# Convolutional Spatial Attention Model for Reading Comprehension with
Multiple-Choice Questions

Zhipeng Chen†, Yiming Cui†‡, Wentao Ma†, Shijin Wang†, Guoping Hu†
†Joint Laboratory of HIT and iFLYTEK (HFL), iFLYTEK Research, Beijing, China
‡Research Center for Social Computing and Information Retrieval (SCIR),
Harbin Institute of Technology, Harbin, China
<EMAIL_ADDRESS>
Corresponding author

###### Abstract

Machine Reading Comprehension (MRC) with multiple- choice questions requires
the machine to read given passage and select the correct answer among several
candidates. In this paper, we propose a novel approach called Convolutional
Spatial Attention (CSA) model which can better handle the MRC with multiple-
choice questions. The proposed model could fully extract the mutual
information among the passage, question, and the candidates, to form the
enriched representations. Furthermore, to merge various attention results, we
propose to use convolutional operation to dynamically summarize the attention
values within the different size of regions. Experimental results show that
the proposed model could give substantial improvements over various state-of-
the-art systems on both RACE and SemEval-2018 Task11 datasets.

## Introduction

Owing to the rapid release of various large-scale datasets, Machine Reading
Comprehension (MRC) has become enormously popular in Natural Language
Processing. For example, cloze-style MRC (such as CNN/DailyMail (?),
Children’s Book Test (CBT) (?)), span-extraction MRC (such as SQuAD (?)), and
multiple-choice MRC (such as MCTest (?), RACE (?), SemEval-2018 Task11 (?)).

In this paper, we mainly focus on solving the reading comprehension with
multiple-choice questions. At the beginning of the reading comprehension
study, this type of reading comprehension task was not that popular because
there is no large-scale dataset available and thus we cannot apply neural
network approaches to solve them. To bring more challenges to reading
comprehension task and mitigate the absence of large-scale multi-choice
reading comprehension dataset, ?(?) propose a new dataset called RACE.
Compared to the earlier MCTest (?), the RACE dataset is made from the English
examinations for Chinese middle and high school students, consisting near
100,000 questions generated by human experts, and is far more challenging than
the MCTest.

In this paper, we propose a novel model called Convolutional Spatial Attention
(CSA) to fully utilize the hierarchical attention information for reading
comprehension with multiple-choice questions. The proposed model first encode
the passage, question, and candidates into word representations which are
enhanced by the additional POS-tag and matching features. Then we concentrate
on enriching the representation of the candidates by incorporating the
passage, question information, and further calculate the attentions between
the passage, question, and candidates, forming the spatial attentions. To
further extract the representative features in the spatial attentions, we
propose to use convolutional neural network to dynamically conclude adjacent
regions with different window size. We mainly test our CSA model on two
multiple-choice reading comprehension datasets: RACE and SemEval-2018 Task11,
and our model achieves state-of-the-art performances on both of them. The
examples of each dataset are given in Figure 1. The main contributions of our
paper can be summarized as follows.

  * •

We focus on modeling different semantic aspects of the candidates, by
integrating the passage and question information, forming the 3D spatial
attention among the passage, question, and candidates.

  * •

We propose a Convolutional Spatial Attention (CSA) mechanism to dynamically
extract representative features from the spatial attentions.

  * •

The proposed model gives substantial improvements over various state-of-the-
art systems on both RACE and SemEval-2018 Task11 datasets, showing its
generalization and extensibility to other NLP tasks.

RACE |  SemEval-2018 Task11
---|---
Passage |  Passage
Is it important to have breakfast every day? A short time ago, a test was given in the United States. People of different ages, from 12 to 83, were asked to have a test. During the test, these people were given all kinds of breakfast, and sometimes they got no breakfast at all. … |  I was thirsty so I decided to make a cup of tea. I looked through my box of teas and rifled through the assorted flavors. I settled on Earl Gray, which is a black tea flavored with bergamot orange. I filled the kettle with water and placed it on the stove, turning on the burner so that it would heat up and begin boiling. …
Question |  Question
What do the results show? |  Why did they use a kettle?
Candidates |  Candidates
A They show that breakfast has affected on work and study. |  A to drink from
B Breakfast has little to do with a person’s work. |  B to boil water
C A person will work better if he only has fruit and milk. |
D They show that girl students should have less for breakfast. |
Figure 1: Example of RACE and SemEval-2018 Task11 dataset. The correct answer
is depicted in bold face.

## Related Works

Massive progress has been made on machine reading comprehension field in
recent years. The booming of the MRC can trace back to the release of the
large-scale datasets, such as CNN/DailyMail (?) and CBT (?)). After the
release of these datasets, various neural network approaches (?; ?; ?; ?) have
been proposed and become fundamental components in the future studies. Another
representative dataset is SQuAD (?), which was difficult than the cloze-style
reading comprehension and requires the machine to generate a span in the
passage to answer the questions. With rapid progress on designing effective
neural network models (?; ?; ?; ?), recent works on this datasets have
surpassed the average human performance, such as QANet (?) etc.

However, current machine reading comprehension models are still struggling
with solving the questions that need reasoning over multiple sentences or even
passage. To solve the reading comprehension with multiple-choice questions,
various approaches have been proposed, and most of them are focusing on
designing effective attentions or persuing enriched representations for
prediction. When releasing the RACE dataset, ?(?) also adopted and modified
two models of the cloze-style reading comprehension: Gated Attention Reader
(?) and Stanford Attentive Reader (?). However, experimental results show that
these models are not capable of this task. ?(?) introduced ElimiNet which use
a combination of elimination and selection to get refined representation of
the candidates. ?(?) proposed the Dynamic Fusion Networks (DFN), which uses
multi-hop reasoning mechanism for this task. ?(?) proposed the Hierarchical
Attention Flow model, which leverage candidate options to model the
interactions among passage, questions, and candidates.

Though various efforts have been made, we believe there is still a large room
for designing effective neural networks for better characterizing the
relations between the passage, questions, and candidates. To this end, we
summarize the main differences between our model and existing models for this
task into three aspects. First, we calculate various representations of the
candidate to better characterize it for prediction. Second, when calculating
attention, we apply additional trainable weights to dynamically adjust the
attention values, which is more flexible. Third, unlike the previous works
only utilize the final-level hierarchical attentions, we propose to use every
attention in each hierarchy, and use convolutional neural network to capture
features for predicting the answer.

## Convolutional Spatial Attention Model

### Task Definition

The RACE dataset (?) for reading comprehension with multiple-choice questions
was proposed, which consists of 28,000+ passages and near 100,000 questions
generated by human experts for the English examinations of Chinese middle and
high school students. Different from the earlier MCTest dataset (?), the RACE
dataset is significantly larger, and thus we can apply deep learning
approaches for this task. As all the questions and choices are generated by
human experts, RACE dataset provides more comprehensive and realistic
evaluation on machine reading comprehension than the other popular datasets
such as CNN/DailyMail, SQuAD datasets, whose answer should appear in context.
Also according to the analysis by ?(?), a large portion of the questions in
RACE need reasoning over various clues, which makes it more challenging and
suitable to evaluate the ability of the reading comprehension systems.
SemEval-2018 Task11 is closely the same with the RACE dataset but with two
candidates and small size. The examples of each dataset are given in Figure 1.

### The Model

In this section, we will give a detailed description on the proposed model.
The main neural architecture of our model is depicted in Figure 2. Throughout
this section, we will use $P$ for representing the passage, $Q$ for the
question, $C$ for the candidates. Note that, as the operations on each
candidate are the same, for simplicity, we only take one of the candidates for
illustration.

Figure 2: Main neural architecture of the Convolutional Spatial Attention
(CSA) model.

### Word Representation

We transform each word in the passage, question, and candidates into
continuous representations. In this paper, there are three components in the
embedding layer, which can be listed as follows.

  * •

Word Embedding $E_{word}$: We use traditional pre-trained GloVe embedding for
initialization (?) and keep fixed during the training process.

  * •

ELMo Embedding $E_{elmo}$: For this part, we use pre-trained ELMo (?)
embedding.

  * •

Feature Embedding $E_{feat}$: We also utilize three additional features to
enhance the word representations.

    1. 1.

POS-tag Embedding $E_{pos}$: We use NLTK (?) for part-of-speech tagging for
each word. Similar to traditional word embeddings, we assign different
trainable vectors for each part-of-speech tag.

    2. 2.

Word Matching $F_{match}$: Take the text as an example, if the word in text
also appears in question or candidate, we set the value as one, otherwise set
it as zero. In this way, we can also add this feature to the question and
candidate.

    3. 3.

Fuzzy Word Matching $F_{fuzzy}$: Similar to the word matching feature, but we
loosen the matching criteria as partial matching. For example, we regard
‘teacher’ and ‘teach’ as fuzzy matching, because the string ‘teacher’ is
partially matched by ‘teach’.

We concatenate three embedding components to form the final word
representations for the text $E_{P}\in{\mathbb{R}}^{|P|\times e}$, question
$E_{Q}\in{\mathbb{R}}^{|Q|\times e}$, and candidates
$E_{C}\in{\mathbb{R}}^{|C|\times e}$, where $|P|,|Q|,|C|$ are the length of
the passage, question, and candidates, $e$ is the final embedding size
(including all three components).

$\displaystyle E=[E_{word};E_{elmo};E_{feat}]$ (1) $\displaystyle
E_{feat}=[E_{pos};F_{match};F_{fuzzy}]$ (2)

After obtaining embedding representations, we further feed each word embedding
into a shared highway network (?) with $\tanh$ output activation (denoted as
$\sigma$). In this paper, we apply two consecutive highway networks with
shared weights. Then we use Bi-Directional LSTM (?) to model the contextual
information, forming $H_{P}\in{\mathbb{R}}^{|P|\times h}$,
$H_{Q}\in{\mathbb{R}}^{|Q|\times h}$, and $H_{C}\in{\mathbb{R}}^{|C|\times h}$
($h$ is hidden size of Bi-LSTM). Note that, we use different Bi-LSTMs for the
passage, question, and candidates.

$\displaystyle\widetilde{H}=\sigma(\text{2-Highway}(E))$ (3) $\displaystyle
H=\text{Bi-LSTM}(\widetilde{H})$ (4)

### Enriched Representation

Calculating attention and generating enriched representation play very
important roles in machine reading comprehension. In our model, we will
calculate various types of enriched representations for better characterizing
the candidate and question, which are the essential components in this task.
The procedure for generating enriched representation is illustrated in
Algorithm 1.

For example, we wish to embed the passage information into the candidate
representation to better aware the relevant part in the passage and obtain the
passage-aware candidate representation $R_{CP}$. According to Algorithm 1,
$R_{CP}$ can be generated as follows.

  * •

[Line 1] Given the Bi-LSTM representations of passage $H_{P}$ and candidate
$H_{C}$, we first calculate the attention matrix where each element indicate
the matching information between them. In this paper, we adopt the attention
mechanism used in FusionNet (?). where two representations are transformed by
individual fully-connected layer with an output activation $f$. Also, a
trainable diagonal weight matrix $D$ is applied. The activation function $f$
is defined as RELU throughout this paper.

  * •

[Line 2] Then we apply an element-wise weight matrix $W$ to the attention
matrix $M^{\prime}$. This is designed to let the model flexibly adjust the
attention values.

  * •

[Line 3] We apply softmax to the weighted attention matrix $M$ to the last
dimension of it, which calculates the passage-level attention vector w.r.t.
each candidate word.

  * •

[Line 4] After obtaining the normalized attention matrix $M_{att}$, we make a
dot product between the $M_{att}$ and the passage $H_{P}$ to extract
candidate-related passage.

  * •

[Line 5] Finally, we concatenate candidate-related passage $Y_{CP}^{\prime}$
and Bi-LSTM candidate representation $H_{C}$, and feed them to a Bi-LSTM to
fully integrate passage information into the candidate representations.

Algorithm 1 Enriched Representation.

0: Time-Distributed representation $X_{1}$ Time-Distributed representation
$X_{2}$

0: Random weight matrix $W_{1}\in{\mathbb{R}}^{h\times h_{att}}$ Random weight
matrix $W_{2}\in{\mathbb{R}}^{h\times h_{att}}$ Diagonal weight matrix
$D\in{\mathbb{R}}^{h_{att}\times h_{att}}$ All-one weight matrix
$W\in{\mathbb{R}}^{|X_{1}|\times|X_{2}|}$

0: $X_{2}$-aware $X_{1}$ representation $Y$

1: Calculate attention matrix
$M^{\prime}\in\mathbb{R}^{|X_{1}|\times|X_{2}|}$:
$M^{\prime}=f(W_{1}X^{1})^{T}\cdot D\cdot f(W_{2}X^{2})$

2: Apply element-wise weight: $M=M^{\prime}\odot W$

3: Apply softmax function to the last dimension of $M$: $M_{att}=softmax(M)$

4: Calculate raw representation $Y^{\prime}\in{\mathbb{R}}^{|X_{2}|\times h}$:
$Y^{\prime}={M_{att}}^{T}\cdot X_{1}$

5: Concatenate raw representation $Y^{\prime}$ and raw input $X_{1}$, then
apply Bi-LSTM: $Y=\text{Bi-LSTM}([X_{1};Y^{\prime}])$

6: return $Y$

By applying the proposed algorithm $g(X_{1},X_{2})$, we can calculate the
passage-aware and question-aware candidate representation $R_{CP}$ and
$R_{CQ}$. Besides, as the question information is also important, we also
calculate passage-aware question representation $R_{QP}$ and self-attended
question representation $R_{self\text{-}Q}$. Note that, we use Bi-LSTM output
$B_{Q}$ and passage-aware question representation $R_{QP}$ to obtain the
(almost-)self-attended question representation, which combines different
levels of representations.

$\displaystyle R_{CQ}$ $\displaystyle=g(B_{C},B_{Q})$ (5) $\displaystyle
R_{CP}$ $\displaystyle=g(B_{C},B_{P})$ (6) $\displaystyle R_{QP}$
$\displaystyle=g(B_{Q},B_{P})$ (7) $\displaystyle R_{self\text{-}Q}$
$\displaystyle=g(B_{Q},R_{QP})$ (8)

### Convolutional Spatial Attention

With previously generated representations, we can calculate the matching
matrix to measure the similarity between them. In this paper, we adopt simple
dot product to obtain the matching matrix. As the candidate information is
important to answer the question, firstly we calculate the matching matrix
using various candidate representations to the self-attended question
representation $R_{self\text{-}Q}$. The motivation is to use the question
information as the key to extracting candidate information in different
levels. We use question-aware candidate representation $R_{CQ}$, passage-aware
candidate representation $R_{CP}$, and candidate Bi-LSTM representation
$H_{C}$, as shown below.

$\displaystyle M_{11}$ $\displaystyle=R_{CQ}\cdot R_{self\text{-}Q}$ (9)
$\displaystyle M_{12}$ $\displaystyle=R_{CP}\cdot R_{self\text{-}Q}$ (10)
$\displaystyle M_{13}$ $\displaystyle=H_{C}\cdot R_{self\text{-}Q}$ (11)

In a similar way, we can also replace the self-attended question
representation $R_{self\text{-}Q}$ with passage-aware question representation
$R_{QP}$ in Equation 9, 10, 11, to obtain $M_{21},M_{22},M_{23}$. Then we
concatenate all matrices on the channel dimension to form a Spatial Attention
Cube $M\in\mathbb{R}^{6\times|C|\times|Q|}$, which is similar to an ‘image’
with 6 channels.

$\displaystyle M=[M_{11};M_{12};M_{13};M_{21};M_{22};M_{23}]$ (12)

In order to extract high-level features inside the spatial attention cube, we
use CNN-MaxPooling operation to dynamically conclude adjacent attention
information, which is similar to the traditional operation on the image.
Formally, we first apply convolutional operation on the $M$ to summarize
different length of adjacent elements We adopt three convolutional kernels: 5,
10, and 15, along with the dimension of the question length.

After convolution operation, we could get three features maps w.r.t. different
convolutional kernels. The procedure can be illustrated as the following
equations.

$\displaystyle O_{1}$ $\displaystyle=\text{Max-Pooling}_{1\times
3}\\{CNN_{1\times 5}(M)\\}$ (13) $\displaystyle O_{2}$
$\displaystyle=\text{Max-Pooling}_{1\times 2}\\{CNN_{1\times 10}(M)\\}$ (14)
$\displaystyle O_{3}$ $\displaystyle=\text{Max-Pooling}_{1\times
1}\\{CNN_{1\times 15}(M)\\}$ (15)

In this way, we used CNN-MaxPooling operation to obtain three feature vectors
$O_{1},O_{2},O_{3}$ by using different convolutional kernels and max-pooling
intervals.

### Final Prediction

After obtaining three feature vectors, we flatten, concatenate, and feed them
into a fully-connected layer to get a scalar value denoting the possibility of
being the correct answer. Recall that, we have several candidates for a given
question, so we will get $N$ candidate scores in this stage. We apply softmax
function to these scores to obtain the final probability distributions over
the candidates.

$\displaystyle s_{i}={\bf w^{T}}\cdot[O_{1};O_{2};O_{3}]$ (16) $\displaystyle
Pr(A|P,Q,C)=softmax([s_{1};...;s_{N}])$ (17)

To train our model, we use traditional cross entropy loss to minimize the gap
between the prediction and the ground truth.

Model | RACE-M | RACE-H | RACE
---|---|---|---
Sliding Window (?) | 37.3 | 30.4 | 32.2
Stanford AR (?) | 44.2 | 43.0 | 43.3
GA Reader (?) | 43.7 | 44.2 | 44.1
ElimiNet (?) | N/A | N/A | 44.5
Hierarchical Attention Flow (?) | 45.0 | 46.4 | 46.0
Dynamic Fusion Network (?) | 51.5 | 45.7 | 47.4
CSA Model (single model) | 51.0 | 47.3 | 48.4
CSA Model + ELMo (single model) | 52.2 | 50.3 | 50.9
GA Reader (6-ensemble) | - | - | 45.9
ElimiNet (6-ensemble) | - | - | 46.5
GA + ElimiNet (12-ensemble) | - | - | 47.2
Dynamic Fusion Network (9-ensemble) | 55.6 | 49.4 | 51.2
CSA Model (7-ensemble) | 55.2 | 52.4 | 53.2
CSA Model + ELMo (9-ensemble) | 56.8 | 54.8 | 55.0
Table 1: Experimental results on RACE. The best previous results are in
italics, and overall best results are in bold face.

## Experiments

### Experimental Setups

To evaluate our system, we carried out experiments on the following two public
datasets.

  * •

RACE: English examinations for Chinese middle and high school students. The
questions are generated by human experts, which has four candidates for each
question. The test set consists of 4,934 instances, where RACE-M (middle
school) has 1,436 instances, and RACE-H (high school) for 3,498 instances.

  * •

SemEval-2018 Task11: The dataset provided by the SemEval-2018 Task11
organizer, which mainly focus on solving commonsense reading comprehension.
Each question has two candidates to choose from.

The data are tokenized and lower-cased by using Natural Language Toolkit
(NLTK) (?), and all punctuations are removed. The main hyper-parameters of our
model are listed in Table 2. Note that, except for the candidate numbers, all
hyper-parameters are identical among two datasets. The word embeddings are
initialized by the pre-trained GloVe word vectors (Common Crawl, 6B tokens,
100-dimension) (?), and keep fixed during training. The words that do not
appear in the pre-trained word vectors are set to the unk token and
initialized accordingly. We use Adam (?) for weight optimizations with an
initial learning rate of 0.001. In order to prevent overfitting, we apply
dropout of 0.35 to all the representation layers. The models are built on
Keras platform (?) with Tensorflow backend (?).

Symbol | Descriptions | Size
---|---|---
$|P|$ | Passage max length | 300
$|Q|$ | Question max length | 20
$|C|$ | Candidate max length | 10
$e$ | Word embedding | 200
$h$ | Bi-LSTM hidden size | 250
$h_{att}$ | Attention hidden size | 80
$es$ | ELMo embedding size | 1024
$p$ | POS-tag embedding | 16
Table 2: Hyper-parameter settings.

### Overall Results

RACE. The experimental results are shown in Table 1. As we can see that, our
CSA model shows significant improvements over various state-of-the-art systems
by a large margin. We also compared our model to the recent unpublished work
DFN, while our model gives an absolute gain of 1.0% in the overall test set,
and further gains can be obtained by incorporating ELMo, demonstrating the
effectiveness of our proposed model. Also, when compared to the Hierarchical
Attention Flow model (?), our CSA model show substantial improvements,
indicating that utilizing the attentions in each hierarchy and using
convolutional neural network to extract the most representative features from
the spatial attentions are useful in this task.

When it comes to the ensemble results, though we only use 7 models in ensemble
with majority voting approach, the overall results show an absolute gain of 2%
over the previous state-of-the-art result by DFN. Also we observed that our
model shows slightly worse result in RACE-M but significantly better in RACE-H
regardless of single model or ensemble. These results indicate that our model
is more capable of solving difficult question (high school). While on the
contrary, it may hurt the performance on the relatively easier question
(middle school). We will give a detailed analysis for illustrating this
phenomenon in the next section.

SemEval-2018 Task11. The distributions of question type is quite different
between the RACE and SemEval-2018 Task11 datasets. To test if our model could
generalize to other reading comprehension dataset, we also carried out
experiments on the very recent SemEval-2018 Task11 dataset. The results are
shown in Table 3. As we can see that, our CSA model could give moderate
improvements over the top-ranked SemEval systems in both single model and
ensemble, and set up new state-of-the-art performance on this task,
demonstrating the proposed model is powerful and showing its potential
possibility to generalize to other NLP tasks. Note that, we directly train the
model using the hyper-parameter settings of the RACE experiments without
finding other hyper-parameter combinations, indicating further improvements
may be obtained through fine-tuning.

Model | Dev | Test
---|---|---
HMA (?) | 84.48 | 80.94
TriAN (?) | 83.84 | 81.94
CSA Model (single model) | 83.63 | 82.20
CSA Model + ELMo (single model) | 83.84 | 83.27
TriAN (ensemble) | 85.27 | 83.95
HMA (ensemble) | 86.46 | 84.13
CSA Model (ensemble) | 84.05 | 84.34
CSA Model + ELMo (ensemble) | 85.05 | 85.23
Table 3:  Experimental results on SemEval-2018 Task11. The two top-ranked
systems are listed as baselines.

### Ablation Study

We also carried out model ablations to further demonstrate the effectiveness
of the proposed approaches. The results are shown in Table 4.

Model | RACE
---|---
CSA Model | 48.52
w/o attention weight | 48.18
w/o enriched representation | 47.52
w/o convolutional spatial attention | 47.30
CSA Model + ELMo | 50.89
w/o attention weight | 49.49
w/o enriched representation | 49.78
w/o convolutional spatial attention | 48.47
Table 4: Ablations on several model components.

As shown in the model description, we adopt the attention mechanism used in
FusionNet (?), and applied an element-wise weight to the attention matrix. By
applying additional trainable weights to the attention matrix could give
moderate improvements suggesting that these weights are useful in dynamically
adjusting the attention values. When we remove all the enriched candidate
representations, there is an absolute drop of 1.0%. This suggests that it is
necessary to incorporate various information (such as the passage and
question) into the candidate representation. We also removed our convolutional
spatial attention mechanism in the model. The results show a significant drop
in performance by 1.22%, indicating that the proposed convolutional spatial
attention is effective in extracting most representative values among the
various attention matrices. The same results drop is shown by incorporating
ELMo. Further more, whether we use ELMo or not, our convolutional spatial
attention has improved significantly. That also proved that our model can get
father improvement after some big data trained toolkit like ELMo or some other
transformer.

The details of the ablations are shown below.

  * •

w/o attention weight: We change $M=M^{\prime}\odot W$( in Alg 1 line 2) to
$M=M^{\prime}$, where $W$ is the attention weight.

  * •

w/o enriched representation: We remove interaction of candidate-to-question
($R_{CQ}$ in Figure 2 ) and candidate-to-passage ($R_{CP}$ in Figure 2) and
only use candidate’s LSTM output $R_{C}$.

  * •

w/o convolutional spatial attention: We use two fully-connected layer to
transform the matching matrix to a score. The first fully-connected layer
squeeze the matching matrix to a vector along the question length dimension.
The second fully-connected layer squeeze the vector to a scalar score. As
every candidate has six matching matrices, so we can get six scores. Finally,
we add the six scores as the final prediction.

## Analysis and Discussion

### Quantitative Analysis

In CSA model, we adopt various enriched representation for better
characterizing candidate information. We compare the performance between the
original CSA model and discard enriched candidate representation, as shown in
Figure 3.

Figure 3: Quantitative analysis on different type of questions in RACE
development and test set.

As we can see that our model yields significant improvements on the question
type ‘how’ and ‘why’, which are relatively difficult than the other type of
questions requiring high-level reasoning within the context, demonstrating
that our CSA model performs better on the relatively sophisticated questions.
On the contrary, we find that our CSA model shows relatively inferior
performance on the question type ‘who’, ‘where’, and ‘when’, which are often
answered by a single word or entity. This phenomenon suggests that further
efforts should be made on balancing the word-level attention and highly
abstracted attention. These results also explain why our CSA model shows
significant improvements on RACE-H subset while giving a relatively inferior
performance on RACE-M subset. As the sample number of RACE-H subset is three
times the size of RACE-M, the overall performance of our model still shows
significant improvements over various state-of-the-art systems.

### Case Study

We also randomly sampled one hundred question from RACE test set, and
classified them into three categories according to its difficulty.

  * •

The first level is the question that can be answered by matching a few words
in the passage. If the model finds the right place in the text, the answer is
easy to find. With the attention mechanism, the neural model is especially
good at solving this kind of question.

  * •

The second level is the question that can be answered by using a few sentences
without reasoning, which is relatively difficult than the first level.

  * •

The third level is the question that needed to comprehensive reasoning via
multiple clues in the passage. This kind of question is more complicated. To
answer this question, the intention of the questioner must be captured.

The results are shown in Table 5. As we can see that the current model shows
relatively good performance on the first and second level questions than the
third one, indicating that more investigation should be made on solving those
questions that need reasoning.

State | Total # | Right # | Accuracy
---|---|---|---
All | 100 | 56 | 56.0%
First level | 30 | 18 | 60%
Second level | 30 | 20 | 66.7%
Third level | 40 | 18 | 45.0%
Table 5:  Case analysis on RACE dataset.

### Error Analysis

In this part, a interesting example is shown in Figure 4. Our model choose the
right answer “C”. How can our model choose the right answer? Because the
candidate “C” has a bigest semantic overlap with passage. To improve that, we
change candidate “C” to “ellipsis in talking”, but the model can still choose
the right answer. Even we change candidate “C” to “ellipses in talking”. From
this experiment, we can see that our model can really handle part of high
level semantic matching. Further more, we change the question to “The
conversation is mainly about _ .”, the model still choose the candidate “C”
with a high probability. This reflects the problem that the deeplearning model
is not sensitive to some special important information in passage and
question. That phenomenon more or less approve the weakness of neural model.
So we need to do more in-depth research to design some special component like
CNN in dealing with picture to deal with text in reading comprehension.

RACE Dataset
---
Passage
As is known to all, in daily conversation people often use simple words and
simple sentences, especially elliptical sentences. Here is an interesting
conversation between Mr Green and his good friend Mr Smith, a fisherman. Do
you know what they are talking about?
Question
The text is mainly about _ .
Candidates
A how to catch fish. B how to spend a Sunday
C ellipsis in conversations D joy in fishing
Figure 4: Example of RACE dataset. The correct answer is depicted in bold
face.

## Conclusion

We propose the Convolutional Spatial Attention (CSA) model to tackle the
machine reading comprehension with multiple-choice questions. The proposed
model could fully extract the mutual information among the passage, question,
and candidates, to form the enriched representations using the modified
attention mechanism that has trainable weights. To summarize attention
matrices from various sources, we propose to use convolutional operation to
dynamically summarize the attention values within the different size of
regions, which is beneficial to capture diverse features. Experimental results
show that the proposed CSA model could give substantial improvements over
various state-of-the-art systems on both RACE and SemEval 2018 Task11
datasets. Also, the ablation studies verify the effectiveness of several
proposed components in our model, and analysis also shows that the CSA model
is superior in solving relatively sophisticated questions (such as ‘why’ or
‘how’ questions).

## Acknowledgments

We would like to thank all three anonymous reviewers for their constructive
comments to improve our paper. This work was supported by the National Key R&D
Program of China via grant No. 2016YFC0800806.

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$\\{s,s_{1},\ldots,s_{\ell-1},s^{\prime}\\}$ when $\ell>1$ and to
$\\{s,s^{\prime}\\}$ when $\ell=1$, cf. Figure 9.

In all these cases any irreducible component of ${\mathcal{E}}_{\mathcal{F}}$
meeting ${\mathcal{D}}$ is either contained in ${\mathcal{D}}$ or dicritical.

  3. (3)

There is an irreducible component of ${\mathcal{D}}$ containing at least 3
singular points of ${\mathcal{F}}^{\sharp}$.

$\ell>1$
$\cdots$$s$$s_{1}$$s_{\ell-1}$$D_{1}$${\mathcal{C}}$$D_{2}$$D_{\ell}$$\bullet$$\cdots$$\bullet$$\bullet$$\cdots$
AAA or AAA $\ell=1$ $s$$D_{1}$${\mathcal{C}}$$\bullet$

Figure 7. Situation (2a): there is no nodal singular point on ${\mathcal{D}}$
and some dicritical component must meet ${\mathcal{D}}$.

$\ell>1$
$s$$s_{1}$$\cdots$${\mathcal{C}}$$D_{1}$$D_{2}$$D_{\ell}$$s_{\ell-1}$$s_{\ell}$$\bullet$$\cdots$$\bullet$$\bullet$$\cdots$$\bullet$$\bullet$
AAAorAAA $\ell=1$ $s$${\mathcal{C}}$$D_{1}$$s_{\ell}$$\bullet$$\bullet$

Figure 8. Situation (2b): $s_{\ell}$ is the unique nodal singular point on
${\mathcal{D}}$ and some dicritical components may meet ${\mathcal{D}}$.

$\ell>1$
$s$$s_{1}$$\cdots$${\mathcal{C}}$$D_{1}$$D_{2}$$D_{\ell}$${\mathcal{C}}^{\prime}$$s_{\ell-1}$$s^{\prime}$$\bullet$$\cdots$$\bullet$$\bullet$$\cdots$$\bullet$$\bullet$
A or A $\ell=1$
$s$$s^{\prime}$${\mathcal{C}}$$D_{1}$${\mathcal{C}}^{\prime}$$\bullet$$\bullet$

Figure 9. Situation (2c): there is no nodal singular point on ${\mathcal{D}}$
and some dicritical component must meet ${\mathcal{D}}$.

We say that the dynamical componant ${\mathcal{D}}$ is small in cases (1) and
(2) and that it is big in case (3). We will now go through each case.

###  5.3. Small dynamical components

We denote again by ${\mathcal{C}}$ the strict transform of an isolated
separatrix of ${\mathcal{F}}$ and by $\phi:{\mathcal{F}}\to\mathcal{G}$ a
topological conjugacy between ${\mathcal{F}}$ and $\mathcal{G}$.

In the case (1) the equality

${{\mathrm{CS}}}({\mathcal{F}}^{\sharp},{\mathcal{C}},s)={{\mathrm{CS}}}(\mathcal{G}^{\sharp},{\mathsf{A}}_{\phi}({\mathcal{C}}),{\mathsf{A}}_{\phi}(s))$

is proven by R. Rosas under weak hypothesis [38, Proposition 13], see also
[22, Theorem 1.12].

Index Formula (23) implies that:

  * •

in case (2a) the Camacho-Sad index
${{\mathrm{CS}}}({\mathcal{F}}^{\sharp},{\mathcal{C}},s)$ is a rational number
that can be expressed as a continued fraction in terms of the self-
intersections $D_{i}\cdot D_{i}$ of the irreducible components of
${\mathcal{D}}\cap{\mathcal{E}}_{\mathcal{F}}$; as these ones are invariant by
topological conjugacies, cf. [46] or [20], the equality of Camacho-Sad indices
(22) is satisfied;

  * •

in case (2b) we have
${{\mathrm{CS}}}({\mathcal{F}}^{\sharp},D_{\ell},s_{\ell})\in{\mathbb{R}}\setminus{\mathbb{Q}}$
and consequently
${{\mathrm{CS}}}({\mathcal{F}}^{\sharp},{\mathcal{C}},s)\in{\mathbb{R}}\setminus{\mathbb{Q}}$;
in fact $s$ must be a linearizable saddle of ${\mathcal{F}}^{\sharp}$ (recall
that $s$ is not a node). Then the proof of [22, Theorem 1.12] gives the
equality (22).

In case (2c) any reduced singularity may appear at the point $s$ and for this
reason we will say that ${\mathcal{D}}$ is a free (dynamical) component. All
the free components of Poincaré type, i.e. with
${{\mathrm{CS}}}({\mathcal{F}}^{\sharp},{\mathcal{C}},s)\in{\mathbb{C}}\setminus{\mathbb{R}}$,
are topologically conjugated. Indeed, this claim uses the following fact:

######  Lemma 5.1.

Let $(\lambda_{1},\lambda_{2})$ and $(\mu_{1},\mu_{2})$ be two pairs of
${\mathbb{R}}$-independent complex numbers and $g:{\mathbb{C}}\to{\mathbb{C}}$
be a ${\mathbb{R}}$-linear map such that $g(i\lambda_{j}):=i\mu_{j}$. Then the
maps $\psi_{j}:{\mathbb{C}}\to{\mathbb{C}}$, $j=1,2$, defined by

$\left\\{\begin{aligned} \psi_{j}(z)&:=\exp\left(\mu_{j}^{-1}g(\lambda_{j}\log
z)\right)\hbox{ if }z\neq 0\,,\\\ \psi_{j}(0)&:=0\,,\end{aligned}\right.$

are well defined homeomorphisms and the map

$\Psi:{\mathbb{C}}^{2}\to{\mathbb{C}}^{2}\,,\quad\Psi(z_{1},z_{2})=\big{(}\psi_{1}(z_{1}),\psi_{2}(z_{2})\big{)}$

is a homeomorphism that conjugates the foliation defined by the meromorphic
form $\lambda_{1}\frac{dz_{1}}{z_{1}}+\lambda_{2}\frac{dz_{2}}{z_{2}}$ to that
defined by $\mu_{1}\frac{dz_{1}}{z_{1}}+\mu_{2}\frac{dz_{2}}{z_{2}}$.

###### Proof.

The map

$(\mu_{1}\log z_{1}+\mu_{2}\log z_{2})\circ\Psi=g(\lambda_{1}\log
z_{1}+\lambda_{2}\log z_{2})$

is constant along the leaves of
$\lambda_{1}\frac{dz_{1}}{z_{1}}+\lambda_{2}\frac{dz_{2}}{z_{2}}$. ∎

Thus, in case (2c) when ${\mathcal{F}}^{\sharp}$ is of Poincaré type at $s$,
then it is also of Poincaré type at each singular point of the free component
${\mathcal{D}}$. Using similar methods as those developed by E. Paul in [34]
one can see that the local conjugacies given by Lemma 5.1 at the singular
points of the free component can be glued into a global
${\mathcal{C}}^{0}$-conjugacy along the whole free component. Finally, for two
topologically conjugated foliations whose reductions contain a Poincaré type
free component, the equality (22) may not be satisfied at the singular points
of the free component. This phenomenon forces us to exclude this case (2c)
with the following assumption that there is no free dynamical component:

  1. (NFC)

There is no dynamical component of $S_{\mathcal{F}}^{\sharp}$ without nodal
singular points of ${\mathcal{F}}^{\sharp}$ whose compact irreducible
components contain exactly two singular points of ${\mathcal{F}}^{\sharp}$.

Notice that if $\phi:{\mathcal{F}}\to\mathcal{G}$ is a topological conjugacy
between generalized curves and ${\mathcal{F}}$ satisfies (NFC) then
$\mathcal{G}$ also satisfies (NFC) thanks to Corollary 1.12.

###  5.4. Big dynamical components

To obtain equality (22) in case (3) we will ask ${\mathcal{F}}$ to fulfill the
following transverse rigidity assumption:

  1. (TR)

If a dynamical component of $S_{\mathcal{F}}^{\sharp}$ contains an irreducible
component with at least $3$ singular points of ${\mathcal{F}}^{\sharp}$, it
also contains an irreducible component whose holonomy group is topologically
rigid, for instance unsolvable.

We recall that a group $G$ of germs of ${{\mathrm{Diff}}}({\mathbb{C}},0)$ is
topologically rigid if any orientation preserving homeomorphism germ that
conjugates this group with another subgroup of
${{\mathrm{Diff}}}({\mathbb{C}},0)$ is necessarily holomorphic. D. Cerveau and
P. Sad showed that non abelian subgroups with generic linear part are
topologically rigid [8] and by classical results of Y. Ilyashenko, A.A.
Shcherbakov and I. Nakai [13, 41, 31] any unsolvable subgroup is topologically
rigid. Condition (TR) above is Krull generic [17]. We recall that a property
$\mathscr{P}$ on the space $\Omega$ of germs of holomorphic differential
$1$-forms on $({\mathbb{C}}^{2},0)$ is Krull generic if it is open dense for
the Krull topology on $\Omega$, i.e.

  1. (a)

for any $\omega\in\Omega$ and for any $p\in\mathbb{N}$ there is
$\eta\in\Omega$ satisfying $\mathscr{P}$ with same $p$-jet at $0$ as $\omega$,

  2. (b)

if $\omega$ satisfies $\mathscr{P}$ then there is $q\in\mathbb{N}$ such that
every $\xi\in\Omega$ with same $q$-jet as $\omega$ also satisfies
$\mathscr{P}$.

The interest of Condition (TR) is that we can apply the main theorem of J.
Rebelo in [37], whose proof remains valid for the extended statement below:

######  Theorem 5.2 (Transverse Rigidity Theorem).

Let ${\mathcal{F}}$ be a generalized curve and let ${\mathcal{D}}$ be a
dynamical component of $S_{\mathcal{F}}^{\sharp}$ containing an irreducible
component with topologically rigid holonomy group. Then ${\mathcal{F}}$ is
transversally rigid along ${\mathcal{D}}$ in the following sense: for any
homeomorphism $\phi:({\mathbb{C}}^{2},0)\to({\mathbb{C}}^{2},0)$ that
conjugates ${\mathcal{F}}$ to another germ of foliation preserving the
orientations of the ambient space and of the leaves, there is an open
neighbourhood $W$ of ${\mathcal{D}}\setminus{\mathcal{N}}$ such that $\phi$ is
transversally holomorphic at each point of
$E_{\mathcal{F}}(W)\setminus\\{0\\}$, where
${\mathcal{N}}\subset{\mathcal{E}}_{\mathcal{F}}$ is the set of nodal singular
points of ${\mathcal{F}}^{\sharp}$.

We recall that a germ of homeomorphism $\phi$ which conjugates two foliations
${\mathcal{F}}$ and $\mathcal{G}=\phi({\mathcal{F}})$, is transversally
holomorphic at a point $p$, if for any germs of submersion first integrals
$u:({\mathbb{C}}^{2},p)\to({\mathbb{C}},0)$ and
$v:({\mathbb{C}}^{2},\phi(p))\to({\mathbb{C}},0)$ there exists a germ of
biholomorphism $\phi^{\flat}:({\mathbb{C}},0)\to({\mathbb{C}},0)$ such that
$\phi^{\flat}\circ u=v\circ\phi$.

Notice that for a ${\mathcal{C}}^{0}$-conjugacy defined only on a neighborhood
of a punctured separatrix $C\setminus\\{0\\}$, transversal holomorphy property
is not suficient to obtain the invariance of the Camacho-Sad index along the
strict transform of $C$. A trivial counter-example with $C$ regular is given
by performing the blowing-up $E$ with center $\\{0\\}$ and by taking for
$\mathcal{G}$ the germ of $E^{-1}{\mathcal{F}}$ at the meeting point $p$ of
the strict transform $C^{\prime}$ of $C$ by $E$ and the exceptional divisor
$E^{-1}(0)$. Restricted to a neighborhood $V$ of $C^{\prime}\setminus\\{p\\}$,
$E$ is a biholomorphism that conjugates $\mathcal{G}_{|V}$ to
${\mathcal{F}}_{|E(V)}$. However, taking $\phi=(E_{|V})^{-1}$ we have

${{\mathrm{CS}}}({\mathcal{F}},C,0)={{\mathrm{CS}}}(\phi({\mathcal{F}}),C^{\prime},p)+1\,,$

see [7, Proposition 2.1]. We thus must impose as additional hypothesis the
compatibility of the conjugacy with the peripheral structure of $C$ in the
fundamental group of the complementary of the ${\mathcal{F}}$-appropriate
curve $S$.

###  5.5. Peripheral structure and Index Invariance Theorem

We fix local coordinates $(u,v)$ at the meeting point $s$ of the strict
transform ${\mathcal{C}}$ of $C$ with the exceptional divisor
${\mathcal{E}}_{\mathcal{F}}$, such that near $s$ the equation $v=0$ defines
${\mathcal{C}}$ and $u=0$ defines ${\mathcal{E}}_{\mathcal{F}}$; we consider
the loop $m$, resp. $p$, with common origin
$a:=(\varepsilon_{1},\varepsilon_{2})$, defined by $u(t)=\varepsilon_{1}$,
$v(t)=\varepsilon_{2}e^{2\pi it}$, resp. by $u(t)=\varepsilon_{1}e^{2\pi it}$,
$v(t)=\varepsilon_{2}$, $t\in[0,1]$; we choose $\varepsilon_{1}$ and
$\varepsilon_{2}$ small enough so that the loops $E_{\mathcal{F}}\circ m$ and
$E_{\mathcal{F}}\circ p$ are contained in a Milnor ball $B$ for the foliation
${\mathcal{F}}$; let us fix $b\in B\setminus S$ and a path $\alpha$ in
$B\setminus S$ joining $a$ to $b$. We call peripheral structure of $C$ in
$\pi_{1}(B\setminus S,b)$ the data of the two classes $\mathbf{m}$,
$\mathbf{p}\in\pi_{1}(B\setminus S,b)$, respectively called meridian and
parallel, defined by the path

$\alpha^{-1}{\vee}(E_{\mathcal{F}}\circ m){\vee}\alpha\quad\hbox{ and
}\quad\alpha^{-1}{\vee}(E_{\mathcal{F}}\circ p){\vee}\alpha\,.$

A consequence of the fact [12] that the subgroup
$\langle\mathbf{m},\mathbf{p}\rangle\subset\pi_{1}(B\setminus S,b)$ is equal
to its normalizer in $\pi_{1}(B\setminus S,b)$, is that $\mathbf{m}$ and
$\mathbf{p}$ do not depend on the choice of the path $\alpha$, see [21,
Corollary 6.1.4].

######  Theorem 5.3.

Let $\phi$ be a homeomorphism defined on an open tubular neighborhood $V$ of
$C\setminus\\{0\\}$ in ${\mathbb{C}}^{2}$ that conjugates ${\mathcal{F}}_{|V}$
to $\mathcal{G}_{|\phi(V)}$. If $\phi$ is transversally holomorphic and sends
respectively the meridian and the parallel of the peripheral structure of $C$
to the meridian and the parallel of $\phi(C)$, then the Camacho-Sad index of
${\mathcal{F}}^{\sharp}$ along the strict transform of $C$ by
$E_{\mathcal{F}}$ is equal to that of $\mathcal{G}^{\sharp}$ along the strict
transform of $\phi(C)$ by $E_{\mathcal{G}}$.

###### Proof.

Let $(u,v)$ (resp. $(u^{\prime},v^{\prime})$) be local coordinates centered at
the intersection point $s$ (resp. $s^{\prime}$) of the strict transform
${\mathcal{C}}=\\{v=0\\}$ (resp. ${\mathcal{C}}^{\prime}=\\{v^{\prime}=0\\}$)
of $C$ (resp. $\phi(C)$) with the exceptional divisor
${\mathcal{E}}_{\mathcal{F}}=\\{u=0\\}$ (resp.
${\mathcal{E}}_{\mathcal{G}}=\\{u^{\prime}=0\\}$). By the transversal
holomorphy of $\phi$ the holonomies of ${\mathcal{C}}$ and
${\mathcal{C}}^{\prime}$ are analytically conjugated and consequently

${{\mathrm{CS}}}({\mathcal{F}}^{\sharp},{\mathcal{C}},s)-{{\mathrm{CS}}}(\mathcal{G}^{\sharp},{\mathcal{C}}^{\prime},s^{\prime})\in{\mathbb{Z}}.$
(24)

Let $\Gamma_{n}$ be the lift of the loop $t\mapsto(\varepsilon_{1}e^{2i\pi
nt},0)$ to a leaf of ${\mathcal{F}}^{\sharp}$. Formula (21) implies that

${{\mathrm{CS}}}({\mathcal{F}}^{\sharp},{\mathcal{C}},s)=\lim\limits_{n\to\infty}\frac{1}{2i\pi
n}\int_{\Gamma_{n}}\frac{dv}{v}\,.$

Let $\xi_{n}$ be a path contained in $\\{u=\varepsilon_{1},0<|v|<1/n\\}$
joining the points $\Gamma_{n}(1)$ and $\Gamma_{n}(0)$ in such a way that the
variation of the argument of $v\circ\xi_{n}$ is bounded, consequently

$\lim\limits_{n\to\infty}{{\mathrm{Re}}}\left(\frac{1}{2i\pi
n}\int_{\xi_{n}}\frac{dv}{v}\right)=0$

and

${{\mathrm{Re}}}({{\mathrm{CS}}}({\mathcal{F}}^{\sharp},{\mathcal{C}},s))=\lim\limits_{n\to\infty}\frac{I_{n}}{n},\quad\text{where}\quad
I_{n}:=\frac{1}{2i\pi}\int_{\Gamma_{n}\vee\xi_{n}}\frac{dv}{v}\in{\mathbb{Z}}\,.$
(25)

The homotopy class of $E_{\mathcal{F}}(\Gamma_{n}\vee\xi_{n})$ can be
decomposed as $I_{n}\mathbf{m}_{n}+n\mathbf{p}_{n}$ using the peripheral
structure $\mathbf{m}_{n},\mathbf{p}_{n}$ of $C$ in $\pi_{1}(V\setminus
C,b_{n})\hookrightarrow\pi_{1}(B\setminus S,b_{n})$,
$b_{n}=E_{\mathcal{F}}(\Gamma_{n}(0))$. Up to isotopy we can assume that
$\phi^{\sharp}\circ u=u^{\prime}\circ\phi^{\sharp}$ in a neighbourhood of the
circle $\\{|u|=\varepsilon_{1},v=0\\}$, where
$\phi^{\sharp}=E_{\mathcal{G}}^{-1}\circ\phi\circ E_{\mathcal{F}}$. As the
variation of the argument of $v^{\prime}\circ\phi^{\sharp}\circ\xi_{n}$ is
bounded, because $\phi$ is transversely holomorphic, we have

${{\mathrm{Re}}}({{\mathrm{CS}}}(\mathcal{G}^{\sharp},{\mathcal{C}}^{\prime},s^{\prime}))=\lim\limits_{n\to\infty}\frac{I_{n}^{\prime}}{n},\quad\text{where}\quad
I_{n}^{\prime}:=\frac{1}{2i\pi}\int_{\phi^{\sharp}(\Gamma_{n}\vee\xi_{n})}\frac{dv^{\prime}}{v^{\prime}}\in{\mathbb{Z}}\,,$
(26)

and the homotopy class of $\phi(E_{\mathcal{F}}(\Gamma_{n}\vee\xi_{n}))$ can
be decomposed as
$I_{n}^{\prime}\mathbf{m}^{\prime}_{n}+n\mathbf{p}^{\prime}_{n}$, where
$\mathbf{m}^{\prime}_{n},\mathbf{p}^{\prime}_{n}$ is the peripheral structure
of $C^{\prime}$ in $\pi_{1}(\phi(V\setminus
C),\phi(b_{n}))\hookrightarrow\pi_{1}(B^{\prime}\setminus
S^{\prime},\phi(b_{n}))$. By hypothesis the group morphism

$\phi_{*}:\pi_{1}(V\setminus C,b_{n})\to\pi_{1}(\phi(V\setminus
C),\phi(b_{n}))$

sends $\mathbf{m}_{n}$ into $\mathbf{m}_{n}^{\prime}$ and $\mathbf{p}_{n}$
into $\mathbf{p}^{\prime}_{n}$. Since

$\phi_{*}(I_{n}\mathbf{m}_{n}+n\mathbf{p}_{n})=I_{n}\mathbf{m}_{n}^{\prime}+n\mathbf{p}_{n}^{\prime}\,,$

we obtain that $I_{n}=I_{n}^{\prime}$. Hence relations (24), (25) and (26)
give the equality
${{\mathrm{CS}}}({\mathcal{F}}^{\sharp},{\mathcal{C}},s)={{\mathrm{CS}}}(\mathcal{G}^{\sharp},{\mathcal{C}}^{\prime},s^{\prime})$.
∎

In addition global homeomorphisms always fulfill the peripheral structure
assumption:

######  Proposition 5.4.

[21, Theorem 6.2.1] Let us consider Milnor balls $B$ and $B^{\prime}$ for
germs of curves $S$ and $S^{\prime}$ at the origin of ${\mathbb{C}}^{2}$ and
$b\in B\setminus S$. For any homeomorphism $\phi:B\to\phi(B)\subset
B^{\prime}$ such that $\phi(S\cap B)=S^{\prime}\cap\phi(B)$, the map
$\phi_{\ast}:\pi_{1}(B\setminus S,b)\to\pi_{1}(B^{\prime}\setminus
S^{\prime},\phi(b))$ sends respectively the meridian and the parallel of the
peripheral structure in $\pi_{1}(B\setminus S,b)$ of any irreducible component
$C$ of $S$ to the meridian and the parallel of the peripheral structure of
$\phi(C)$ in $\pi_{1}(B^{\prime}\setminus S^{\prime},b)$.

Theorem 5.3, Proposition 5.4 and the previous analysis of small dynamical
components give us:

######  Theorem 5.5 (Index Invariance Theorem).

For generalized curves satisfying assumptions (NFC) and (TR), the Camacho-Sad
indices after reduction of singularities are topological invariants in the
sense that equalities (22) are satisfied for any ${\mathcal{C}}^{0}$-conjugacy
$\phi:{\mathcal{F}}\to\mathcal{G}$.

The hypothesis of this theorem are minimal. Condition (NFC) is necessary as we
have already seen in §5.3. Counter-examples not fulfilling Condition (TR) are
given by families of logarithmic foliations, whose topological classification
is obtained by E. Paul:

######  Theorem 5.6 ([35, Théorème 3.5]).

Let $\omega_{t}$ be a family of germs of logarithmic forms

$f_{1}\cdots f_{p}\sum_{j=1}^{p}\lambda_{j,t}\frac{df_{j}}{f_{j}},\quad
t\in({\mathbb{C}},0),$

on $({\mathbb{C}}^{2},0)$ whose residues $\lambda_{j,t}\in{\mathbb{C}}$ are
without relation with positive integer coefficients. If there is a family germ
$\\{g_{t},t\in({\mathbb{C}},0)\\}$ of ${{\mathrm{GL}}}_{2}^{+}({\mathbb{R}})$
such that each map $g_{t}$ sends $2i\pi\lambda_{j,0}$ into
$2i\pi\lambda_{j,t}$, then the family germ of foliations
$\\{\omega_{t},t\in({\mathbb{C}},0)\\}$ is topologically trivial.

##  6\. Excellence Theorem and topological moduli space

###  6.1. Excellence Theorem

On the set of foliation germs satisfying (NFC) and (TR) the equivalence
relations of ${\mathcal{C}}^{0}$-conjugacy and
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy coincide:

######  Theorem 6.1 (Excellence Theorem).

Let ${\mathcal{F}}$ and ${\mathcal{G}}$ be two topologically equivalent germs
of foliations at $0\in{\mathbb{C}}^{2}$ fulfilling (NFC) and (TR) properties
stated in §5.3 and §5.4. Then the following assertions are equivalent:

  1. (1)

there exists a homeomorphism germ
$\phi:({\mathbb{C}}^{2},0)\to({\mathbb{C}}^{2},0)$ that conjugates
${\mathcal{F}}$ to $\mathcal{G}$;

  2. (2)

there exists a homeomorphism germ
$\Phi:(M_{\mathcal{F}},{\mathcal{E}}_{\mathcal{F}})\to(M_{\mathcal{G}},{\mathcal{E}}_{\mathcal{G}})$
that conjugates ${\mathcal{F}}^{\sharp}$ to $\mathcal{G}^{\sharp}$; moreover
$\Phi$ is holomorphic at each non-nodal singular point of
${\mathcal{F}}^{\sharp}$ and is transversely holomorphic at each point of
${\mathcal{E}}_{\mathcal{F}}$ that is regular for ${\mathcal{F}}^{\sharp}$ and
not contained in a dicritical component.

###### Proof.

It suffices to use Index Invariance Theorem 5.5, Classification Theorem 4.15
and Transverse Rigidity Theorem 5.2 as indicated in Figure 10.

topological conjugationanalytical conjugation of monodromiesoutside nodal
separatorsequality of Camacho-Sad indicestransversely holomorphic
conjugationoutside nodal separatorsexcellent conjugationunder(TR) and
(NFC)Transverse RigidityTheorem 5.2trivialtrivialClassificationTheorem
4.15Index InvarianceTheorem 5.5true for allgeneralized curves Figure 10.
Strategy of the proof of Excellence Theorem 6.1.

∎

###  6.2. Classification Problem: Complete families and Moduli space

Excellence Theorem reduces the problem of topological classification of germs
of foliations at the origin of ${\mathbb{C}}^{2}$ to the problem of
classification under excellent germs of homeomorphisms. This one can be
formulated as follows. Let us call semi-local invariants of a foliation germ
${\mathcal{F}}$ the data of the topology of its separatrices curve, the
Camacho-Sad indices of ${\mathcal{F}}^{\sharp}$ and the holonomy
representation morphisms of the invariant irreducible components of
${\mathcal{E}}_{\mathcal{F}}$. All these data are
${\mathcal{C}}^{{{\mathrm{ex}}}}$-invariants and therefore topological
invariants under the hypothesis of Excellence Theorem 6.1. Let us denote by
$\text{SL}({\mathcal{F}})$ the set of all foliation germs with same semi-local
invariants as ${\mathcal{F}}$. The goal of ${\mathcal{C}}^{0}$-classification
is to construct a natural set of foliation germs whose intersection with the
topological class of any foliation in ${{\mathrm{SL}}}({\mathcal{F}})$ is non-
empty and totally discontinuous. For a large class of foliation germs a
complete answer to this problem is given in the papers [23, 24, 25] that lead
to a family of foliations in $\text{SL}({\mathcal{F}})$ depending
holomorphically on a parameter that varies in a complex manifold of minimal
dimension. This family has natural properties of factorization of any other
holomorphic family in $\text{SL}({\mathcal{F}})$; moreover the redundancy of
the topological types in this family is minimal and can be precisely
described.

In the spirit of Teichmüller theory, it is convenient for our moduli problem
to endow each foliation germ with a marking. For this, let us fix a foliation
germ ${\mathcal{F}}_{0}$ that is a generalized curve and let us simply denote
by ${\mathcal{E}}^{\diamond}:=({\mathcal{E}},\Sigma)$ the pair
$({\mathcal{E}}_{{\mathcal{F}}_{0}},{{\mathrm{Sing}}}({\mathcal{F}}_{0}^{\sharp}))$.
A marked by ${\mathcal{E}}^{\diamond}$ foliation germ is a pair
$({\mathcal{F}},f)$ where ${\mathcal{F}}$ is a foliation germ at
$({\mathbb{C}}^{2},0)$ and $f$ is a homeomorphism from ${\mathcal{E}}$ into
${\mathcal{E}}_{\mathcal{F}}$ such that

$f(\Sigma)={{\mathrm{Sing}}}({\mathcal{F}}^{\sharp})\quad\hbox{and}\quad
f(D)\cdot f(D^{\prime})=D\cdot D^{\prime}$

for any pair $(D,D^{\prime})$ of irreducible components of ${\mathcal{E}}$.
Two ${\mathcal{E}}^{\diamond}$-markings $f$ and $g$ of a same foliation
${\mathcal{F}}$ are equivalent if $g^{-1}\circ f$ is isotopic to the identity
map, by an isotopy fixing each point of $\Sigma$. Two marked by
${\mathcal{E}}^{\diamond}$ foliation germs $({\mathcal{F}},f)$ and
$(\mathcal{G},g)$ are ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated if there
exists a ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy germ
$\Phi:(M_{\mathcal{F}},{\mathcal{E}}_{\mathcal{F}})\to\
(M_{\mathcal{G}},{\mathcal{E}}_{\mathcal{G}})$ such that $g$ and $\Phi\circ f$
are equivalent ${\mathcal{E}}^{\diamond}$-markings of $\mathcal{G}$.

######  Definition 6.2.

We say that two marked by ${\mathcal{E}}^{\diamond}$ foliation germs
$({\mathcal{F}},f)$ and $(\mathcal{G},g)$ have same semi-local type if

  1. (1)

for any irreducible component $D$ of ${\mathcal{E}}$ and any $s\in\Sigma\cap
D$, we have the equality of Camacho-Sad indices
${{\mathrm{CS}}}(\mathcal{G}^{\sharp},g(D),g(s))={{\mathrm{CS}}}({\mathcal{F}}^{\sharp},f(D),f(s))$

  2. (2)

for any irreducible component $D$ of ${\mathcal{E}}$ with
$D\cap\Sigma\neq\emptyset$, the compositions with the group morphisms induced
by the markings

$H_{D}^{\mathcal{F}}\circ f_{*}\,\hbox{ and }\;\;H_{D}^{\mathcal{G}}\circ
g_{*}\;:\;\pi_{1}(D\setminus\Sigma,\cdot)\longrightarrow{{\mathrm{Diff}}}({\mathbb{C}},0)$

of the holonomy morphisms $H_{f(D)}^{\mathcal{F}}$ and
$H_{g(D)}^{\mathcal{G}}$ of the foliations ${\mathcal{F}}^{\sharp}$ and
$\mathcal{G}^{\sharp}$ along $f(D)$ and $g(D)$, are equal up to composition by
an inner automorphism of ${{\mathrm{Diff}}}({\mathbb{C}},0)$.

To obtain a “parametrization” by a finite dimensional manifold of the set of
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy classes of marked foliation germs
with fixed semi-local type, we need an additional hypothesis called finite
type. This one was first introduced in [23] as a combinatorial property of an
appropriate orientation of the dual graph ${\mathsf{A}}_{\mathcal{F}}$ and
then reinterpreted in [23, 24] as the finiteness dimension
$\tau({\mathcal{F}})$ of a naturally associated cohomological space. This
assumption is not very restrictive because in the set of $1$-differential
forms defining generalized curves, the subset of finite type foliations
contains an open dense set for the Krull topology, see [29, Theorem 6.2.1] and
[23, Remark 2.5].

######  Theorem 6.3.

[25, Theorem 3.7] Let $({\mathcal{F}},f)$ be a marked finite type foliation
germ which is a generalized curve. Then there exists an equisingular marked
family of foliation germs

$\left({\mathcal{F}}_{z,d},f_{z,d}\right)_{(z,d)\in{\mathbb{C}}^{\tau}\times\mathbf{D}}$

such that

  1. (1)

$\mathbf{D}$ is a quotient of a finite product of totally disconnected
subgroups of the circle ${\mathbb{U}}(1)$ and $\tau=\tau({\mathcal{F}})$,

  2. (2)

for any marked foliation germ $(\mathcal{G},g)$ with same semi-local type as
$({\mathcal{F}},f)$, there exists
$(z,d)\in{\mathbb{C}}^{\tau}\times\mathbf{D}$ such that $(\mathcal{G},g)$ is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to
$({\mathcal{F}}_{z,d},f_{z,d})$,

  3. (3)

if $(\mathcal{G}_{t},g_{t})_{t\in P}$ is an equisingular marked family of
foliation germs with parameter space a connected manifold $P$ satisfying
$H^{1}(P,{\mathbb{Z}})=0$, then for any $t_{0}\in P$ and
$(z_{0},d_{0})\in{\mathbb{C}}^{\tau}\times\mathbf{D}$ such that
$(\mathcal{G}_{t_{0}},g_{t_{0}})$ is
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated to ${\mathcal{F}}_{z_{0},d_{0}}$,
there exists a unique holomorphic map
$\lambda:P\to{\mathbb{C}}^{\tau}\times\mathbf{D}$ with
$\lambda(t_{0})=(z_{0},d_{0})$ such that for any $t\in P$ the marked foliation
germs $(\mathcal{G}_{t},g_{t})$ and
$({\mathcal{F}}_{\lambda(t)},f_{\lambda(t))})$ are
${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugated,

  4. (4)

in a neighborhood $W_{t^{\prime}}$ of any point $t^{\prime}\in P$, the
${\mathcal{C}}^{{\mathrm{ex}}}$-conjugacies between the marked foliation germs
$(\mathcal{G}_{t},g_{t})$ and $({\mathcal{F}}_{\lambda(t)},f_{\lambda(t))})$
in Assertion (3) can be realized by excellent homeomorphisms $\Phi_{t}$
depending continuously on $t\in W_{t^{\prime}}$.

The notion of marked family of foliation germs, introduced in [25, §2.2], is a
compatibility property of the markings with the local topological triviality
in the parameter of the whole exceptional divisor. That of equisingularity is
precisely defined in [24, §3]; basically it means that the family is locally
equireducible and all the foliation germs in the family have same semi-local
type. Assertion (4) in Theorem 6.3 follows from [25, Theorem 4.4].

We finally describe the redundancy of the
${\mathcal{C}}^{{{\mathrm{ex}}}}$-classes in the family
$\left({\mathcal{F}}_{z,d},f_{z,d}\right)_{(z,d)\in{\mathbb{C}}^{\tau}\times\mathbf{D}}$.
Let us define the ${\mathcal{C}}^{{{\mathrm{ex}}}}$-moduli space of
$({\mathcal{F}},f)$ as the set

${{\mathrm{Mod}}}({\mathcal{F}},f):=\\{\;[\mathcal{G},g]\;|\;\text{$(\mathcal{G},g)$
and $({\mathcal{F}},f)$ have same semi-local type}\\}$

of ${\mathcal{C}}^{{{\mathrm{ex}}}}$-conjugacy classes $[\mathcal{G},g]$ of
marked by ${\mathcal{E}}^{\diamond}$ foliations $(\mathcal{G},g)$. To any
equisingular marked family $\mathcal{G}_{P}:=(\mathcal{G}_{t},g_{t})_{t\in P}$
of foliation germs with same semi-local type as $({\mathcal{F}},f)$ we
associate its moduli map

$\text{mod}_{\mathcal{G}_{P}}:P\longrightarrow{{\mathrm{Mod}}}({\mathcal{F}},f)\,,\qquad
t\mapsto[\mathcal{G}_{t},g_{t}]\,.$

######  Theorem 6.4.

[23, Theorem D] Let $({\mathcal{F}},f)$ be a marked finite type foliation germ
which is a generalized curve. There is a natural abelian group structure on
${{\mathrm{Mod}}}({\mathcal{F}},f)$ given by an exact sequence

$0\to{\mathbb{Z}}^{p}\xrightarrow{\alpha}{\mathbb{C}}^{\tau}\xrightarrow{\Lambda}{{\mathrm{Mod}}}({\mathcal{F}},f)\xrightarrow{\beta}\mathbf{D}\rightarrow
0$

and for any section $\zeta$ of $\beta$ there exists an equisingular marked
family

${\mathcal{F}}_{{\mathbb{C}}^{\tau}\times\mathbf{D}}:=({\mathcal{F}}_{z,d},f_{z,d})_{(z,d)\in{\mathbb{C}}^{\tau}\times\mathbf{D}}$

of foliation germs with same semi-local type as $({\mathcal{F}},f)$ such that

  1. (1)

all assertions in Theorem 6.3 are satisfied,

  2. (2)

the moduli map
${{\mathrm{mod}}}_{{\mathcal{F}}_{{\mathbb{C}}^{\tau}\times\mathbf{D}}}$ is
surjective and for any $(z,d)\in{\mathbb{C}}^{\tau}\times\mathbf{D}$ we have

${{\mathrm{mod}}}_{{\mathcal{F}}_{{\mathbb{C}}^{\tau}\times\mathbf{D}}}(z,d)=\Lambda(z)\cdot\zeta(d)$

where $\cdot$ is the group operation in ${{\mathrm{Mod}}}({\mathcal{F}},f)$.

Notice that the surjectivity of
$\text{mod}_{{\mathcal{F}}_{{\mathbb{C}}^{\tau}\times\mathbf{D}}}$ is
equivalent to property (2) in Theorem 6.3.

A direct consequence of Theorem 6.4 is the explicit description of the
redundancy of the ${\mathcal{C}}^{{\mathrm{ex}}}$-classes in the family
${\mathcal{F}}_{{\mathbb{C}}^{\tau}\times\mathbf{D}}$:

######  Corollary 6.5.

Two marked foliation germs $({\mathcal{F}}_{z,d},f_{z,d})$ and
$({\mathcal{F}}_{z^{\prime},d^{\prime}},f_{z^{\prime},d^{\prime}})$ in the
marked family ${\mathcal{F}}_{{\mathbb{C}}^{\tau}\times\mathbf{D}}$ are
${\mathcal{C}}^{{\mathrm{ex}}}$-conjugated if and only if $d=d^{\prime}$ and
there exists $N\in{\mathbb{Z}}^{p}$ such that $z^{\prime}=z+\alpha(N)$.

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