row. Figure 6:
Second example from IRIS raster #6 showing the fits between the observed and
the synthetic spectra for a pixel in a moss region, in the same format as
Supp. Fig. 5.  Figure 7: Third example from IRIS raster #10 showing the fits
between the observed and the synthetic spectra for a pixel in a moss region in
the same format as Supp. Fig. 5.

Figure 8: Same as Fig. 2 but for ROI-A.

Figure 9: Same as Fig. 2 but for ROI-C. Figure 10: Heatmap showing the depth
dependent correlation coefficients for ROI-A in the same format as Fig. 3.
Figure 11: Heatmap showing the depth dependent correlation coefficients for
ROI-C in the same format as Fig. 3.  Figure 12: Coronal and photospheric
scenery of the 3D MHD plage simulation. Top left: Synthetic Fe ix 171 Å
emission showing bright, reticulated moss patches. Top right: Synthetic Fe xvi
335 Å emisison showing hot (2.5 MK) loops overlying the moss. Lower left:
Synthetic photospheric radiation temperature at 2780 Å. Lower right: Vertical
velocity (uz) map at $z$=0 Mm.