# Targeted Augmentation for Low-Resource Event Extraction

Sijia Wang, Lifu Huang
Virginia Tech
<EMAIL_ADDRESS>
###### Abstract

Addressing the challenge of low-resource information extraction remains an
ongoing issue due to the inherent information scarcity within limited training
examples. Existing data augmentation methods, considered potential solutions,
struggle to strike a balance between weak augmentation (e.g., synonym
augmentation) and drastic augmentation (e.g., conditional generation without
proper guidance). This paper introduces a novel paradigm that employs targeted
augmentation and back validation to produce augmented examples with enhanced
diversity, polarity, accuracy, and coherence. Extensive experimental results
demonstrate the effectiveness of the proposed paradigm. Furthermore,
identified limitations are discussed, shedding light on areas for future
improvement111The source code, model checkpoints, and data are publicly
available at https://github.com/VT-NLP/TALOR-EE..

##  1 Introduction

Event extraction (EE) Grishman (1997); Chinchor and Marsh (1998); Ahn (2006)
is the task of identifying and categorizing event mentions in natural language
text. While supervised methods deliver impressive performance, they depend
heavily on extensive manual annotations Chen et al. (2020); Du and Cardie
(2020); Lin et al. (2020); Liu et al. (2020); Li et al. (2020a); Lyu et al.
(2021). Generalizing these approaches to low-resource learning setting poses
challenges Pasupat and Liang (2014); Huang et al. (2016); Huang and Ji (2020);
Lai et al. (2020b); Shen et al. (2021b); Lyu et al. (2021); Zhang et al.
(2021b); Wang et al. (2023b).

Figure 1: Talor-EE framework overview.

Data augmentation is one direction for efficiently addressing the low-resource
event extraction problem. However, it’s remained unexplored what data
augmentation strategies are the best for low-resource event extraction given
its unique challenges. Previous studies show that weak augmentations, such as
synonym augmentation (Wei and Zou, 2019) or through back translation (Edunov
et al., 2018), contribute minimally to distribution enrichment, while drastic
augmentations can leading to misguided acquisitions Cao et al. (2015); Gao et
al. (2022). Drastic augmentations usually undermine existing event structure,
resulting in grammatical incorrectness, structure misalignment, or semantic
drifting Wang et al. (2023a).

In this work, we explore several dimensions for data augmentation, including
diversity, polarity, accuracy, and coherence. Our focus revolves around
enhancing diversity in the context of targeted augmentation for low-resource
event extraction (Talor-EE). This involves enriching event structures with
entities drawn from a targeted subset Gao et al. (2022). Simultaneously, we
address the issue of polarity by not only generating positive event mentions
based on actual occurrences but also incorporating negative event mentions,
e.g., hypothetical event mentions Linguistic Data Consortium (2005). This
approach is particularly valuable for overcoming limitations in generative
event extraction models Hsu et al. (2022); Liu et al. (2022). To ensure both
accuracy and coherence in our generated content, we introduce a back-and-forth
validation module Back-Validation. The rationale behind this module is that an
accurate generation should align with the given event structure, while
coherent generation should seamlessly integrate with the same structure.

Our research encompasses a series of comprehensive experiments conducted
across various low-resource learning scenarios, including zero-shot and few-
shot learning settings. These experiments span different event extraction
models. The outcomes of these experiments consistently highlight the
effectiveness of targeted augmentation in low-resource event extraction.
Notably, among all the dimensions investigated, diversity emerges as the most
crucial factor. Additionally, we meticulously scrutinize the quality of the
generated sentences, shedding light on the limitations inherent in the
proposed framework.

The contributions of this work are as follows:

  * •

We explore the application of data augmentation techniques for low-resource
event extraction.

  * •

We develop a novel augmentation method that incorporates enriched event
structures and contextual entities, retrieved from external corpus. The
generated examples are validated through a back-validation module, ensuring
accuracy and coherence.

  * •

Comprehensive experiments are conducted to assess the effectiveness of the
proposed paradigm across various models and datasets.

##  2 Related Work

#### Low-resource Event Extraction

Although some studies have employed meta-learning Kang et al. (2019); Li et
al. (2021); Xiao and Marlet (2020); Yan et al. (2019); Chowdhury et al.
(2021), or metric learning Sun et al. (2021); Wang et al. (2020a); Zhang et
al. (2021a); Agarwal et al. (2021) to align candidate event semantics with a
few examples of novel event types for few-shot event detection, their
performance is inherently constrained by the limited examples provided Lai et
al. (2020a); Deng et al. (2020); Lai et al. (2020b); Cong et al. (2021); Chen
et al. (2021); Shen et al. (2021b). Recent studies Wei et al. (2023); Han et
al. (2023); Li et al. (2023) have explored in-context learning by providing
task instructions and a handful of in-context examples. Nevertheless, their
experimental findings reveal a notable performance gap between in-context
learning and approaches based on fine-tuning.

#### Data Augmentation

creates synthetic data from the existing data. Traditional data augmentation
approaches focus on expanding lexical diversity (Wei and Zou, 2019; Feng et
al., 2020; Ng et al., 2020) or syntax variation (Kim et al., 2022; Loem et
al., 2022; Hussein et al., 2022; Wang et al., 2023a). Post selection (Yang et
al., 2020) or representative selection (Edwards et al., 2021) helps to prevent
a waste of resources and time in generating new documents. Yet existing
augmentation methods suffer from gradual drift problem (Hu et al., 2021a, b).
The previous work Ma et al. (2023) utilizes language models for training data
synthesis but lacks assurance in the soundness and naturalness of event
structures due to the random combination of sampled triggers and arguments.
Additionally, it falls short by primarily relying on the self-reflection
capability of language models, without fully leveraging annotations for
existing event annotations. Thus, in addition to the lexical and syntactical
diversity, we leverage the large-scale pre-trained autoregressive models to
generate contextually diversified free texts.

#### Controlled Text Generation

approaches (Ghosh et al., 2021) generate text with specific constraint.
Approaches that promote similarity (Guan et al., 2021) or coherence (Shen et
al., 2021a; Wang et al., 2021a) towards the original sentences lack contextual
diversity and might produce over-confident probability estimation (Wang et
al., 2021a; Gowda and May, 2020). Rule-based constraint generation might
generate meaningless tokens to meet constraints (Wang et al., 2021b), while
template-based constraint generation (Cao and Wang, 2021) is difficult to
generalize to new domains without human effort.

#### Learning with noisy labels

Many works learn with noisy labels by detecting corrupted instances, e.g., Han
et al. (2018); Yu et al. (2019); Huang et al. (2019); Yao et al. (2020); Wei
et al. (2020); Jiang et al. (2020); Zhang et al. (2021c), and their
application to low-resource learning setting Wang et al. (2020b); Li et al.
(2020b); Cheng et al. (2021). However, joint training of the sample selection
module and the target task model takes considerable iterations to converge.
Traditional data-centric methods Zhu et al. (2022) face limitations in low-
resource settings due to biased neighbor information. This study demonstrates
that training with relatively fair-quality labels can be effective.

##  3 Model

###  3.1 Problem Formulation

Given a sentence, the Event Extraction (EE) task aims to extract event
mentions, represented by an event trigger and a set of event arguments.
Formally, given a sentence $w=\\{w_{1},...,w_{n}\\}$, and a target event type
$e_{i}$, if there is an event occurrence of $e_{i}$ in $w$, a EE system aims
to extract an event trigger $t$ and its argument mentions
$a=\\{a_{1},...,a_{g}\\}$. In this work, we focus on zero-shot and few-shot
learning settings of EE. For few-shot EE (FSEE), training data contains two
parts: (1) A large-scale data set
$\mathcal{D}_{base}=\\{(\mathbf{x}_{i},\mathbf{y}_{i})\\}_{i=1}^{M}$ that
covers the seen event types (named base types), where $M$ denotes the number
of base event types; (2) a smaller data set
$\mathcal{D}_{novel}=\\{(\mathbf{x}_{j},\mathbf{y}_{j})\\}_{j=1}^{N\times K}$
that covers $N$ novel event types, with $K$ examples each. Note that the base
and novel event types are disjoint except for the Other class, indicating non-
event type. In zero-shot event extraction (ZSEE), the training data set only
contains a large-scale set
$\mathcal{D}_{base}=\\{(\mathbf{x}_{i},\mathbf{y}_{i})\\}_{i=1}^{M}$ for the
base event types. The model $f$ will be optimized on base event types and
evaluated on the novel types. Following previous work, we set $N=5,10$ and
$K=0,1,5,10$ in this work.

###  3.2 Targeted Augmentation [Diversity]

In contrast to previous data augmentation approaches (Wei and Zou, 2019; Feng
et al., 2020; Ng et al., 2020; Kim et al., 2022; Loem et al., 2022; Hussein et
al., 2022; Wang et al., 2023a), we have improved upon the conventional
conditional generation method by transitioning from random sampling to a
targeted selection strategy. The targeted augmentation module serves as a
mechanism to ensure diversity. Theoretically, it can retrieve an infinite
number of entities from the external corpus, seamlessly incorporating these
entities into the given event structure. Consequently, the module can generate
an infinite variety of new event structures. Thus, the targeted augmentation
provides a theoretical framework for sampling and augmenting an extensive
array of entities, particularly beneficial when working with a limited set of
annotated event mentions.

#### Dependent Context Retrieval

For a given event structure, we retrieve context candidates from the corpus
that share tokens with the event structure. In our experiments, we gathered
sentences containing the mention of the event trigger. To extract context
information from the sampled sentences, we utilized the spaCy Named Entity
Recognition (NER) parser222https://spacy.io/usage/linguistic-features to
identify entity mentions. Consequently, the extracted entity mentions from
each sampled sentence serve as context candidates for the given event
structure. The context corpus employed in this study is the NYT Annotated
Corpus333https://catalog.ldc.upenn.edu/LDC2008T19.

#### Targeted Generation

Given an event structure ${e}_{i}=\\{{t}_{i},{a}_{1},...,{a}_{p}\\}$ and a
sampled context candidate ${c}=\\{{c}_{1},...,{c}_{q}\\}$, a generator is
leveraged to generate a corresponding sentence. If the sampled context
entities could potentially serve as argument roles in the original event
structures, we employ an add-or-replace strategy, to further tailor the event
structure. The feasibility of integrating an entity into the event structure
depends on its entity type. If the argument role is vacant in the original
structure, and the entity type of the sampled entity aligns with the argument
role, we add the entity to the event structure. If the argument role is
already populated, we substitute it with the sampled entity.

For example, given an annotation on the sentence "now it ’s up to the appeals
court and the board of pardon and paroles to officially clear their names.", a
Justice:Pardon event is represented by the event structure {Trigger: clear,
Adjudicator: court, Adjudicator: board of pardon and paroles}. A complete
Justice:Pardon structure may also include two argument roles, namely Defendant
and Place. From the sampled context entities [Paul Laxalt, 1988, Nevada],
Nevada is added to the event structure as an Place role, and Paul Laxalt is
added as a Defendant role. Note that "Nevada" is added because it is a GPE
entity and a GPE entity is one of the possible entity types for a Place role.
Similarly, Paul Laxalt is added as a Defendant because it is a PER entity.
Here we present a generated sentence with the enriched event structure: "The
court in Nevada clear Paul Laxalt, as advised by the board of pardon and
paroles." The process is illustrated in Figure 1.

Figure 2: Event mention accuracy verification module.

###  3.3 Negative Augmentation [Polarity]

Polarity is maintained through the negative augmentation design. This process
generates not only positive event mentions but also negative mentions,
including hypothetical mentions and believed event mentions. For event
extraction, we focus on identifying event that occurs, and also negative
mentions. For example, in the sentence “John Hinkley denied his attempt to
assassinate Ronald Reagan.”, a model, especially generative models, might
overlook this Conflict:Attack mention triggered by the token assassinate,
because this is not an actual event that happens. More specifically, negative
event mentions include (1) explicit negative mentions: expressed with a
negative word such as not or never, or a negative lexical context such deny,
refuse or disobey, (2) asserted mentions: including hypothetical events,
believed events, or promised events, etc Linguistic Data Consortium (2005).

Thus in addition to augmenting high-quality positive training examples,
particular attention is paid to augmenting negative training examples. In this
work, we write negative/asserted expression prompts to guide their generation.
Prompts and generated negative augmentation examples are listed in Table 6 and
Table 7 in Appendix B.

###  3.4 Back-Validation

Given noisy training examples, previous research has utilized methods to
detect and rectify corrupted data during training Han et al. (2018); Yu et al.
(2019); Huang et al. (2019); Yao et al. (2020); Wei et al. (2020); Jiang et
al. (2020); Zhang et al. (2021c), but such approaches necessitate extensive
training. In our context, where the generated data is considered of reasonable
quality, we propose the incorporation of a back-and-forth validation module.
This module aims to ensure the accuracy and coherence of the generated
content, thereby enhancing the reliability of the augmented examples.

#### Event Mention Accuracy Verification [Accuracy]

For each generated example, its accuracy can be verified through an entailment
verification module. As shown in Figure 2, given the generated sentence and
its source event structure, we first textualize the event structure into a
passage to express the event structure, by a pre-defined template Hsu et al.
(2022). Then the two texts will be passed into an NLI entailment verification
module. The intuition is that, for a valid generation, it should entail the
template passage with the event structure.

Figure 3: Event mention coherence verification module.

#### Event Mention Coherence Verification [Coherence]

In addition to ensuring generation accuracy, we aim for the generated sentence
to exhibit strong coherence with the provided event structure. Specifically,
there should be no extraneous or omitted arguments when compared to the given
event structure. The intuition is that if the generated sentence aligns
coherently with the provided event structure, a template passage incorporating
the event structure should entail the generated sentence, and vice versa. A
distinctive scenario arises when the event structure is incomplete. In such
instances, we adapt the missing argument role in the template with the
expression "an unspecific [argument role]." Illustrated in Figure 3, if the
Place argument role is absent, we want to ensure that the generated event
mention does not introduce an extraneous arbitrary Place argument role.
Consequently, we substitute "[Place]" with "[an unspecific Place]." This
modification ensures that the generated sentence fails the forward-and-
backward entailment test in such scenarios.

###  3.5 Generative Event Extraction Model

DEGREE Hsu et al. (2022) is a generative event extraction model that
conceptualizes event extraction as a conditional generation problem. Given a
sentence and a crafted prompt, DEGREE generates an output following a
specified format. The predictions for event triggers and argument roles can be
then parsed from the generated output using a deterministic algorithm. In
contrast to earlier classification-based models, the generation framework
offers a versatile approach to incorporate supplementary information and
guidance. Through the creation of suitable prompts, DEGREE can better capture
the dependencies between entities and, consequently diminish the requisite
number of training examples.

The EE template defines the anticipated output format and is organized into
two main parts. The initial segment is referred to as the trigger template,
structured as “Event trigger is <Trigger>”, with “<Trigger>” acting as a
placeholder for event trigger in the original passage. The subsequent section
is the argument template, and its composition varies based on the specific
event type. For instance, the argument template for a Conflict:Attack event is
“some people or some organization in somewhere was ordered by some adjudicator
to pay a fine.” Each underlined string, beginning with "some-," serves as a
placeholder corresponding to an argument role for a Justice:Fine event. For
example, "somewhere" corresponds to the Place where the event occurs. Note
that every event type has its own argument template. Event extraction
templates and the construction details can be found in Hsu et al. (2022).

###  3.6 Robust Fine-tuning

Given the synthesized training samples $\mathcal{D}_{gen}$ that augment
$\mathcal{D}_{train}$ for fine-tuning a classification $M$. The primary
concern is the presence of label noise, where some generated samples may
inaccurately align with their corresponding labels, potentially degrading
model performance when using standard supervised learning. To address this
challenge, we employ a noise-robust training procedure to enhance stability.
We first fine-tune the back-validator $V$ with the training data constructed
from the base dataset. For negative examples, we construct two datasets: (1)
sample unpaired event structures and sentences within the corpus and (2)
replace argument roles in the template with "an unspecific [argument role]".
Then we validate the augmented examples with the fine-tuned validator $V$, and
validated examples are then used for fine-tuning the EE model $M$. Finally, we
employ a random sample selection on the base data set $\mathcal{D}_{base}$ and
the synthesized training set $\mathcal{D}_{gen}$, along with the entire few
shot training set $\mathcal{D}_{novel}$ to update the EE model $M$. The
algorithm is shown in Algorithm 1.

Algorithm 1 Robust Fine-tuning

Input: Base data set $\mathcal{D}_{base}$; few shot training set
$\mathcal{D}_{novel}$; synthesized training set $\mathcal{D}_{gen}$.
Output: Model $M$, validator $V$

fine-tune $V$ with back-validation data constructed from $\mathcal{D}_{train}$

pass $\mathcal{D}_{gen}$ into $V$, collect $\mathcal{D}^{\prime}_{gen}$ that
pass back-validation

for each epoch $t$ do

Sample meta batch $D^{t}_{base}$ from $\mathcal{D}_{base}$

Sample noisy batch $D^{t}_{gen}$ from $\mathcal{D}^{\prime}_{gen}$

Update model $M$ with $D^{t}_{train}$, $\mathcal{D}_{novel}$, and
$D^{t}_{gen}$

Discard corrupted data by semantic distance to the center instances

end for

Model | Time/Sentence(s) | Cost/Sentence($)
---|---|---
Vicuna-7B | 2.7 | 0
LLaMA2-7B | 8.7 | 0
GPT-3.5-turbo | 2.4 |  $\sim$0.0035
Table 1: Augmentation cost per sentence. Method | K-shot | Common 5 | Common 10
---|---|---|---
Tri-I | Tri-C | Arg-I | Arg-C | Tri-I | Tri-C | Arg-I | Arg-C
Matching Baseline | full | 42.7 | 42.1 | - | - | 46.3 | 46.3 | - | -
Lemmatization Baseline | full | 51.5 | 50.2 | - | - | 56.0 | 56.0 | - | -
OneIE | full | 72.7 | 70.5 | 52.3 | 49.9 | 74.5 | 73.0 | 51.2 | 48.9
DEGREE | full | 68.4 | 66.0 | 51.9 | 48.7 | 72.0 | 69.8 | 52.5 | 49.2
BERT_QA | 1-shot | 10.0 | 1.4 | 1.3 | 1.3 | 8.2 | 1.6 | 1.1 | 1.1
5-shot | 14.0 | 12.6 | 11.1 | 10.8 | 20.8 | 15.4 | 14.6 | 13.9
10-shot | 37.8 | 11.3 | 22.9 | 22.1 | 32.0 | 27.8 | 19.5 | 18.6
OneIE | 1-shot | 4.2 | 4.2 | 1.5 | 1.5 | 4.1 | 2.7 | 2.0 | 2.0
5-shot | 39.3 | 38.5 | 24.8 | 22.8 | 41.9 | 41.9 | 29.7 | 27.2
10-shot | 54.8 | 53.3 | 36.0 | 34.9 | 61.5 | 57.8 | 41.4 | 39.2
DEGREE | 0-shot | 53.3 | 46.8 | 29.6 | 25.1 | 60.9 | 54.5 | 42.0 | 31.4
1-shot | 60.1 | 53.3 | 38.8 | 31.6 | 61.2 | 60.9 | 41.1 | 34.7
5-shot | 57.8 | 55.5 | 40.6 | 36.1 | 65.8 | 64.8 | 45.3 | 42.7
10-shot | 63.8 | 61.2 | 46.0 | 42.0 | 72.1 | 68.8 | 52.5 | 48.4
Talor-EE (Vicuna) | 0-shot | 66.1 | 62.3 | 38.7 | 32.9 | 71.6 | 68.7 | 40.7 | 35.9
1-shot | 63.5 | 55.7 | 37.5 | 32.0 | 69.2 | 64.5 | 47.8 | 43.2
5-shot | 67.0 | 65.2 | 46.6 | 43.1 | 72.7 | 70.0 | 50.1 | 44.9
10-shot | 70.4 | 66.2 | 46.4 | 42.7 | 73.9 | 71.7 | 49.2 | 44.9
Talor-EE (LLaMA) | 0-shot | 65.0 | 62.5 | 41.0 | 36.5 | 65.6 | 64.8 | 47.5 | 43.8
1-shot | 66.5 | 61.0 | 42.3 | 34.4 | 71.5 | 66.7 | 45.4 | 42.4
5-shot | 70.2 | 63.9 | 46.3 | 42.4 | 71.7 | 70.1 | 50.5 | 46.7
10-shot | 70.0 | 67.6 | 46.2 | 43.3 | 70.5 | 70.2 | 51.2 | 49.5
Talor-EE (GPT) | 0shot | 67.9 | 66.1 | 46.1 | 40.0 | 72.5 | 70.3 | 46.9 | 42.8
1-shot | 68.5 | 64.8 | 42.1 | 35.6 | 72.5 | 68.1 | 46.5 | 42.8
5-shot | 67.9 | 64.2 | 44.6 | 42.6 | 73.6 | 70.6 | 48.5 | 44.7
10-shot | 70.2 | 67.4 | 43.0 | 41.4 | 74.2 | 70.5 | 48.3 | 47.7
Table 2: Low-resource EE results on ACE05-E. Bold represents the highest score
for the current setting.

##  4 Experiments

Figure 4: Experimental results on ACE05-E. (a-b) are visualizations for Common 5, and (c-d) for Common 10. Method | K-shot | Common 5 | Common 10
---|---|---|---
Tri-I | Tri-C | Arg-I | Arg-C | Tri-I | Tri-C | Arg-I | Arg-C
DEGREE | full | 54.7 | 53.1 | 45.4 | 44.7 | 58.8 | 58.2 | 51.3 | 50.8
DEGREE | 0-shot | 32.2 | 26.8 | 16.1 | 15.5 | 47.7 | 45.4 | 28.7 | 28.0
1-shot | 34.4 | 33.8 | 28.0 | 26.2 | 39.4 | 39.4 | 30.7 | 29.9
5-shot | 44.8 | 39.2 | 28.9 | 28.7 | 56.3 | 55.5 | 44.5 | 42.7
10-shot | 48.4 | 45.8 | 39.3 | 38.8 | 59.3 | 57.8 | 48.4 | 47.8
Talor-EE (Vicuna) | 0-shot | 41.9 | 40.2 | 31.0 | 28.9 | 50.6 | 49.0 | 37.9 | 36.6
1-shot | 48.5 | 38.7 | 31.3 | 30.4 | 47.8 | 41.6 | 35.9 | 34.8
5-shot | 45.8 | 43.0 | 35.8 | 33.4 | 56.2 | 53.7 | 42.5 | 41.0
10-shot | 55.7 | 52.0 | 40.6 | 37.6 | 58.2 | 56.7 | 47.8 | 44.9
Talor-EE (LLaMA) | 0-shot | 40.8 | 34.7 | 26.2 | 23.8 | 51.6 | 45.4 | 37.8 | 36.4
1-shot | 47.4 | 39.1 | 33.4 | 33.2 | 47.3 | 44.4 | 46.2 | 44.6
5-shot | 48.9 | 44.5 | 37.7 | 34.8 | 55.3 | 54.6 | 48.5 | 47.8
10-shot | 58.1 | 55.7 | 45.5 | 42.5 | 58.2 | 57.5 | 52.2 | 48.4
Talor-EE (GPT) | 0-shot | 49.3 | 41.9 | 34.0 | 32.4 | 57.1 | 55.8 | 43.1 | 40.8
1-shot | 50.3 | 42.0 | 34.5 | 32.1 | 51.6 | 44.3 | 43.7 | 42.1
5-shot | 52.9 | 48.2 | 39.1 | 37.3 | 57.5 | 56.0 | 49.4 | 45.5
10-shot | 56.9 | 54.6 | 43.5 | 43.0 | 62.4 | 61.7 | 53.4 | 49.6
Table 3: Low-resource EE results on ERE. Bold represents the highest score for
the current setting.

We perform experiments on three public benchmark datasets, including ACE05-E
(Automatic Content Extraction)444https://catalog.ldc.upenn.edu/LDC2006T06 and
ERE (Entity Relation Event) Song et al. (2015). To showcase the effectiveness
of the proposed method under low resource settings, experiments are conducted
under $N$way-$K$shot learning setting, where $N\in\\{5,10\\}$, and
$K\in\\{0,1,5,10\\}$.

#### Compared baselines

We consider the following baselines: (1) Matching baseline555(1) and (2) are
baselines for event detection tasks, thus only trigger detection results are
reported., a proposed baseline that makes trigger predictions by performing
string matching between the input passage and the event keywords. (2)
Lemmatization baseline, another proposed baseline that performs string
matching on lemmatized input passage and the event keywords. (3) BERT_QADu and
Cardie (2020), (4) OneIE Lin et al. (2020), (5) DEGREE Hsu et al. (2022) and
(6) QueryExtract Wang et al. (2022). The implementation details can be found
in Appendix A.

#### Generation Agents

Three generation agents are experimented in this work, including
vicuna-7b-v1.3 (Vicuna), Llama-2-7b (LLaMA), and gpt-3.5-turbo (GPT). For each
agent, we list the augmentation cost in Table 1, where two factors are listed
including generation time and cost per sentence.

Figure 5: Experimental results on ERE. (a-b) are visualizations for Common 5, and (c-d) for Common 10. Method | K-shot | Common 5 | Common 10
---|---|---|---
Tri-I | Tri-C | Arg-I | Arg-C | Tri-I | Tri-C | Arg-I | Arg-C
Talor-EE (LLaMA) | 1-shot | 66.5 | 61.0 | 42.3 | 34.4 | 71.5 | 66.7 | 45.4 | 42.4
5-shot | 70.2 | 63.9 | 46.3 | 42.4 | 71.7 | 70.1 | 50.5 | 46.7
10-shot | 70.0 | 67.6 | 46.2 | 43.3 | 70.5 | 70.2 | 51.2 | 49.5
\- enriched context | 1-shot | 61.2 | 52.1 | 35.9 | 28.3 | 72.9 | 64.6 | 46.2 | 40.6
5-shot | 68.5 | 64.2 | 43.5 | 41.1 | 73.2 | 70.0 | 45.7 | 44.6
10-shot | 67.0 | 63.4 | 43.1 | 39.5 | 74.7 | 71.7 | 46.4 | 43.2
\- negative augmentations | 1-shot | 70.5 | 65.1 | 41.8 | 34.4 | 74.1 | 67.4 | 44.4 | 38.8
5-shot | 69.3 | 62.6 | 41.8 | 39.3 | 77.4 | 73.4 | 48.4 | 42.8
10-shot | 69.1 | 61.3 | 40.8 | 39.6 | 74.1 | 70.5 | 46.6 | 44.3
\- back-validation | 1-shot | 61.2 | 52.1 | 35.9 | 28.3 | 72.7 | 66.0 | 47.3 | 42.2
5-shot | 68.0 | 62.8 | 43.1 | 38.6 | 76.1 | 74.6 | 48.6 | 44.4
10-shot | 67.2 | 65.2 | 42.1 | 40.2 | 75.3 | 71.2 | 47.3 | 46.7
Table 4: Ablation study on ACE05-E.

###  4.1 Main results

The experimental results for low-resource Event Extraction (EE) are presented
in Table 2 and Figure 4 for ACE05-E, and Table 3 and Figure 5 for ERE,
respectively. From the experiment results, several conclusions can be drawn:
(1) With the augmented examples, the performance of low-resource EE generally
exhibits improvement, evident in both zero-shot learning and few-shot learning
settings. This improvement is consistent across different generation agents
(Vicuna, LLaMA, and GPT) and backbone EE models. Table 8 displays experimental
results on ACE05-E with QueryExtract as the backbone model, highlighting the
effectiveness of augmented training examples across various EE models. (2) The
observed improvement is more pronounced in extremely low-resource scenarios,
particularly in zero-shot, 1-shot, and 5-shot scenarios. The impact is less
significant when more clean training examples are available, such as in the
10-shot setting. (3) We observe that the performance of zero-shot augmented
training can surpass that of 1-shot training with clean examples. This
discrepancy arises because some sampled clean training examples may not
straightforwardly express event information. For instance, the token “open”
could trigger a Start-Organization event, introducing confusion in the
semantics of the Start-Organization event type. (4) Augmented examples
generated by different generation agents consistently enhance low-resource EE
performance. Notably, greater performance gains are achieved with examples
generated by LLaMA and GPT.

Additionally, we have evaluated the generation quality and the effectiveness
of the proposed modules. Notably, for diversity, there is a substantial
increase in unique argument roles compared to the few-shot examples. For
example, in the common 10 and 5-shot settings, the count of unique argument
roles surged from 142 to 1184, marking a remarkable increase of 2502
percentage points, on average across the generation models. Regarding
polarity, among the 30 sampled augmentations verified through human
evaluation, the generated event mention expressions consistently align with
the targeted negative expression types. In terms of back-validation, the
evaluation involved two annotators who each assessed 200 randomly sampled
generations (100 for with back-validation generations and 100 for generations
without back-validation). On average, seven generations were deemed not fluent
when utilizing the back-validation module, while 19 generations were
identified as not fluent without the back-validation module.

ID |  Text |  GTH |  Predictions
---|---|---|---
(a) |  Hoon said Saddam ’s regime was crumbling under the pressure of a huge air assault . |  crumbling; End-Org; regime: Org; |  None
(b) |  The combination of the banking operations of Barclays Spain and Zaragozano will bring together two complementary businesses. |  combination; Transfer-Ownership; Barclays Spain: Buyer; Zaragozano: Artifact; |  combination, Merge-Org; businesses, Org
(c) |  Married for the second time , Hariri has five children. |  Married, Marry; Hariri: Person; |  None
(d) |  However the firm announced on Friday that it had reached a deal with the British arm of French distributors Pathe to show four releases. |  None |  releases; Release-Parole; firm: Entity;
Table 5: Case study for challenging examples

###  4.2 Ablation Studies

An ablation study was conducted to assess the effectiveness of each proposed
module, and the experimental results are presented in Table 4. (Omitting the
enriched context in the setting entails bypassing the Dependent Context
Retrieval module, resulting in the absence of newly generated event
structures.) On average, across all settings, the performance of trigger
classification decreased by 2.5% and 1.9%, and argument classification
decreased by 8.3% and 7.1%, in the absence of enriched context or back-
validation, respectively. Without negative augmentations, the argument
classification decreases by 7.5%, while trigger classification performance is
on par with Talor-EE (LLaMA). This highlights that the designed modules have a
more pronounced impact on argumentation classification than on trigger
detection. The absence of enriched context led to the most significant
decrease in argument classification performance, emphasizing the crucial role
of augmentation diversity in mitigating low-resource argument extraction.

###  4.3 Error Analysis

Table 5 illustrates several challenging examples. For event trigger detection,
most of the errors are from the insufficient understanding of the trigger
phrase. For example in example (a) in Table 5, linking the phrase “crumbling”
to the End-Org(anization) event is challenging given the limited trigger
training examples from either clean data or augmented data. Example (b) is
challenging because the token “combination” entails closer semantic relation
to the Merge-Org event. Example (c) illustrates a case where the current data
augmentation model falls short in generating intricate event expressions.
Example (d) illustrates a scenario in which the use of augmented data could
potentially cause confusion. In this case, the actual event pertains to a film
release rather than a judicial release. Despite inadequate context
information, there is a likelihood that the augmented data might have
generated a false prediction with increased confidence. One potential solution
to this challenge is the ability to distinguish between multiple meanings of
the same word.

In contrast to event trigger detection, argument extraction presents greater
challenges, as improvements in argument extraction prove less pronounced than
those in trigger detection. Our conclusion stems from a meticulous analysis of
the generated outputs and prediction results, revealing two primary reasons.
The first reason is the lack of clear and comprehensive explanations for
certain argument roles, for example, the argument role “agent” in the Start-
Org event type. According to the definition Linguistic Data Consortium (2005),
an “agent” in a Start-Org event is a “PER”, “ORG”, or “GPE” entity responsible
for the “START-ORG” Event. However, it requires tremendous expert knowledge to
write precise instructions for argument roles like this. The second reason
pertains to the lack of clear distinctions among argument roles in generation
prompts. We recognize that elucidating the purpose and differentiation of each
argument role can be intricate. For instance, we observed minimal or even
adverse effects of augmented data on the event type “Transfer-Ownership”. This
complexity arises from the potential confusion surrounding three specific
argument roles: “Beneficiary”, “Buyer”, and “Seller”, particularly when the
trigger involves terms like “sell” or “acquire”. Notably, altering the trigger
from “sell” to “acquire” induces a substantial change in the sentence’s entire
syntactic structure.

##  5 Conclusion

In conclusion, this study proposes a new paradigm for tackling low-resource
event extraction tasks. Generation agents are employed to create a diverse
training dataset for event structures enriched with domain-invariant entities.
The generated examples undergo a thorough back-and-forth validation process to
assess accuracy and coherence. Our research encompasses extensive experiments
in diverse low-resource learning scenarios, such as zero-shot and few-shot
learning settings, across various event extraction models. The outcomes of
these experiments highlight the effectiveness of the proposed framework.
Furthermore, our proposed methodology can inspire researchers from diverse
domains to embrace a comparable paradigm or delve into the investigation of
data augmentation methods as a means of enriching their training datasets.

## Limitations

Talor-EE establishes a powerful starting point for advancing few-shot learning
research, offering a flexible framework for framing new tasks through our
proposed augmentation method. It encourages a systematic exploration of
general and resilient enhancements for low-resource event extraction systems.
However, augmenting non-event examples takes appropriate attention, as the
proposed system may tend to predict additional event mentions. The absence of
a clear distinction between an actual event and a non-event mention, due to
the lack of a precise definition, underscores the need for appropriate action.
We extend a warm invitation to future low-resource research endeavors and
augmentation methods to delve into the structural aspects of event generation
within a contrastive setting.

## Acknowledgements

This research is supported by the award No. 2238940 from the Faculty Early
Career Development Program (CAREER) of the National Science Foundation (NSF).
The views and conclusions contained herein are those of the authors and should
not be interpreted as necessarily representing the official policies, either
expressed or implied, of the U.S. Government. The U.S. Government is
authorized to reproduce and distribute reprints for governmental purposes
notwithstanding any copyright annotation therein.

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Event Expression Type |  Instruction Prompt
---|---
Negative Events |  An Event is NEGATIVE when it is explicitly indicated that the Event did not occur. Negative example 1: His wife was sitting in the backseat and was ’not hurt’. Negative example 2: Yeltsin ordered Skuratov’s suspension, but parliament repeatedly ’refused to sack’ him. Given the generated sentence, “[SENT]”, change it into a negative expression that the Event did not occur.
Believed Events |  Believed Events are event mentions that some people or organizations think or believe would happen but are not necessarily real or true event occurrences. Example 1: Rumors of ’arrests’ circulated in Vancouver. Example 2: The charity was suspected of ’giving’ money to al Qaeda. Given the generated sentence you provide, ’[SENT]’, change it into a believed event sentence:
Hypothetical Events |  Hypothetical events are event mentions that are supposed to happen but are not necessarily real or true event occurrences. Example 1: Should he not ’pay’ the money, they would ’kill’ him. Example 2: A demonstration of how he would behave if he were to ’become’ President. Given the generated sentence you provide, ’[SENT]’, change it into a hypothetical event sentence:
Promised Events |  Promised Events are event mentions that are promised to happen but are not necessarily real or true event occurrences. Example 1: He said he would ’leave’ town. Example 2: Promises of ’aid’ made by Arab and European countries. Given the generated sentence you provide, ’[SENT]’, change it into a promised event sentence:
Desired Event |  Desired events are event mentions that are desired to happen but not necessarily real or true event occurrences. Example: They wanted to ’acquire’ the company last year. Given the generated sentence you provide, “[SENT]”, change it into a Desired event sentence:
Table 6: Negative/asserted expression generation template. “[SENT]” is a placeholder for the generated sentence with a positive event expression. The instruction is adapted from Linguistic Data Consortium (2005). id | Note |  Content
---|---|---
1 | Event Structure |  Trigger: bankruptcy. Org: Hazelhurst & Associates Inc.
Context |  10 percent, yesterday, $22.5 million
Positive mention |  Hazelhurst & Associates Inc. declared bankruptcy yesterday, with $22.5 million in debts.
Negative mention |  Hazelhurst & Associates Inc. did not declare bankruptcy yesterday, with $22.5 million in debts.
Asserted mention: |  It is believed that Hazelhurst & Associates Inc. will declare bankruptcy tomorrow, with $30 million in debts.
2 | Event Structure |  Trigger: pardon, Place: Jordan, Adjudicator: Abdullah II, Defendant: Rich
Context |  Republicans, today, his darkest hours
Positive mention |  Rich received a pardon from Abdullah II during his darkest hours , as Republicans gathered today to offer their support .
Negative mention |  Rich’s pardon from Abdullah II was canceled during his darkest hours, as Republicans did not gather.
Asserted mention |  Rich desired to receive a pardon from Abdullah II during his darkest hours, as Republicans gathered last year to offer their support.
Table 7: Negative Augmentation Example

##  Appendix A Implementation

For a fair comparison with baseline approaches, we use the pre-trained bert-
large-uncased model for fine-tuning and optimizing our model with BertAdam. We
optimize the parameters with grid search: training epoch 10, learning rate
$\in[3e\text{-}6,1e\text{-}4]$, training batch size $\in\\{8,12,16,24,32\\}$,
dropout rate $\in\\{0.4,0.5,0.6\\}$. Our experiments run on one Quadro RTX
8000. For trigger detection, the average runtime is 3.0 hours. For argument
detection, the average runtime is 1.3 hours. We use Spacy to generate POS
tags. We use three random seed 0, 39, 42 for all experiments, and report the
mean scores.

#### Sampling Strategy

Note that in the context of few-shot learning with an $N$way-$K$shot setting,
the variable K denotes the number of event mentions rather than training
examples. The original corpus contains numerous instances where a single
sentence includes multiple event mentions, presenting a challenge for the few-
shot example sampling process. Without regularization, the sampled examples
may probably exceed the specified K event mentions.

To address this issue and ensure that, for every setting, the sampled examples
with novel event types do not surpass K, we employ a sorting mechanism based
on the frequency of event types in decreasing order. This involves sorting the
event types and then sampling in the sorted order. For instance, consider the
examples with "Justice:Acquit" mentions, one of which also includes a
"Justice:Convict" mention. If we were to first sample examples for
"Justice:Convict" and this particular example is omitted, we would miss the
opportunity to include this crucial instance for "Justice:Acquit." This
becomes especially significant in settings such as 5-shot or 10-shot, where
"Justice:Acquit" has a total of four examples. Without this sampling approach,
the mentioned example may be excluded from the training procedure, impacting
the model’s performance.

#### Generation Instruction

The following instruction are used to prompt generations given the event
structure: “You are a helpful assistant in generating fluent and reasonable
sentences with event mentions. An Event is a specific occurrence involving
participants. An Event is something that happens. An Event can frequently be
described as a change of state. Please be sure the given event information is
in the generated sentence. However, the given context information is optional
in generation. Generate a sentence with {event_type_name} event, with optional
context information: {list_of_context_entitites}. {event_template}.” The
{event_template} refers to the textual representation given the event
structure, as presented in Hsu et al. (2022).

Method | K-shot | Common 5 | Common 10
---|---|---|---
Tri-I | Tri-C | Arg-I | Arg-C | Tri-I | Tri-C | Arg-I | Arg-C
QE | 1-shot | 58.6 | 48.7 | 33.1 | 29.3 | 58.6 | 51.2 | 37.5 | 30.1
5-shot | 61.9 | 57.1 | 37.6 | 33.1 | 66.7 | 61.1 | 41.7 | 36.5
10-shot | 64.1 | 62.2 | 40.3 | 38.6 | 72.0 | 67.2 | 45.6 | 45.2
Tolar-QE (Vicuna) | 1-shot | 60.6 | 58.0 | 41.8 | 34.2 | 60.4 | 58.0 | 41.4 | 35.0
5-shot | 65.4 | 62.1 | 44.3 | 35.8 | 70.8 | 68.8 | 47.2 | 41.6
10-shot | 65.7 | 64.0 | 43.4 | 39.6 | 69.5 | 68.1 | 50.8 | 43.7
Tolar-QE (LLaMa) | 1-shot | 64.7 | 57.6 | 39.3 | 28.3 | 57.8 | 54.9 | 43.5 | 33.9
5-shot | 61.6 | 59.4 | 42.3 | 37.1 | 71.2 | 65.1 | 46.2 | 40.9
10-shot | 66.0 | 64.9 | 44.1 | 39.8 | 68.2 | 67.4 | 49.4 | 44.9
Tolar-QE (GPT) | 1-shot | 64.8 | 58.7 | 38.4 | 31.3 | 62.8 | 61.2 | 43.8 | 36.1
5-shot | 67.5 | 59.6 | 41.4 | 36.5 | 66.1 | 66.1 | 47.5 | 43.6
10-shot | 67.4 | 65.2 | 42.7 | 39.1 | 71.1 | 70.4 | 49.2 | 46.5
Table 8: Few-shot Event Extraction results with data augmentation on ACE05-E
with QueryExtract (QE).

##  Appendix B Negative Event Mentions Prompts

Table 6 list generation instructions of negative event mentions for generation
agents. Table 7 shows negative augmentation examples.

##  Appendix C Experimental Results with QE

Table 8 shows Experimental results for ACE05-E with QueryExtract (QE) as the
baseline model.

##  Appendix D Features Contributed by Augmented Data

The features that are better captured by the proposed approach include (1) The
mapping between candidate triggers and event types. The presence of a greater
variety of event mention expressions within diverse contexts enhances the
robustness and comprehensiveness of the mapping between candidate triggers and
event types. (2) The mapping between negative expressions and event types. Due
to the limited occurrence of negative events in the training data, their
availability as few-shot examples is restricted. With the integration of the
negative augmentation module, the mapping between negative expressions and
event types becomes clearer. (3) The relation between candidate triggers and
arguments. The generated sentences exhibit a comparatively higher prevalence
of straightforward event expressions than those present in annotated data,
such as ACE2005. These less complex expressions contribute to a good fit for
features related to the relation between candidate triggers and arguments, in
the low-resource settings.


# A Generalization of Whyburn’s Theorem, and Aperiodicity for Abelian
$C^{*}$-Inclusions

Vrej Zarikian  Department of Mathematics, U. S. Naval Academy, Annapolis, MD
21402, USA<EMAIL_ADDRESS>
(Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz.)

###### Abstract.

Let $j:Y\to X$ be a continuous surjection of compact metric spaces. Whyburn
proved that $j$ is _irreducible_ , meaning that $j(F)\subsetneq X$ for any
proper closed subset $F\subsetneq Y$, if and only if $j$ is _almost one-to-
one_ , in the sense that

$\overline{\\{y\in Y:j^{-1}(j(y))=y\\}}=Y.$

In this note we prove the following generalization: There exists a unique
minimal closed set $K\subseteq Y$ such that $j(K)=X$ if and only if

$\overline{\\{x\in X:\operatorname{\operatorname{card}}(j^{-1}(x))=1\\}}=X.$

Translated to the language of operator algebras, this says that if
$\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$ is a unital
inclusion of separable abelian $C^{*}$-algebras, then there exists a unique
_pseudo-expectation_ (in the sense of Pitts) if and only if the _almost
extension property_ of Nagy-Reznikoff holds. More generally, we prove that a
unital inclusion of (not necessarily separable) abelian $C^{*}$-algebras has a
unique pseudo-expectation if and only if it is _aperiodic_ (in the sense of
Kwaśniewski-Meyer).

###### Key words and phrases:

irreducible map, almost one-to-one map, quasicontinuous selection, abelian
$C^{*}$-algebra, unique pseudo-expectation, almost extension property,
aperiodicity

###### 2010 Mathematics Subject Classification:

Primary 54E40; Secondary 46J10.

##  1\. Introduction

Let $j:Y\to X$ be a continuous surjection of compact Hausdorff spaces.
Following [2], we say that $j$ is _almost one-to-one_ if

$\overline{\\{y\in Y:j^{-1}(j(y))=y\\}}=Y.$

We say that $j$ is _irreducible_ if $j(F)\subsetneq X$ for all proper closed
subsets $F\subsetneq Y$. It is easy to see that

$\text{$j$ one-to-one}\implies\text{$j$ almost one-to-one}\implies\text{$j$
irreducible}.$

In general, an irreducible map need not be almost one-to-one (see Example 5.1
below). If, however, $Y$ is metrizable, then Whyburn’s Theorem [11, Theorem 2]
states that irreducible maps are necessarily almost one-to-one, so that the
two notions coincide.

Irreducible maps and almost one-to-one maps arise naturally in topological
dynamics. Indeed, a minimal self-map of a compact Hausdorff space must be
irreducible [5, Lemma 2.1]. Thus a minimal self-map of a compact metric space
must be almost one-to-one, by Whyburn’s Theorem (see also [5, Theorem 2.7]).

In Theorem 4.2 below, we generalize Whyburn’s Theorem, proving that for a
continuous surjection $j:Y\to X$ of compact metric spaces, there exists a
unique minimal closed set $K\subseteq Y$ such that $j(K)=X$ if and only if

$\overline{\\{x\in X:\operatorname{\operatorname{card}}(j^{-1}(x))=1\\}}=X.$

To see that this is a generalization, suppose that $j$ is irreducible. Then
there exists a unique minimal closed set $K\subseteq Y$ such that $j(K)=X$,
namely $K=Y$. By Theorem 4.2, $\overline{X_{1}}=X$, where

$X_{1}=\\{x\in X:\operatorname{\operatorname{card}}(j^{-1}(x))=1\\}.$

If

$Y_{1}=\\{y\in Y:j^{-1}(j(y))=y\\},$

then $j(Y_{1})=X_{1}$ and $j(\overline{Y_{1}})=\overline{X_{1}}=X$. Thus
$\overline{Y_{1}}=Y$, by irreducibility. So $j$ is almost one-to-one.

As the following example shows, Theorem 4.2 is a strict generalization of
Whyburn’s Theorem.

######  Example 1.1.

Let

$Y=([0,1]\times\\{0\\})\cup(\\{1\\}\times[0,1])\subseteq\operatorname{\mathbb{R}}^{2},$
$X=[0,1]\subseteq\operatorname{\mathbb{R}},$

and $j:Y\to X$ be defined by

$j(x,y)=x.$

Then there exists a unique minimal closed set $K\subseteq Y$ such that
$j(K)=X$, namely

$K=[0,1]\times\\{0\\},$

and

$\overline{\\{x\in
X:\operatorname{\operatorname{card}}(j^{-1}(x))=1\\}}=\overline{[0,1)}=[0,1]=X.$

Clearly $j$ is not irreducible. It is not almost one-to-one because

$\overline{\\{(x,y)\in
Y:j^{-1}(j(x,y))=(x,y)\\}}=\overline{[0,1)\times\\{0\\}}=[0,1]\times\\{0\\}\subsetneq
Y.$

When translated into the language of operator algebras, Theorem 4.2 says that
for a unital inclusion
$\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$ of separable
abelian $C^{*}$-algebras, there exists a unique _pseudo-expectation_ (in the
sense of Pitts) if and only if the _almost extension property_ of Nagy-
Reznikoff holds. (All these terms will be defined in Section 6 below.) This
answers a question posed to the author by Ruy Exel and Bartosz Kwaśniewski
following his 2019 talk at the International Workshop on Operator Theory and
its Applications in Lisbon, Portugal.

More generally, we prove (Theorem 6.2) that a unital inclusion
$\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$ of (not
necessarily separable) abelian $C^{*}$-algebras has a unique pseudo-
expectation if and only if the inclusion is _aperiodic_ (in the sense of
Kwaśniewski and Meyer). Aperiodicity is weaker than the almost extension
property [7, Theorem 5.5], but stronger than having a unique pseudo-
expectation [7, Theorem 3.6]. For separable $C^{*}$-inclusions, aperiodicity
and the almost extension property coincide [7, Theorem 5.5]. Thus Theorem 4.2
is actually a corollary of Theorem 6.2, modulo a non-trivial operator-
algebraic result, as illustrated in the diagram below:

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Nonetheless, we include a direct, purely-topological proof of Theorem 4.2 that
does not rely on Theorem 6.2 nor on [7, Theorem 5.5].

Acknowledgements: We thank Alexis Alevras and Marcos Valdes for helpful
discussions about this material.

##  2\. Preliminaries

Let $j:Y\to X$ be a continuous surjection of compact Hausdorff spaces. In this
section we show that there must exist a minimal closed set $K\subseteq Y$ such
that $j(K)=X$ (Proposition 2.1) and we establish a criterion for determining
when $K$ is unique (Proposition 2.2).

######  Proposition 2.1.

Let $j:Y\to X$ be a continuous surjection of compact Hausdorff spaces. Then
there exists a minimal closed set $K\subseteq Y$ such that $j(K)=X$.

###### Proof.

Let

$\operatorname{\mathcal{F}}=\\{F\subseteq Y:\text{$F$ is closed and
$j(F)=X$}\\},$

a nonempty set partially ordered by reverse inclusion. Suppose
$\mathcal{L}\subseteq\operatorname{\mathcal{F}}$ is a nonempty totally ordered
subset and define

$F_{0}=\bigcap\\{F:F\in\mathcal{L}\\}.$

If $x\in X$, then

$\\{F\cap j^{-1}(x):F\in\mathcal{L}\\}$

is a collection of closed subsets of $Y$ with the finite intersection
property. Since $Y$ is compact,

$F_{0}\cap j^{-1}(x)=\bigcap\\{F\cap
j^{-1}(x):F\in\mathcal{L}\\}\neq\emptyset.$

Since $x$ was arbitrary, $j(F_{0})=X$. So $F_{0}\in\operatorname{\mathcal{F}}$
is an upper bound for $\mathcal{L}$. By Zorn’s Lemma,
$\operatorname{\mathcal{F}}$ has a maximal element. ∎

######  Proposition 2.2.

Let $j:Y\to X$ be a continuous surjection of compact Hausdorff spaces. Then
the following are equivalent:

  1. i.

There exists a unique minimal closed set $K\subseteq Y$ such that $j(K)=X$.

  2. ii.

If $F_{0}=\bigcap\\{F\subseteq Y:\text{$F$ is closed and $j(F)=X$}\\}$, then
$j(F_{0})=X$.

###### Proof.

Let

$F_{0}=\bigcap\\{F\subseteq Y:\text{$F$ is closed and $j(F)=X$}\\}.$

(i $\implies$ ii) Suppose there exists a unique minimal closed set $K\subseteq
Y$ such that $j(K)=X$. If $F\subseteq Y$ is closed and $j(F)=X$, then
$K\subseteq F$. Thus $K\subseteq F_{0}$. In fact, $F_{0}=K$, since
$F_{0}\subseteq K$ by definition. Therefore $j(F_{0})=j(K)=X$. (ii $\implies$
i) Conversely, suppose $j(F_{0})=X$. Let $K\subseteq Y$ be a minimal closed
set such that $j(K)=X$. Then $F_{0}\subseteq K$ by definition and so $K=F_{0}$
by minimality. ∎

##  3\. Quasicontinuous Selections

Let $j:Y\to X$ be a surjection. A _selection_ of $j$ is a function
$\alpha:X\to Y$ such that $j\circ\alpha=\operatorname{\operatorname{id}}_{X}$.
It is well-known that continuous surjections of compact metric spaces have
Borel selections (see [1, Theorem 3.4.1.], for example). On the other hand, a
continuous selection is usually impossible. In this section, we draw attention
to a theorem of Crannell, Frantz, and LeMasurier, which states that continuous
surjections of compact metric spaces have _quasicontinuous_ selections. This
result plays a critical role in our proof of Theorem 4.2 below.

######  Definition 3.1 (quasiopen).

Let $Y$ be a topological space. We say that $A\subseteq Y$ is quasiopen if
$A\subseteq\overline{A^{\circ}}$.

######  Definition 3.2 (quasicontinuous).

We say that a mapping $j:Y\to X$ of topological spaces is quasicontinuous if
for every open set $U\subseteq X$, $j^{-1}(U)\subseteq Y$ is quasiopen.

######  Theorem 3.3 ([3], Corollary 5).

Let $j:Y\to X$ be a continuous surjection of compact metric spaces. Then $j$
admits a quasicontinuous selection $\alpha:X\to Y$.

##  4\. A Generalization of Whyburn’s Theorem

In this section we state and prove our main result, Theorem 4.2. We start with
an elementary lemma.

######  Lemma 4.1.

Let $j:Y\to X$ be a continuous surjection of compact metric spaces. Then the
mapping

$X\to\operatorname{\mathbb{R}}:x\mapsto\operatorname{\operatorname{diam}}(j^{-1}(x))$

is upper semi-continuous.

###### Proof.

Let $\varepsilon>0$. Suppose that $x_{n}\to x$, where
$\operatorname{\operatorname{diam}}(j^{-1}(x_{n}))\geq\varepsilon$ for all
$n\in\operatorname{\mathbb{N}}$. For each $n\in\operatorname{\mathbb{N}}$,
there exist $y_{n},y_{n}^{\prime}\in j^{-1}(x_{n})$ such that
$d(y_{n},y_{n}^{\prime})>\varepsilon-1/n$. Passing to a subsequence, if
necessary, we may assume that $y_{n}\to y$ and $y_{n}^{\prime}\to y^{\prime}$.
Then $y,y^{\prime}\in j^{-1}(x)$ and $d(y,y^{\prime})\geq\varepsilon$. Thus
$\operatorname{\operatorname{diam}}(j^{-1}(x))\geq\varepsilon$. ∎

######  Theorem 4.2.

Let $j:Y\to X$ be a continuous surjection of compact metric spaces. Then the
following are equivalent:

  1. i.

There exists a unique minimal closed set $K\subseteq Y$ such that $j(K)=X$.

  2. ii.

$\\{x\in X:\operatorname{\operatorname{card}}(j^{-1}(x))=1\\}$ is dense in
$X$.

###### Proof.

(i $\implies$ ii) Suppose there exists a unique minimal closed set $K\subseteq
Y$ such that $j(K)=X$. Let $\varepsilon>0$. Assume that $\\{x\in
X:\operatorname{\operatorname{diam}}(j^{-1}(x))<\varepsilon\\}$ is not dense
in $X$. Then there exists $x_{0}\in X$ and $r>0$ such that

$B_{r}(x_{0})\subseteq\\{x\in
X:\operatorname{\operatorname{diam}}(j^{-1}(x))\geq\varepsilon\\}.$

Let $\alpha:X\to Y$ be a quasicontinuous selection of $j$. Then
$\alpha(x_{0})\in j^{-1}(B_{r}(x_{0}))$ and so there exists $\delta>0$ such
that $B_{\delta}(\alpha(x_{0}))\subseteq j^{-1}(B_{r}(x_{0}))$. Then
$\alpha^{-1}(B_{\delta}(\alpha(x_{0}))$ is a quasiopen set containing $x_{0}$.
Thus

$U=\alpha^{-1}(B_{\delta}(\alpha(x_{0})))^{\circ}$

is a nonempty open subset of $X$. Now

$\alpha^{-1}(B_{\delta}(\alpha(x_{0})))=j(B_{\delta}(\alpha(x_{0}))\cap\alpha(X)),$

and so there exists $A\subseteq B_{\delta}(\alpha(x_{0}))\cap\alpha(X)$ such
that $j(A)=U$. In particular, $\operatorname{\operatorname{diam}}(A)\leq
2\delta$. On the other hand,

$U=j(A)\subseteq j(B_{\delta}(\alpha(x_{0})))\subseteq
B_{r}(x_{0})\subseteq\\{x\in
X:\operatorname{\operatorname{diam}}(j^{-1}(x))\geq\varepsilon\\}.$

Thus for every $y\in A$, there exists $y^{\prime}\in Y$ such that
$j(y^{\prime})=j(y)$ and $d(y^{\prime},y)>\varepsilon/4$. It follows that
$d(y^{\prime},A)\geq\varepsilon/4-2\delta$. Setting
$A^{\prime}=\\{y^{\prime}:y\in A\\}$, we see that $j(A^{\prime})=U$ and
$d(A^{\prime},A)\geq\varepsilon/4-2\delta$. Choosing $\delta<\varepsilon/16$,
we conclude that $d(A^{\prime},A)>\varepsilon/8$, which implies
$\overline{A}\cap\overline{A^{\prime}}=\emptyset$. But then
$F=\overline{A}\cup j^{-1}(U^{c})$ and $F^{\prime}=\overline{A^{\prime}}\cup
j^{-1}(U^{c})$ are closed sets such that $j(F)=X$, $j(F^{\prime})=X$, and
$j(F\cap F^{\prime})=U^{c}\subsetneq X$, which contradicts Proposition 2.2. By
Lemma 4.1, $\\{x\in
X:\operatorname{\operatorname{diam}}(j^{-1}(x))<\varepsilon\\}$ is an open
dense subset of $X$. Thus

$\\{x\in
X:\operatorname{\operatorname{card}}(j^{-1}(x))=1\\}=\bigcap_{n=1}^{\infty}\\{x\in
X:\operatorname{\operatorname{diam}}(j^{-1}(x))<1/n\\}$

is dense in $X$.

(ii $\implies$ i) Suppose $X_{1}=\\{x\in
X:\operatorname{\operatorname{card}}(j^{-1}(x))=1\\}$ is dense in $X$. Define
$K=\overline{j^{-1}(X_{1})}$. If $F\subseteq Y$ is closed and $j(F)=X$, then
$j^{-1}(X_{1})\subseteq F$, which implies $K\subseteq F$. Since

$X=\overline{X_{1}}=\overline{j(j^{-1}(X_{1}))}\subseteq\overline{j(K)}=j(K),$

we see that $K$ is the unique minimal closed subset of $Y$ such that $j(K)=X$.
∎

##  5\. The Split Interval

In Whyburn’s theorem, the assumption of metrizability cannot be omitted [4,
Exercise 3.1.C]. Of course, the same is true for Theorem 4.2, as the following
example illustrates.

######  Example 5.1.

Let $\ddot{\operatorname{\mathbb{I}}}$ be the “split interval”, i.e., the set

$((0,1]\times\\{0\\})\cup([0,1)\times\\{1\\})\subseteq\operatorname{\mathbb{R}}^{2}$

equipped with the order topology arising from the lexicographic order

$(x,y)<(x^{\prime},y^{\prime})\iff(x<x^{\prime})\vee((x=x^{\prime})\wedge(y<y^{\prime})).$

Let $\operatorname{\mathbb{I}}=[0,1]$ be the unit interval equipped with its
usual topology. Then
$j:\ddot{\operatorname{\mathbb{I}}}\to\operatorname{\mathbb{I}}$ defined by

$j(x,y)=x$

is an irreducible continuous surjection of compact Hausdorff spaces, but

$\\{x\in\operatorname{\mathbb{I}}:\operatorname{\operatorname{card}}(j^{-1}(x))=1\\}=\\{0,1\\},$

which is not dense in $\operatorname{\mathbb{I}}$. Of course,
$\ddot{\operatorname{\mathbb{I}}}$ is not metrizable (it is not second
countable).

##  6\. Aperiodicity for Abelian $C^{*}$-Inclusions

Let $\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$ be a
unital inclusion of $C^{*}$-algebras (cf. [1]). Then
$\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$ has the _pure
extension property_ (PEP) if every pure state on $\operatorname{\mathcal{A}}$
extends uniquely to a pure state on $\operatorname{\mathcal{B}}$. A useful
relaxation of the PEP is the _almost extension property_ (AEP) of Nagy and
Reznikoff [8]. Instead of insisting that every pure state on
$\operatorname{\mathcal{A}}$ extends uniquely to a pure state on
$\operatorname{\mathcal{B}}$, the requirement is that a weak-* dense
collection of pure states on $\operatorname{\mathcal{A}}$ extends uniquely to
pure states on $\operatorname{\mathcal{B}}$. A _conditional expectation_ (CE)
of $\operatorname{\mathcal{B}}$ onto $\operatorname{\mathcal{A}}$ is a
completely positive map
$E:\operatorname{\mathcal{B}}\to\operatorname{\mathcal{A}}$ such that
$E|_{\operatorname{\mathcal{A}}}=\operatorname{\operatorname{id}}_{\operatorname{\mathcal{A}}}$.
A _pseudo-expectation_ (PsExp) for the $C^{*}$-inclusion
$\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$ is a
completely positive map $\theta:\operatorname{\mathcal{B}}\to
I(\operatorname{\mathcal{A}})$ such that
$\theta|_{\operatorname{\mathcal{A}}}=\operatorname{\operatorname{id}}_{\operatorname{\mathcal{A}}}$
[9, Definition 1.3]. Here $I(\operatorname{\mathcal{A}})$ is Hamana’s
_injective envelope_ of $\operatorname{\mathcal{A}}$, the minimal injective
operator system containing $\operatorname{\mathcal{A}}$. Every conditional
expectation is a pseudo-expectation, but there are $C^{*}$-inclusions with no
conditional expectations, while pseudo-expectations always exist. We have the
relations

(6.1) $\text{PEP}\implies\text{AEP}\implies\text{$\exists!$
PsExp}\implies\text{$\exists$ at most one CE}.$

Only the middle implication is not immediate from the definitions.

It is somewhat surprising that the differences between the conditions in (6.1)
above, in particular the difference between the almost extension property and
having a unique pseudo-expectation, can be witnessed by abelian
$C^{*}$-inclusions. Indeed, let
$\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$ be a unital
inclusion of abelian $C^{*}$-algebras. By the Gelfand-Naimark Theorem,
$\operatorname{\mathcal{A}}\cong C(X)$, the continuous complex-valued
functions on a compact Hausdorff space $X$ (cf. [1, Theorem 1.7.3]). Likewise
$\operatorname{\mathcal{B}}\cong C(Y)$ for some compact Hausdorff space $Y$.
Under these identifications, the inclusion map
$\operatorname{\mathcal{A}}\to\operatorname{\mathcal{B}}$ corresponds to the
map $C(X)\to C(Y):f\mapsto f\circ j$ for some continuous surjection $j:Y\to
X$. It is easy to see that
$\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$ has the almost
extension property if and only if

$\overline{\\{x\in X:\operatorname{\operatorname{card}}(j^{-1}(x))=1\\}}=X.$

By [10, Corollary 3.21],
$\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$ has a unique
pseudo-expectation if and only if there exists a unique minimal closed set
$K\subseteq Y$ such that $j(K)=X$. Thus the abelian $C^{*}$-inclusion
$C(\operatorname{\mathbb{I}})\subseteq C(\ddot{\operatorname{\mathbb{I}}})$
corresponding to the continuous surjection
$j:\ddot{\operatorname{\mathbb{I}}}\to\operatorname{\mathbb{I}}$ described in
Example 5.1 above has a unique pseudo-expectation, but not the almost
extension property. On the other hand, by Theorem 4.2, for an abelian
$C^{*}$-inclusion $C(X)\subseteq C(Y)$ with $Y$ compact metric, the almost
extension property and having a unique pseudo-expectation are equivalent. The
metrizability of $Y$ corresponds precisely to the separability of $C(Y)$, and
so Theorem 4.2 says that for separable abelian $C^{*}$-inclusions
$\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$, the almost
extension property is equivalent to having a unique pseudo-expectation.

Recently, Kwásniewski and Meyer substantially advanced our understanding of
the relationship between the almost extension property and having a unique
pseudo-expectation. Namely they introduced the notion of _aperiodicity_ for
(arbitrary) $C^{*}$-inclusions [6, Definition 5.14], proved in [7, Theorems
5.5 and 3.6] that

(6.2) $\text{AEP}\implies\text{aperiodicity}\implies\text{$\exists!$ PsExp},$

and showed that for separable $C^{*}$-inclusions, the almost extension
property is equivalent to aperiodicity [7, Theorem 5.5]. In Theorem 6.2 below,
we prove that for abelian $C^{*}$-inclusions, aperiodicity is equivalent to
having a unique pseudo-expectation. Combined with [7, Theorem 5.5], this gives
another proof of Theorem 4.2 above, our generalization of Whyburn’s Theorem.

For a $C^{*}$-algebra $\operatorname{\mathcal{A}}$ we denote by
$\operatorname{\mathcal{A}}_{1}^{+}$ the positive norm-one elements of
$\operatorname{\mathcal{A}}$, and by
$\operatorname{\mathbb{H}}(\operatorname{\mathcal{A}})$ the non-zero
hereditary subalgebras of $\operatorname{\mathcal{A}}$.

######  Definition 6.1 (aperiodic bimodules and $C^{*}$-inclusions).

Let $\operatorname{\mathcal{A}}$ be a unital $C^{*}$-algebra. Following [6]:

  * •

We say that a normed $\operatorname{\mathcal{A}}$-bimodule
$\operatorname{\mathcal{X}}$ is aperiodic if for all
$x\in\operatorname{\mathcal{X}}$, $\varepsilon>0$, and
$\operatorname{\mathcal{D}}\in\operatorname{\mathbb{H}}(\operatorname{\mathcal{A}})$,
there exists $d\in\operatorname{\mathcal{D}}_{1}^{+}$ such that

$\|dxd\|<\varepsilon.$

  * •

We say that an inclusion
$\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$ of unital
$C^{*}$-algebras is aperiodic if
$\operatorname{\mathcal{B}}/\operatorname{\mathcal{A}}$ is aperiodic as a
normed $\operatorname{\mathcal{A}}$-module. Equivalently, if for all
$b\in\operatorname{\mathcal{B}}$, $\varepsilon>0$, and
$\operatorname{\mathcal{D}}\in\operatorname{\mathbb{H}}(\operatorname{\mathcal{A}})$,
there exists $d\in\operatorname{\mathcal{D}}_{1}^{+}$ and
$a\in\operatorname{\mathcal{A}}$ such that

$\|dbd-a\|<\varepsilon.$

######  Theorem 6.2.

For an abelian $C^{*}$-inclusion
$\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$, aperiodicity
is equivalent to having a unique pseudo-expectation.

###### Proof.

Let $\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$ be an
abelian $C^{*}$-inclusion with unique pseudo-expectation
$E:\operatorname{\mathcal{B}}\to I(\operatorname{\mathcal{A}})$. Then $E$ is a
$*$-homomorphism and $\operatorname{\mathcal{J}}=\ker(E)$ is the unique
maximal $\operatorname{\mathcal{A}}$-disjoint ideal of
$\operatorname{\mathcal{B}}$ [10, Corollary 3.21]. To show that
$\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$ is aperiodic,
we must show that $\operatorname{\mathcal{B}}/\operatorname{\mathcal{A}}$ is
aperiodic as a normed $\operatorname{\mathcal{A}}$-module. Since the canonical
maps

$(\operatorname{\mathcal{J}}+\operatorname{\mathcal{A}})/\operatorname{\mathcal{A}}\to\operatorname{\mathcal{B}}/\operatorname{\mathcal{A}}\to\operatorname{\mathcal{B}}/(\operatorname{\mathcal{J}}+\operatorname{\mathcal{A}})$

form an exact sequence of normed $\operatorname{\mathcal{A}}$-modules, it
suffices to show that
$(\operatorname{\mathcal{J}}+\operatorname{\mathcal{A}})/\operatorname{\mathcal{A}}\cong\operatorname{\mathcal{J}}$
and
$\operatorname{\mathcal{B}}/(\operatorname{\mathcal{J}}+\operatorname{\mathcal{A}})\cong(\operatorname{\mathcal{B}}/\operatorname{\mathcal{J}})/\operatorname{\mathcal{A}}$
are aperiodic as normed $\operatorname{\mathcal{A}}$-modules [6, Lemma 5.12].

The aperiodicity of the normed $\operatorname{\mathcal{A}}$-module
$(\operatorname{\mathcal{B}}/\operatorname{\mathcal{J}})/\operatorname{\mathcal{A}}$
is equivalent to the aperiodicity of the $C^{*}$-inclusion
$\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}/\operatorname{\mathcal{J}}$.
That, in turn, follows from the aperiodicity of the $C^{*}$-inclusion
$\operatorname{\mathcal{A}}\subseteq I(\operatorname{\mathcal{A}})$ [7, Remark
3.17] and the fact that the induced map
$\dot{E}:\operatorname{\mathcal{B}}/\operatorname{\mathcal{J}}\to
I(\operatorname{\mathcal{A}})$ is a $*$-monomorphism.

To show that $\operatorname{\mathcal{J}}$ is aperiodic as a normed
$\operatorname{\mathcal{A}}$-module, we identify the $C^{*}$-inclusion
$\operatorname{\mathcal{A}}\subseteq\operatorname{\mathcal{B}}$ with the
$C^{*}$-inclusion $C(X)\subseteq C(Y)$ arising from the continuous surjection
$j:Y\to X$. Then

$\operatorname{\mathcal{J}}=\\{g\in C(Y):g|_{K}=0\\},$

where $K\subseteq Y$ is the unique minimal closed set such that $j(K)=X$. Let
$g\in\operatorname{\mathcal{J}}$, $\varepsilon>0$, and
$\operatorname{\mathcal{D}}\in\operatorname{\mathbb{H}}(\operatorname{\mathcal{A}})$.
Then

$\operatorname{\mathcal{D}}=\\{f\in C(X):f|_{F}=0\\},$

where $F\subsetneq X$ is a closed set. Define $U=\\{y\in
Y:|g(y)|<\varepsilon\\}$, an open subset of $Y$ containing $K$. Assume that
$j(U^{c})\cup F=X$. Then $K^{\prime}=U^{c}\cup j^{-1}(F)$ is a closed subset
of $Y$ such that $j(K^{\prime})=X$. It follows that $K\subseteq K^{\prime}$,
which implies $K\subseteq j^{-1}(F)$, which in turn implies $j(K)\subseteq F$,
a contradiction. So $j(U^{c})\cup F\subsetneq X$. By Urysohn’s Lemma, there
exists $f\in C(X)_{1}^{+}$ such that $f|_{j(U^{c})\cup F}=0$. Then
$f\in\operatorname{\mathcal{D}}_{1}^{+}$ and $\|(f\circ j)g(f\circ
j)\|<\varepsilon$, which completes the proof. ∎

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