# Utilizing Background Knowledge for Robust Reasoning over Traffic Situations

Jiarui Zhang1, Filip Ilievski1, Aravinda Kollaa1,
Jonathan Francis2, Kaixin Ma3, Alessandro Oltramari2

# Utilizing Background Knowledge for Robust Reasoning over Traffic Situations

Jiarui Zhang1, Filip Ilievski1, Aravinda Kollaa1,
Jonathan Francis2, Kaixin Ma3, Alessandro Oltramari2

###### Abstract

Understanding novel situations in the traffic domain requires an intricate
combination of domain-specific and causal commonsense knowledge. Prior work
has provided sufficient perception-based modalities for traffic monitoring, in
this paper, we focus on a complementary research aspect of Intelligent
Transportation: traffic understanding. We scope our study to text-based
methods and datasets given the abundant commonsense knowledge that can be
extracted using language models from large corpus and knowledge graphs. We
adopt three knowledge-driven approaches for zero-shot QA over traffic
situations, based on prior natural language inference methods, commonsense
models with knowledge graph self-supervision, and dense retriever-based
models. We constructed two text-based multiple-choice question answering sets:
BDD-QA for evaluating causal reasoning in the traffic domain and HDT-QA for
measuring the possession of domain knowledge akin to human driving license
tests. Among the methods, Unified-QA reaches the best performance on the BDD-
QA dataset with the adaptation of multiple formats of question answers.
Language models trained with inference information and commonsense knowledge
are also good at predicting the cause and effect in the traffic domain but
perform badly at answering human-driving QA sets. For such sets, DPR+Unified-
QA performs the best due to its efficient knowledge extraction. ††Accepted to
Workshop on Knowledge Augmented Methods for Natural Language Processing, in
conjunction with AAAI 2023.

## Introduction

Figure 1: Overview of our study framework, which evaluates three knowledge-
enhanced language methods adapted with different knowledge sources on two
evaluation datasets in a zero-shot manner.

As humans, we are able to make sense of novel situations in any domain even
with limited domain-specific knowledge. The domain of intelligent traffic
monitoring is a particularly attractive one, given the large market and the
long list of car manufacturers that innovate in this domain (Int 2022). Being
able to understand novel situations in traffic requires a complex association
of domain-specific and causal commonsense knowledge. Prior work on evaluation
tasks (Xu et al. 2017; Kim et al. 2018; Xu, Huang, and Liu 2021) and
corresponding methods (Halilaj et al. 2021; Muppalla et al. 2017; Chowdhury et
al. 2021; Wickramarachchi, Henson, and Sheth 2020) has focused on the
perception-based modality, with many of the input exemplars coming from
sensors like street video cameras.

Perception (e.g., tracking of participants over time, counting cars, etc.) is
certainly a key aspect of traffic monitoring. However, a holistic
understanding of situations in traffic requires other, complementary skills
such as causal and commonsense reasoning (e.g., red light causes cars to stop,
direct sunlight does not), and possession of rich domain knowledge (e.g., the
driving speed in school zones is limited to 30mph). A large body of work has
focused on general commonsense reasoning (Ma et al. 2021b; Zhang et al. 2022),
but the role of commonsense and causal reasoning has not been explored
thoroughly. Complementarily, ontologies for describing traffic participants
(Ope 2020) exist, but it is unclear whether connecting them to neural models
can bring the desired generalizability. Given that natural language is a
common medium for human communication, and considering the success of large
language models (LMs), it is intuitive to turn to using the textual modality
to develop and test robust traffic understanding models. To the best of our
knowledge, understanding traffic situations presented in natural language has
not been explored in depth so far. This brings up a natural question of
whether large pretrained models can effectively solve realistic traffic
understanding tasks dominantly presented in language, such as human driving
license tests. We also observe that prior work has not systematically
investigated the role of different types of background knowledge, like causal
commonsense knowledge and traffic domain knowledge, in reasoning over traffic
situations.

In this paper, we focus on an evaluation for robust reasoning over traffic
situations described in natural language. This allows us to reduce complexity,
understand the behavior of various neural and neuro-symbolic models, and
measure the impact of various sources of background knowledge. We construct
two realistic text-based multiple-choice question answering (MCQA) sets: BDD-
QA for evaluating causal reasoning in the traffic domain and HDT-QA for
measuring the possession of domain knowledge akin to human driving license
tests. We adopt three knowledge-driven approaches for zero-shot QA over
traffic situations, based on prior methods for natural language inference,
commonsense models with knowledge graph self-supervision, and dense retriever-
based models. By doing so, we investigate the impact of extracting commonsense
and domain knowledge from structured and unstructured data sources. An
overview of our study framework is shown in Figure 1.

We summarize our contributions as follows:

  1. 1.

We formulate two novel text-based tasks and benchmarks for situational
reasoning in the traffic domain: BDD-QA, which evaluates whether systems can
apply causal reasoning to make decisions in arbitrary traffic situations, and
HDT-QA, which tests the ability of systems to answer questions that require
domain knowledge from driving manuals. We divide the datasets into meaningful
partitions to allow balanced and fine-grained analysis.

  2. 2.

We adapt different knowledge-aware methods that have been shown to generalize
well across natural language reasoning tasks in prior work: natural language
inference models, knowledge graph-based commonsense reasoners, and a novel
retrieval-based QA method with an open-book access to official driving
manuals.

  3. 3.

We perform extensive experiments of the three methods with different language
models as backbones against the two datasets in a zero-shot evaluation setup.
We perform an in-depth analysis of model performance on data partitions and
look closer into the model predictions to provide useful insights into the
role of background knowledge for building robust traffic understanding
methods.

## Related work

### Traffic Understanding

CADP (Shah et al. 2018) is a spatio-temporally annotated dataset for accident
forecasting using traffic camera views. The Berkeley Deep-Drive dataset (BDD)
is a video dataset consisting of real driving videos containing abundant
driving scenarios (Xu et al. 2017). The follow-up work, BDD-X (Kim et al.
2018), provides the action description of the BDD video and their
explanations. TrafficQA (Xu, Huang, and Liu 2021) consists of over 60K QA sets
based on over 10k traffic scenes. The QA set of TrafficQA includes 6 different
aspects of reasoning problems: basic understanding, attribution,
introspection, counterfactual inference, event forecasting, and reverse
reasoning. All QA pairs are based on visual inputs.

The traffic benchmarks have inspired corresponding methods. By using knowledge
to enhance traffic understanding, Wickramarachchi, Henson, and Sheth (2020)
provides a demonstration of the process of creating and evaluating knowledge
graph embedding for autonomous driving data. Chowdhury et al. (2021) enhance
the knowledge graph for autonomous driving and the task of scene entity
prediction. ITSKG (Muppalla et al. 2017) is a knowledge graph framework for
extracting actionable information from raw sensor data in traffic. CoSI
(Halilaj et al. 2021) proposes a knowledge graph-based approach for
representing information sources relevant to traffic situations.

Different from traffic monitoring, which mainly requires the perception
modality, in this work, we explore a shift in the paradigm from monitoring to
understanding, which requires abundant domain knowledge and commonsense
reasoning methods. However, perceptual annotations provide valuable
information that can be reused for creating comprehension tasks. We utilize
the textual descriptions of actions and their explanations associated with the
BDD-X dataset to construct our BDD-QA dataset for causal reasoning.

### Situational Reasoning in Natural Language

MultiNLI (Williams, Nangia, and Bowman 2017) is a large (433k) NLI corpus with
ten distinct genres of written and spoken English. MNLI has become a popular
benchmark for most of the language models, such as BERT (Devlin et al. 2018),
RoBERTa (Liu et al. 2019), BART (Lewis et al. 2019), and DeBERTa (He et al.
2020). These methods have been widely used, but have not been applied to the
traffic domain, to the best of our knowledge. In our work, we use such models
that are pretrained with the MNLI dataset, to evaluate their zero-shot
performance on inference tasks in the traffic domain based on the assumption
that the models are capable of causal reasoning generally.

ATOMIC (Sap et al. 2019) provides a large atlas of everyday commonsense
reasoning with a focus on inferential if-then knowledge. CauseNet (Heindorf et
al. 2020) provides a large-scale knowledge base of claimed causal relations
between concepts which can be used for casual reasoning and question-answering
tasks. Ma et al. (2021a) developed a method that utilizes commonsense
knowledge that is extracted from the knowledge graph to train language models
via self-supervision and such models are proven to have high accuracy across
several zero-shot tasks (Zhang et al. 2022). Given that the knowledge-driven
models perform well on many general-domain zero-shot commonsense tasks, in
this work, we evaluate their performance on reasoning tasks in the traffic
domain to investigate the effect of commonsense knowledge on answering
traffic-domain questions.

DPR (Dense Passage Retrieval) (Karpukhin et al. 2020) is a passage retrieval
method using dense representations of sentences and passages for QA. Trained
on a Wikipedia dump and several QA datasets, the DPR model overperforms
several traditional context retrieval methods like TF-IDF and BM25. Unified-QA
(Khashabi et al. 2020) is a generalized QA system across 4 formats of QA and
is proven to perform well on all of the benchmarks. In our work, we retrieve
relevant domain knowledge with DPR to help answer domain-specific questions
with the Unified-QA model.

Table 1: Examples of BDD-QA and HDT-QA. (*) denotes the correct answer. BDD-
QA-EP questions ask for the correct effect given the cause and BDD-QA-CP for
the correct cause given the effect. For HDT-QA, we show one example for each
category (having 2,3,4,5 candidate answers) BDD-QA-EP
---
C: The light has turned green and traffic is flowing smoothly.
E1: The driver jerks abruptly to the right;
E2: The car is driving straight into the parking lot;
E3: The car accelerates slowly to a maintained speed(*)
BDD-QA-CP
E: The car is driving forward slowly.
C1: The light ahead was red at the intersection;
C2: The passenger is seated in the car and the lane is clear;
C3: There is slow traffic in front of the car(*)
HDT-QA (2 candidates)
Q: Yield also means stop if you cannot merge safely
into the flow of traffic
A1: True(*); A2: False
HDT-QA (3 candidates)
Q: You just sold your vehicle.
You must notify the DMV within  days.
A1: 15; A2: 5(*); A3: 10;
HDT-QA (4 candidates)
Q: You have the right of way when you are:
A1: Entering a traffic circle;
A2: Backing out of a driveway;
A3: Leaving a parking space;
A4: Already in a traffic circle(*)
HDT-QA (5 candidates)
Q: Why should you cover your cargo during its transportation?
A1: To protect the cargo from weather only;
A2: There is no reason;
A3: To protect people from the cargo’s contents if it spills;
A4: To protect people from the cargo’s contents if it spills
and to protect the cargo from the weather(*);
A5: To help your vehicle accelerate smoothly

## Tasks & Datasets

We refer to situational reasoning as the task of providing the best possible
justification or decision for a given situation. For instance, in the traffic
domain, the justification for “Why did the car accelerate?” could be the light
turning green or the absence of traffic. We focus on situational reasoning in
the traffic domain. Formally, we pose traffic situational reasoning as a
multiple-choice question-answering task. Each task entry consists of a natural
language question $Q$, and $n$ candidate answers $\\{A_{1},...,A_{n}\\}$,
consisting of $1$ correct answer and $n-1$ distractors. The task of an AI
model is to select the most probable answer among the candidates.

To evaluate whether AI models comprehensively understand the traffic domain
knowledge, we introduce two diverse tasks: BDD-QA and HDT-QA. BDD-QA tests
model understanding of the causality of traffic participant behaviors and
their reasons, while HDT-QA focuses on driving policies, and legible or
recommended drivers’ decisions. Examples of the two datasets are shown in
Table 1. Next, we describe the details of these two QA datasets.

### BDD-QA dataset

Data Source We create a multiple choice question set from the Berkeley Deep-
Drive Explanation Dataset (Kim et al. 2018) (BDD-X), which includes over 26k
QA sets and covers an extensive range of weather and road types. Given that
the dataset consists of an action and its justification (reason), we treat the
action as an effect and the justification as a cause.

Sampling Strategy We use sentence transformers (Reimers and Gurevych 2019) to
represent the sentences as sentence embeddings. The model we use, all-mpnet-
base-v2, is based on MPNet (Song et al. 2020) and is pretrained on over 1B
sentence pairs and reportedly reached a state-of-art performance. All of the
sentence embeddings are normalized and we compute their distance using cosine
similarity. To make the sentence embedding more accurate, we transform each
sentence into a declarative and grammatically complete one. For example, if a
justification sentence is to turn left, it will be transformed to The car
wants to turn left by filling in the subject. If the justification sentence is
because the light turns red, we will remove the because, resulting in the
sentence The light turns red. To avoid having causes that are too similar, we
remove the QA pairs that have a cause that has a similarity to others higher
than a threshold, in our work, we set this threshold to 0.9.

Table 2: Examples of 5 classes of effects (car actions) discovered in the BDD-QA dataset. Classes | Sentences
---|---
Accelerate | The car accelerates
The car inches forward
The car is traveling down the road.
Slow | The car slows slightly
The car moves forward slowly
The car maintains a slow speed
Stop | The car stops
The car is parked at the right curb
The car is stationary
Merge | The car merges into the lane to its left
The car is moving to the left lane
The car is switching lanes to the left
Turn | The car slows down and pulls to the right
The car turns left and drives forward
The car swerves to the left and slows

To generate multiple-choice questions, for each cause-and-effect pair, we
employ two different approaches: (1) Sampling QA-pairs by dissimilar effects
for effects-prediction, (BDD-QA-EP), and (2) Sampling QA-pairs by dissimilar
causes for causes-prediction, (BDD-QA-CP). We randomly sample cause/effect
from other pairs, then we set an upper bound of sentence similarity to avoid
sampling too similar answers that are also correct for a question. The upper
bound threshold $t$ is set to 0.4 in our work, all the randomly sampled
cause/effect whose similarity with the true answer is higher than the
threshold would be removed and we will re-sample until it satisfies the
threshold restriction. For example, given the cause The road doesn’t have much
traffic and effect The car accelerates down the road, a good effect distractor
would be The vehicle is stopped rather than The car is accelerating faster. To
improve the diversity of the sampled distractors, we also make sure that the
dissimilarity between the distractors is also lower than the same threshold
$t$.

Action Classification The project (Ope 2020) presents an ontology that
describes the concepts in the traffic domain, including car maneuvers. In the
real world, the possible actions performed by cars are few and lead to a
limited set of effects. Inspired by this, we classify the effects of BDD-QA
into several classes, each of which presents a type of car action. After we
compute the sentence embeddings for each action, we use k-means (MacQueen
1967) for the action sentence clustering. After trying different numbers of
classes, we noticed that the clusters are intuitive. We decide to use 5
classes that can represent the car’s actions best from a human perspective,
namely: accelerate, slow, stop, merge and turn. Table 2 shows examples for
each of the five classes.

### HDT-QA dataset

We also develop a complementary QA set by using driving tests intended to test
the driving knowledge of humans. We call this dataset HDT-QA (Human Driving
Test QA).

Data Source We scrap the questions of HDT-QA from a website that contains the
driving test data and driving manuals from all 51 states of the USA (Dri
2022). For each state, there are three subjects of driving tests and manuals
Motorcycle, Car and Commercial Driver, resulting in 153 state-subject
combinations. The test data are all multiple-choice questions, where the
number of answer candidates ranges between 2 and 5. The statistics of our HDT-
QA set for each number of candidate answers are shown in Table 3.

Filtering To make our test data fair for text-based methods, we only keep
questions that can fully be answered based on the textual content. We exclude
all questions with images (e.g., traffic signs). We have also filtered out the
questions where one of the candidate answers is all of the above, as answering
such questions is not the focus of our work.

Table 3: Statistics of the partitions of our two datasets. Dataset | Partition | Size
---|---|---
BDD-QA | Event Prediction | 3,139
Cause Prediction | 3,139
HDT-QA | 2 answers | 131
3 answers | 2,398
4 answers | 7,558
5 answers | 40

## Method

A key strength of neural language models is their ability to provide answers
for open-world reasoning tasks (Devlin et al. 2018; Radford et al. 2019).
While the models can be tuned on a dataset to enhance their accuracy, prior
work has shown that this leads to over-fitting (Ma et al. 2021b). Anticipating
similar over-fitting to QA sets in the traffic domain, we focus on developing
models that include commonsense and domain knowledge to enable them to better
understand situations in traffic and generalize across unseen traffic QA
tasks. We propose three methods to enrich language models with background
knowledge for the traffic domain: NLI-based models, Knowledge-based models,
and Retrieval-based models. We use readily available models and run them
directly on our two benchmarks in a zero-shot setting.

### Inference-based reasoning models

We propose an NLI method based on semantic-level reasoning for the zero-shot
evaluation of our tasks. Given a premise and its hypothesis that can be from
any domain, the NLI model we use is expected to capture coherence information
between them. It takes two sentences as the input and outputs a 3-digit vector
which represents confidence letting the sentence pair’s relation be
entailment, neutral, and contradiction.

To avoid the effect of the data format, we keep our QA data as declarative
sentence pairs. As is described in the last paragraph, given a sentence pair
$S_{1},S_{2}$, the model is asked to give the score of entailment, neutral,
and contradiction of these two sentences as $P_{e},P_{n},P_{c}$, respectively.
Then the model chooses the candidate with the highest margin score of
$P_{e}-P_{c}$ as the true answer. Formally, the final prediction of the model
will be:

$pred=\arg\max_{i}(P(e|p_{i},MNLI)-P(c|p_{i},MNLI))$

Where $p_{i}$ is the $i_{th}$ candidate sentence pair for the model to choose.
$e$ stands for entailment and $c$ stands for contradiction. $MNLI$ refers to
the inference knowledge that is extracted from the MNLI dataset by the
language model.

### KG-based reasoning models

Self-supervision with structured knowledge has proven to improve the language
model’s ability to solve reasoning problems in a zero-shot setting (Ma et al.
2021a; Zhang et al. 2022; Dou and Peng 2022). Thus, we evaluate the
performance of language models enhanced by commonsense knowledge of traffic
domain question-answering problems.

Our knowledge-based QA framework follows the setup of Ma et al. (2021b); Zhang
et al. (2022). Given a question and several candidates, the model chooses the
question-candidate pair that best entails the knowledge extracted from the
knowledge graph:

$pred=\arg\max_{i}(R(s_{i}|CSKG))$

where $R(s|CSKG)$ is the reasonable score of a sentence based on the knowledge
that the language model extracted from the Commonsense Knowledge graph.

We chose models that are pretrained on a synthetic dataset that is generated
from the Commonsense Knowledge Graph (Ilievski, Szekely, and Zhang 2021). In
total, Commonsense Knowledge Graph contains 7 million commonsense statements,
which is a significant portion of implicit facts that the commonsense models
have absorbed via self-supervision.

### Retrieval-based QA models

Retrieval-based QA models aim to find evidence for the correct answer for a
given contextualized question in background knowledge documents. This task
aims to capture domain knowledge such as rules and regulations we abide by
while driving in traffic. We pose the input to the retrieval-based models as a
multiple-choice question answering based on context. The main idea is to
extract the most relevant content as evidence that can help answer the
question based on domain documents and use it as a context to find the best
possible choice.

We propose a novel pipeline for domain-based question answering. We use Dense
Passage Retriever (DPR) (Karpukhin et al. 2020) as the relevant domain
extractor module to encode each paragraph and question sentence into vectors.
Given a corpus $C$ include several passages $PA$, we use a cosine similarity
score to extract the most relevant paragraph:

$PA_{select}=\arg\max_{PA_{i}}~{}cos\\_sim(PA_{i},Q),PA_{i}\in C$

The passage extracted then along with the question and choices are passed to a
QA model, then the QA model generates the predicted answer, the probability of
generating an answer is:

$\displaystyle P(GA|Q,A,C)=$
$\displaystyle\prod_{i=i}^{n}P(ga_{i}|Q,A,PA_{select},ga_{1},ga_{2},...,ga_{i-1})$

Where $C$ is the whole corpus, for our task, we use a comprehensive collection
of driving manuals as the domain corpus (Dri 2022). After processing the
manual PDFs, we divide them into sentences by simply using the punctuation ” .
? ! ” as sentence boundaries. Then we use a T5-based grammar correction (Gra
2021) as a data cleaning tool to make our corpus have correct spelling and
grammar. Finally, we combine every 10 neighborhood sentences as paragraphs
that are fed to the QA model.222Paragragh segmentation based on the similarity
between neighboring sentences yielded consistently worse results. In total,
there are $42.979$ paragraphs and each paragraph has an average word count of
$147.5$.

After the model generates the predicted answer, we choose the answer which is
most similar to the $GA$ as our final prediction.

Table 4: Evaluation results on the BDD-QA-EP and BDD-QA-CP dataset of our three methods. The Vanilla Roberta-large model was only pretrained by unlabelled corpus. The Vanilla Unified-QA method doesn’t feed the most relevant corpus to the model. Different from other results, supervised models only use 10% of the whole data for evaluation. The result of the human evaluation is included in the last row. Method | Model | BDD-QA-EP | BDD-QA-CP |  Avg
---|---|---|---|---
Random | - | 33.3 | 33.3 | 33.3
Vanilla models | Roberta-large | 36.1 | 43.6 | 40.0
UnifiedQA-v2 | 75.9 | 63.3 | 69.6
NLI-based | Roberta-large-mnli | 63.2 | 58.6 | 60.9
Bart-large-mnli | 66.3 | 56.8 | 61.5
Deberta-large-mnli | 67.0 | 54.5 | 60.8
Deberta-v2-xlarge-mnli | 71.3 | 59.7 | 65.5
Deberta-v2-xxlarge-mnli | 72.4 | 64.9 | 68.7
Knowledge-based | Roberta-base | 63.3 | 54.4 | 58.9
Roberta-large | 71.7 | 56.8 | 64.3
T5-small | 54.3 | 46.0 | 50.1
T5-large | 66.4 | 55.5 | 61.0
T5-3b | 71.0 | 58.6 | 64.8
Retrieval-based | UnifiedQA-v2 + DPR | 71.4 | 55.7 | 63.6
Supervised | Roberta-large | 93.0 | 81.2 | 86.1
Human | - | 74.0 | 68.0 | 71.0

## Experiments

### Model setup

For Inference-base reasoning models, we select several NLI models that are
trained on the MNLI (Williams, Nangia, and Bowman 2017) dataset. These models
are: Roberta-large, Bart-large, Deberta-large, Deberta-v2-xlarge and
Deberta-v2-xxlarge.

For KG-based reasoning models, we choose the five models from Zhang et al.
(2022) for evaluation: Roberta-base, Roberta-large and T5-small, T5-large, and
T5-3b models.

For Retrieval-based models, we use the sentence transformer model pretrained
in (Karpukhin et al. 2020) to encode the paragraphs and sentences. As a QA
model, we use the Unified-QA T5-based transformer model (Khashabi, Kordi, and
Hajishirzi 2022), to output the most probable candidate answer.

### Other Baselines

To indicate the upper bound of models’ performance on our dataset, we have
also tested a supervised method based on tuning Roberta-large. We have divided
all of the datasets into training and testing parts with a ratio of 90% and
10%. For the HDT-QA dataset with 2 and 5 questions, we do not perform the
supervised method due to the limited size of the dataset.

To investigate the correlation between our two datasets, we also include a
transfer learning experiment that takes the supervised model trained on one
dataset to evaluate the other dataset. We also report the results of an
unsupervised vanilla Roberta-large model and vanilla Unified-QA model as well
as of a random baseline.

Table 5: Evaluation results on the BDD-QA-EP dataset of our models on 5 action classes. Methods | Action Class
---|---
Accelerate |  Slow |  Stop |  Merge |  Turn
NLI-based | Roberta-large | 70.5 | 59.7 | 68.5 | 59.6 | 51.8
Bart-large | 77.1 | 59.5 | 67.5 | 68.6 | 57.1
Deberta-large | 74.7 | 58.4 | 71.6 | 70.1 | 58.7
Deberta-v2-xlarge | 74.5 | 62.5 | 76.2 | 74.5 | 66.2
Deberta-v2-xxlarge | 79.0 | 65.3 | 75.8 | 72.2 | 65.7
Knowledge-based | Roberta-base | 56.5 | 57.4 | 51.5 | 66.5 | 82.4
Roberta-large | 68.7 | 70.4 | 68.5 | 77.0 | 81.3
T5-small | 57.1 | 58.8 | 59.0 | 42.6 | 58.0
T5-large | 59.7 | 60.8 | 65.8 | 65.5 | 80.1
T5-3B | 65.1 | 68.7 | 63.7 | 75.1 | 81.8

## Results

We have tested three methods on our two datasets. We use accuracy as our
evaluation metric. For the BDD-QA dataset, the Unified-QA model reaches the
best performance by adapting multiple formats of question answers. For the
HDT-QA dataset, our proposed retrieval-based model, DPR+Unified-QA outperforms
other methods by nearly 20 points, which shows the effectiveness of retrieving
domain knowledge for question answering.

### BDD-QA Task

Table 4 shows the results for three different types of models (NLI-based, KG-
based, and Retrieval-based QA) on the BDD-QA dataset. The result shows that
the best accuracy is obtained by Unified-QA-v2, which benefits from its
adaption of multiple formats of question answers. Such performance is 29.6
points higher than the unsupervised Roberta-large result, but still, has a
large gap with the supervised model (16.5 points). In general, NLI-based
models have a balanced performance in the two datasets while KG-based models
perform well on the BDD-QA-EP dataset but poorly on BDD-QA-CP.

Zero-shot performance depends on model size and background training data. We
see that the performance of NLI-based models is getting continuously better
when the model size grows, which is expected. Specifically, NLI-based models’
performance on BDD-QA-EP is growing faster when the model size is smaller,
while for BDD-QA-CP the growth is more obvious between bigger models. Given
the sentence embeddings of all the causes and effects, we have computed the
average sentence similarity between causes as $0.403$, while for effects it is
$0.553$, which means causes are more different from each other than the
effects. The difference of similarities is intuitive because the effects
(cars’ actions) are limited while the causes represent factors in the
environment and are therefore more open-ended. By focusing on three Deberta-
large models from Table 4, we observe that the performance on BDD-QA-CP grows
faster than BDD-QA-EP when the model size is bigger, $(64.9-59.7>72.4-71.3$
while $59.7-54.5\approx 71.3-67.0)$. Such a result implies that predicting the
correct cause given an effect needs a language model having a larger capacity,
which is coherent with the observation between cause and effect prediction.

Among all of the knowledge-based models, T5-3b, the best-performing model
across five zero-shot benchmarks in Zhang et al. (2022), which is an encoder-
decoder model trained with sequence-to-sequence language model loss, achieves
the best performance. Roberta-large, the encoder-only model trained on masked
language model loss, also reaches a high performance that is close to T5-3b’s.
Overall, knowledge-based models are not good at predicting the cause, none of
them reaches 60% accuracy in the BDD-QA-CP dataset. To explain this low
performance, we look deeper into the properties of the synthetic training data
that was constructed in (Ma et al. 2021a) from CSKG (Ilievski, Szekely, and
Zhang 2021). The source graph has abundant information about X causes Y, which
we translate into training questions of the form: What does X cause? then the
different effects are sampled. However, there are no questions about the
inverse formulation What is the cause of X? like what we do to the BDD-QA-CP
dataset. Such a result is consistent with the finding in Zhang et al. (2022)
that the language models learn to answer well about what is in the data and do
not generalize well to other aspects. Finally, there is still a large gap
between both models’ accuracy and that of the supervised Roberta model, which
leaves plenty of space for future improvement of the models.

Regarding the retriever-based models, Unified QA itself benefits from its
adaption of multiple formats of question-answer pairs so its performance is
close to the best one among the NLI-based and KG-based models. However, after
we provide the most relevant passage with it, the performance decreases. This
implies that knowledge in the driving handbook provides little help for the
model to answer the BDD-QA questions.

Table 6: Evaluation results on the HDT-QA dataset of 10 models. We only train the HDT-QA-3Q and HDT-QA-4Q for supervised learning because the size of HDT-QA-2Q and HDT-QA-5Q are both limited. Methods | HDT-QA-2Q | HDT-QA-3Q | HDT-QA-4Q | HDT-QA-5Q |  Avg
---|---|---|---|---|---
Random | - | 50.0 | 33.3 | 25.0 | 20.0 | 25.7
Vanilla models | Roberta-large | 55.4 | 43.7 | 36.7 | 9.10 | 36.2
UnifiedQA-v2 | 67.9 | 51.6 | 44.0 | 25.0 | 47.1
NLI-based | Roberta-large-mnli | 63.4 | 41.0 | 33.0 | 15.0 | 38.1
Bart-large-mnli | 65.6 | 46.1 | 34.3 | 22.5 | 42.2
Deberta-large-mnli | 62.6 | 42.8 | 34.5 | 15.0 | 38.7
Deberta-v2-xlarge-mnli | 62.6 | 45.7 | 36.3 | 17.5 | 40.5
Deberta-v2-xxlarge-mnli | 63.4 | 47.5 | 36.1 | 22.5 | 42.4
Knowledge-based | Roberta-base | 60.3 | 36.0 | 29.0 | 25.0 | 37.6
Roberta-large | 63.4 | 38.9 | 30.8 | 27.5 | 40.1
T5-small | 57.3 | 35.4 | 23.8 | 22.5 | 34.7
T5-large | 64.1 | 40.3 | 24.6 | 22.5 | 37.9
T5-3b | 66.4 | 45.0 | 25.1 | 22.5 | 39.7
Retrieval-based | UnifiedQA-v2 + DPR | 80.2 | 57.3 | 47.7 | 60.0 | 61.3
Supervised | Roberta-large | - | 71.7 | 75.8 | - | 73.8
Transfer learning | Roberta-large | 45.8 | 43.2 | 37.9 | 21.8 | 37.2

Overlap analysis. We expect that the language model performance highly depends
on their training data and training methods. To investigate the predictions of
two different kinds of models, we look closer at the predictions of the best-
performing NLI-based model and KG-based model Deberta-v2-xxlarge and Roberta-
large, respectively. Among 3,139 QA pairs in BDD-QA-EP, Deberta-v2-xxlarge can
correctly answer 2,256 questions and Roberta can predict 2,302 QA pairs
correctly. However, there are only 1,826 answers that are correctly answered
jointly, which means that for an ensemble model that can combine the two
models’ correct answers, the accuracy would be 87.0% on BDD-QA-EP. For BDD-QA-
CP, the combined accuracy would be 79.1%. Such results are much higher than
the performance of the two models alone and are close to the result of
supervised learning. Such results encourage us to investigate the granular
performance of two kinds of models by splitting the dataset into several
classes. We discuss this next.

Result on classes of effects. We investigate the granular performance of
different models on the action (cause) classes of the BDD-QA-EP dataset. Table
5 shows that the Knowledge-based models perform best on the Turn, Merge and
Slow classes, while NLI-based models do better on the other two classes (Move
and Stop). In the real world, turning, merging, and slowing are more
complicated actions than moving and stopping because their causes vary a lot.
For example, if a car merges to the left, the cause might be The snow is
taking up too much of the right lane, or There are pedestrians in the right or
The car is preparing to execute a u-turn maneuver, or a long list of other
causes. But for Accelerate, the list of causes is more narrow: it would either
be The light turned green or The road is clear. We conclude that knowledge-
based models can handle complex reasoning tasks in the traffic domain, while
NLI-based models are better at surface-level inference. Here, again we see
that the performance for most action classes grows with the increase in model
size.

Human evaluation.  Since BDD-QA is constructed by an automatic procedure, we
perform a human evaluation for this dataset. For each partition of the
dataset, we randomly select 50 QA pairs. Besides providing an answer for each
question, we also ask the annotators to give confidence in their results on a
5-point Likert scale (1 being low and 5 being high) to check whether the
questions are commonsensical. Each question is answered by 3 humans and we
choose the majority-voted candidate as the correct answer. The result is shown
in the last row of Table 4. We note that the human score is 1.4% points higher
than the best zero-shot result, and the average annotator confidence is
$3.67$. We are surprised by the relatively low human score on this task. Upon
further analysis, we notice that most of the questions that are wrongly
answered by humans have multiple reasonable answers, e.g., given the cause The
cross traffic has cleared and the pedestrian is gone The correct effect is The
car makes the left turn and heads down the street, but another effect The car
very slow accelerates is also reasonable given the cause. For this question,
all three humans chose the latter answer. In the future, we will revise the
dataset to further clean up the noise.

### HDT-QA Task

As shown in Table 6, all of the NLI-based and KG-based models perform
consistently poorly on the HDT-QA dataset. The growth of the performance
together with the model size that was observed on the BDD-QA dataset is not as
clearly visible on the HDT-QA dataset. Such a result implies that obtaining
basic commonsense knowledge and semantic inference information is not enough
for answering human driving test questions. The examples of HDT-QA, shown in
Table 2, are about the details of the policies of driving. It is hard even for
humans to pass the driving exams without learning specific rules and driving
policies captured in domain manuals. As a result, it is intuitive to retrieve
domain knowledge to help language models better answer the human driving test.

As shown in Table 6, UnifiedQA itself performs slightly better than the NLI-
based and KG-based models. After we provide the most relevant paragraph to the
QA model automatically with the DPR retriever, the average performance of the
subsets improves by 14.2% points. This means the QA system is able to capture
relevant domain knowledge efficiently for answering human driving test
questions, especially when combined with a dense retrieval model that can
provide the most relevant set of sentences that serve as evidence.

The performance of unsupervised learning and transfer is close to that of NLI-
based and KG-based models, which means the NLI, KG, and BDD-enhanced models
contribute little for models to answer the HDT-QA questions. Lastly,
supervised learning reaches a high accuracy, which is intuitive because some
of the questions in the training and the test set share the same knowledge.

## Discussion

Our experiments show that language models trained with semantic level
coherence information and commonsense knowledge extracted from knowledge
graphs are both performing well at answering traffic domain reasoning tasks.
Careful domain information extraction and model adaption leads to significant
improvement of language models’ performance on domain-specific QA sets. Next,
we discuss and list future extensions inspired by our results.

More efficient retrieving of domain knowledge In this paper, we use Dense
Passage Retrieval to capture the most relevant paragraph in the driving
manuals and directly fed the paragraph to the QA system. However, the language
models are not learning the policies, they are answering the questions only by
consulting the driving manual simultaneously in an open-book setting, which
can be misled by vague information or context mismatch. Developing methods for
effectively extracting domain knowledge and letting language models learn this
knowledge is an important future work for solving HDT-QA and similar traffic-
specific QA tasks. Selecting informative sentences or building knowledge
graphs based on raw data are possible methods for a more precise injection of
domain knowledge.

Joint reasoning over domain knowledge and causal commonsense knowledge Our
methods have isolated the different knowledge types and have focused on
learning either causal or domain knowledge, but not both. We anticipate that a
robust AI model should be able to reason over both knowledge types
simultaneously and know which questions require which specific knowledge
statements. A key future item is integrating the two knowledge types together
in a joint reasoning system and developing benchmarks that can evaluate both
aspects in a more coherent manner.

From natural language QA to multi-modal traffic QA Our study focused on
understanding traffic situations based on using only text as background
information. However, visual information in images and videos is a key
component of traffic scenes. Automatic driving systems take images as input
and make decisions according to them. Popular datasets like Traffic-QA (Xu,
Huang, and Liu 2021) are also based on background information that is given by
an image. Considering visual information is necessary for building a holistic
robust system that understands traffic situations. Thus, we believe that
multi-modal benchmarks and methods that combine perception and background
knowledge for situational traffic understanding are the next milestones toward
developing real-world neuro-symbolic traffic models.

## Conclusion

Besides robust perception models, a holistic situational understanding of
traffic requires generalizable models equipped with commonsense and domain
knowledge, which has received little attention in prior work. To bridge this
gap, this paper studied the strengths and weaknesses of current zero-shot
adaptation methods and their granular effects on different partitions of
tasks. We constructed two benchmarks for evaluating the ability of zero-shot
causal reasoning (BDD-QA) and accessing domain knowledge (HDT-QA) to answer
questions about traffic situations. We proposed two representative methods
that focus on zero-shot causal reasoning and one novel pipeline for retrieving
domain information. On the BDD-QA dataset, we have observed the limited
prediction overlap between the two models. To deeper investigate that, we
split the actions of the cars into several categories, observing significant
differences in performance between the methods, which provides a convincing
explanation of the result of the limited overlap. On the HDT-QA dataset, we
observed a giant performance improvement after we fed the relevant information
extracted from driving manuals in an open-book setting.

The methods in our work can be easily extended to other traffic and related
tasks in other domains. Our code and data can be downloaded at
https://github.com/saccharomycetes/text-based-traffic. Key future directions
include more efficient knowledge retrieval, better integration of causal and
domain knowledge, and the development of multi-modal traffic benchmarks and
methods.

## Acknowledgements

We would like to thank all of the anonymous reviewers for their valuable
suggestions. We also thank Zhivar Sourati, Yifan Jiang, Jiachen Zhang,
Thiloshon Nagarajah, Xiaoshu Luo, Riley Li, and Prateek Chhikara for
participating in our human evaluation study.

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## Appendix

### Details of chosen models

For the Inference-base reasoning model, we select models trained on the same
dataset for a fair comparison, and we specifically chose MNLI because it
contains 10 different genres and a very large corpus (433k). We directly
download the model from hugging-face, all of them share the same input and
output format (take in a premise and hypothesis, give the confidence of the
sentence pair being ”entail”, ”neutral”, ”contract”). In general, we test the
model’s ability to make the right inference on the traffic domain without any
fine-tuning.

For KG-based reasoning models, the chosen models have two different model
architectures and all have different model sizes. The encoder-only model,
Roberta, uses the Masked Language Model loss to give a score for each
question-candidate pair. After getting the score for each candidate, the
Roberta models are trained with the margin loss function:
$L_{margin}=\frac{1}{n}\sum_{\mbox{$\begin{array}[]{c}i=1\\\ i\neq
y\end{array}$}}^{n}max(0,\eta-S_{y}+S_{i})$, where $S_{y}$ and $S_{i}$ are the
negative averaged loss for the correct answer and the distractor respectively.
During the evaluation, the model chooses the pair with the highest score as
the correct answer. The encoder-decoder model T5 learns to give an output ‘1’
or ‘2’ which represents the confidence of the model in predicting whether the
given sentence pair is reasonable or not. For each candidate, its score is
defined as $S=L_{true}-L_{false}$, then the training and evaluating process is
the same as the encoder-only models. We splice the cause and effect together
as the input of the models when we test the BDD-QA-CP and BDD-QA-EP and HDT-QA
datasets. For the HDT-QA dataset, if there is an underscore appearing in the
question, we fill the underscore with candidates to make complete sentences.

For Retrieval-based models, we choose the sentence transformer version
$facebook-dpr-ctx_{e}ncoder-single-nq-base$, $facebook-dpr-question-encoder-
single-nq-base$ to encode the paragraphs and questions, respectively. For the
unified-QA model, we use the version of Unified-QA-v2, T5-3B, which is
pretrained on 20 QA datasets.

### Implementation Details

Regarding libraries, we used python 3.7.10, PyTorch 1.9.0, and transformers
4.11.3.

Among all the supervised learning sets, we are using a learning rate of
$1e^{-5}$, batch size of 32, weight decay 0.01, training epochs of 10, adam-
epsilon of $1e^{-6}$, $\beta 1=0.9,\beta 2=0.98$, the warm-up proportion of
0.05, the margin of 1.0
For CPUs, we used Intel(R) Xeon(R) Gold 5217 CPU @ 3.00GHz (32 CPUs, 8 cores
per socket, 263GB ram).
For GPUs, we used Nvidia Quadro RTX 8000.


# One numerical obstruction for rational maps between hypersurfaces

Ilya Karzhemanov
Laboratory of AGHA, Moscow Institute of Physics and Technology, 9 Institutskiy
per., Dolgoprudny, Moscow Region, 141701, Russia
E-mail address<EMAIL_ADDRESS>
###### Abstract.

Given a rational dominant map $\phi:Y\dashrightarrow X$ between two generic
hypersurfaces $Y,X\subset\mathbb{P}^{n}$ of dimension $\geq 3$, we prove
(under an addition assumption on $\phi$) a “ Noether – Fano type ” inequality
$m_{Y}\geq m_{X}$ for certain (effectively computed) numerical invariants of
$Y$ and $X$.

MS 2020 classification: 14E05, 14M10, 28D20

Key words: rational map, hypersurface, entropy

##  1\. Introduction

###  1.1. Set - up

Let $X\subset\mathbb{P}^{n}$ be a smooth hypersurface over $\mathbb{C}$ given
by an equation $f=0$ in some projective coordinates $x_{0},\ldots,x_{n}$ on
$\mathbb{P}^{n}$. Identify $x_{i}$ with a basis of $H^{0}(X,L)$ for
$L:=\mathcal{O}(1)$. We will assume in what follows that $n\geq 4$. In
particular, given two such hypersurfaces $X$ and $Y$, we have
$\text{Pic}\,X=\text{Pic}\,Y=\mathbb{Z}\cdot L$ (Lefschetz), so that any
rational map $\phi:Y\dashrightarrow X$ is induced by a self - map of
$\mathbb{P}^{n}$.

######  Definition 1.2.

Call $\phi$ _symplectic_ if the corresponding map
$\mathbb{P}^{n}\dashrightarrow\mathbb{P}^{n}$ preserves, up to a constant, the
$2$ \- form
$\displaystyle\sum_{i=1}^{n}\frac{dx_{i}}{x_{i}}\wedge\frac{d\bar{x}_{i}}{\bar{x}_{i}}$
(cf. 2.1 below). Also, call $X$ _symplectically unirational_ , if
$Y:=\mathbb{P}^{n-1}=(x_{n}=0)$ and $\phi$ is symplectic.

######  Example 1.3.

Take $X=Y=(x_{n}=0)\simeq\mathbb{P}^{n-1}$ and $\phi$ to be the _Frobenius
morphism_ $x_{i}\mapsto x_{i}^{d}$, $0\leq i\leq n-1$, for some integer $d\geq
2$. This $\phi$ is easily seen to be symplectic.

Fix a point $o\in X$ and consider the blowup
$\sigma:\widetilde{X}\longrightarrow X$ of $o$. Let $\Sigma:=\sigma^{-1}(o)$
be the exceptional divisor of $\sigma$. Define the quantity (cf. [1,
Definition 1.6])

$m_{X}(L;o):=\sup\left\\{\varepsilon\in\mathbb{Q}\ :\ \text{the linear
system}\ |N(\sigma^{*}L-\varepsilon\Sigma)|\ \text{is mobile for}\ N\gg
1\right\\}.$

It is called the _mobility threshold_ (of $X$) and was first introduced by A.
Corti in [17] (we will sometimes write simply $m_{X}$ when the point $o$ is
irrelevant).

######  Example 1.4.

Let $f:=x_{n}$ and hence $X\simeq\mathbb{P}^{n-1}$ is a projective subspace.
Then we get $m_{X}(L;o)=1$ because the lines passing through $o$ sweep out a
divisor (in particular, the above $\varepsilon$ is always $\leq 1$, while the
opposite inequality is clear — consider the projection
$X\dashrightarrow\mathbb{P}^{n-2}$ from $o$ to obtain $\varepsilon=1$).

In this paper, we will be mostly using the following equivalent definition of
$m_{X}$ (although the first one is more suitable for computations):

$m_{X}(L;o):=\sup\,\frac{\text{mult}_{o}\,\mathcal{M}}{N},$

where sup is taken over all $N\geq 1$ and mobile linear systems
$\mathcal{M}\subseteq|L^{\otimes N}|$.

Let us assume from now on that $X=(f=0)$ and $Y$ are _generic_. Here is our
main result:

######  Theorem 1.5.

If $\phi$ is symplectic, then there exist points $o\in X$, $o^{\prime}\in Y$
such that $m_{Y}(L;o^{\prime})\geq m_{X}(L;o)$.

Theorem 1.5 implies in particular that $1=m_{Y}\geq m_{X}$ for
$Y:=\mathbb{P}^{n-1}$ and symplectically unirational $X$ (cf. Corollary 4.2
below). Note however that similar estimate _does not_ hold for an arbitrary
unirational $X$ (see 4.4).

###  1.6. Discussion

The main idea behind the proof of Theorem 1.5 is that birational invariants of
$X$ appear from a (hidden) _hyperbolic structure_ on hypersurfaces. Namely, we
employ the so - called _pairs - of - pants decomposition_ $\Pi$ from [15],
which we recall briefly in Section 2. This brings further an analogy with
hyperbolic manifolds (especially surfaces and $3$ \- folds) and their
geometric invariants — most common ones being various types of _volumes_.

This includes, as a basic example, the Euler characteristic of Riemann surface
$M$. A finer invariant is the so - called _conformal volume_ $V_{c}(M)$. One
can show that $2V_{c}(M)\geq\lambda_{1}\text{Vol}(M)$ for the first Laplacian
eigenvalue $\lambda_{1}$ of $M$ (see [14, Theorem 1] and corollaries thereof).
This was used, for instance, to obtain obstructions for existence of maps
between Riemann surfaces (see e. g. the proof of the _Surface Coverings
Theorem_ in [6, §4] or that of [8, Theorem $2.\mathrm{A}_{1}$]).

Further, if $M$ is a $d$ \- dimensional closed oriented hyperbolic manifold,
then there is a _Gromov’s invariant_ $\|[M]\|$ (see [20, Chapter 6]). It is
defined in terms of certain (probability) measures on $M$ and coincides with
$C_{d}\text{Vol}(M)$ for some absolute constant $C_{d}$. The most fundamental
property of this invariant (used in the proof of Mostow’s rigidity theorem for
example) is that for a map $M_{1}\longrightarrow M_{2}$ between two hyperbolic
$M_{i}$ one has $\|[M_{1}]\|\geq\|[M_{2}]\|$. The latter inequality and a
close similarity between the definitions of $V_{c}(~{})$, $\|[~{}]\|$, etc.
and that of $m_{X}$ (cf. 1.1 and 2.3 below) have motivated our approach
towards the proof of Theorem 1.5.

Namely, on replacing $X$ by the complex $\Pi$ mentioned above, we recast
$m_{X}$ in “ probabilistic ” terms (see Section 3). The argument here is an
instance of the _(Bernoulli) law of large numbers_ and allows one to give a
conceptual explanation for the estimate $m_{Y}\geq m_{X}$ (cf. Remark 3.4). On
the other hand, results in 2.3 and 2.7, together with initial definition of
$m_{X}$ in 1.1, yield algebro - geometric applications (see Section 4).

Such line of thought — associating $X\rightsquigarrow\Pi$ and extracting
geometric properties of $X$ from combinatorics of $\Pi$ (and vice versa) — is
not new. Classical case includes the _Brunn – Minkowski inequality_ (see e. g.
[3]). In a modern context (including the _mirror symmetry_) this viewpoint
appears in [2] and [19] for instance. Finally, we mention the “ motivic ” part
of the story, when one assigns to $X$ its _stable birational volume_
$[X]_{\text{sb}}$ (see e. g. [16]) or its class $[X]_{\mathcal{K}}$ in the
_connective $K$ \- theory_ (see [21]). In the latter case, given
$\phi:Y\dashrightarrow X$ as above, the _degree formula_ of [21] relating the
classes $[Y]_{\mathcal{K}}$, $[X]_{\mathcal{K}}$ may be considered as a vast
generalization of _Noether – Fano inequality_ (see e. g. [11, Proposition 2])
and was another motivation for our Theorem 1.5.

##  2\. Preliminaries

###  2.1. Pairs - of - pants complex

Consider the intersection

$X^{0}:=X\cap\bigcap_{i=0}^{n}(x_{i}\neq 0)$

of $X$ with the torus $(\mathbb{C}^{*})^{n}\subset\mathbb{P}^{n}$ equipped
with (affine) coordinates $x_{1},\ldots,x_{n}$. We may identify $f$ with the
Laurent polynomial defining $X^{0}$:

(2.2) $f=\sum_{\scriptscriptstyle\tiny
j\in\Delta\cap\mathbb{Z}^{n}}a_{j}t^{-v(j)}x^{j},$

where $\Delta\subset\mathbb{R}^{n}$ is the _Newton polyhedron_ of $f$,
$v:\Delta\cap\mathbb{Z}^{n}\longrightarrow\mathbb{R}$ is a piecewise affine
function, $a_{j}\in\mathbb{C}$ and $t>0$ is a real parameter (cf. [15, 6.4]).

Let us recall the _balanced maximal dual polyhedral $\Delta$ \- complex_ $\Pi$
associated with $X$. It corresponds to certain (dual) simplicial lattice
subdivision of $\Delta$ associated with the corner locus of the Legendre
transform of $v$; $\Pi$ may also be identified with its moment image (see [15,
2.3 and Proposition 2.4]). Here are some properties of $\Pi$ we will use
below:

  * •

there exists a smooth $T^{n-1}$ \- equivariant map (_stratified
$T^{n-1}$-fibration_) $\pi:X\longrightarrow\Pi$, where
$T^{n-1}:=(S^{1})^{n-1}$ consists of the arguments of $x_{i}$, so that for any
open lattice simplex $\Pi_{0}\subset\Pi$ the preimage $\pi^{-1}(\Pi_{0})$ is
an _open pair - of - pants_ , symplectomorphic to
$H^{\circ}:=(\displaystyle\sum_{i=1}^{n}x_{i}=1)\subset(\mathbb{C}^{*})^{n}$
equipped with the $2$ \- form
$\Omega:=\displaystyle\frac{1}{2\sqrt{-1}}\sum_{i=1}^{n}\frac{dx_{i}}{x_{i}}\wedge\frac{d\bar{x}_{i}}{\bar{x}_{i}}$;

  * •

in fact, $X$ is glued out of the _tailored_ (or _localized_) pants $Q^{n-1}$
(see [15, 6.6 and Proposition 4.6]), isotopic to $H^{\circ}$, so that $\pi$ is
a deformation retraction under the Liouville flow associated with $\Omega$,
and $\pi_{*}\Omega^{n-1}$ induces the Euclidean measure on $\Pi$;

  * •

$\Pi$ is also obtained via the _tropical degeneration_ (compatibly with
$\pi$): recall that $f$ depends on $t$ (see (2.2)) and let $t\to 0$ (for each
$Q^{n-1}$) in the preceding constructions — $\Pi$ is then a Gromov – Hausdorff
limit of amoebas $\mathcal{A}_{t}(X_{0})$ (see [15, 6.4]).

###  2.3. Atomic (probability) measure

Choose a global section $s\in H^{0}(X,L)\setminus{\\{0\\}}$ and a point $o\in
X$. We are going to construct a measure $d\mu_{o,s}$ on $\Pi$ so that
$\mathrm{Vol}\,\big{(}\Pi\big{)}=\text{mult}_{o}\left\\{s=0\right\\}$ with
respect to it and $d\mu_{o,s}$ is supported at the point $\pi(o)$.2)2)2) All
considerations below apply literarily to any $L^{\otimes N}$, $N\geq 1$, in
place of $L$.

Identify $s$ with a holomorphic function on a complex neighborhood $U\subset
X$ of $o$ and consider the $(1,1)$ \- current
$\tau:=\displaystyle\frac{\sqrt{-1}}{2\pi}\,\partial\overline{\partial}\log|s|$.
Then $\tau$ acts on the $L^{1}$ \- forms on $U$ of degree $2n-2$ via (Poincaré
– Lelong)

$\omega\mapsto\tau(\omega)=\displaystyle\int_{s\,=\,0}\omega.$

In particular, if $\omega$ is the $(n-1)$ \- st power of the Fubini – Studi
form on $\mathbb{P}^{n}$ restricted to $U$, one may regard $\tau(\omega)$ as
the volume of $U$ with respect to the measure $dm$ whose density function is
the Hessian of $\displaystyle\frac{\sqrt{-1}}{2\pi}\,\log|s|$.

Let us assume from now that $U:=B_{o}(r)$ is the Euclidean ball of radius $r$
centered at $o$. For all $\tau\in\mathbb{C}$, $|\tau|<1$, we consider the
dilations $z\mapsto\tau z$ of $U$ and the family of pushed - forward measures
$\tau_{*}(dm)$.

The following lemma is standard (cf. Remark 2.6 and Proposition 3.3 below):

######  Lemma 2.4.

The limit measure

$\displaystyle\frac{1}{(r\tau)^{2n-2}}\lim_{\tau\to\,0}\
\tau_{*}(dm):=dm_{o,s}$

exists and
$\displaystyle\int_{U}dm_{o,s}=\mathrm{mult}_{o}\left\\{s=0\right\\}$.

###### Proof.

Indeed, $\mathrm{Vol}\,\big{(}\\{s=0\\}\cap U\big{)}$ with respect to
$\displaystyle\frac{1}{(r\tau)^{2n-2}}\ \tau_{*}(dm)$ tends to
$\text{mult}_{o}\left\\{s=0\right\\}$, as $\tau\to 0$. This implies that the
limit of measures exists. ∎

We may assume that for $U=B_{o}(r)$ the radius $r\to 0$ as $t\to 0$ (cf.
(2.2)). Then it follows from 2.1 and Lemma 2.4 that

$\displaystyle\pi(U)=\text{simplicial
complex}\,\Pi_{0}=\text{Gromov\,--\,Hausdorff limit of}\,\mathcal{A}_{t}(U),$
(2.5) $\displaystyle s=\pi^{*}\ell\ \text{for a piecewise linear
function}\,\ell\,\text{on}\,\Pi_{0},$ $\displaystyle
d\mu_{o,s,\scriptscriptstyle\tiny\Pi_{0}}:=\pi_{*}dm_{o,s}=\text{measure
on}\,\Pi_{0}\,\text{supported at}\,\pi(o)\,\text{and such that}$
$\displaystyle\mathrm{Vol}\,\big{(}\Pi_{0}\big{)}=\int_{\scriptscriptstyle\tiny\Pi_{0}}d\mu_{o,s,\scriptscriptstyle\tiny\Pi_{0}}=\text{mult}_{o}\left\\{s=0\right\\};$

note also that $\pi(o)$ belongs to the corner locus of $\ell$.

Finally, $d\mu_{o,s,\scriptscriptstyle\tiny\Pi_{0}}$ induces a measure
$d\mu_{o,s}$ on $\Pi\supseteq\Pi_{0}$ in the obvious way, which concludes the
construction.

######  Remark 2.6.

The atomic measure $d\mu_{o,s}$ is an instance of the _convexly derived
measure_ from [5] (one may also treat the above “ density function ” $\ell$ as
a discrete version of the Hessian of
$\displaystyle\frac{\sqrt{-1}}{2\pi}\,\log|s|$). The “ mass concentration ”
concept of [5] will be used further to obtain intrinsic (bi) rational
invariants of $X$.

###  2.7. Rational maps: tropicalization

Let $Y\subset\mathbb{P}^{n}$ be another hypersurface, similar to $X$, with the
maximal dual complex $\Pi^{Y}$, projection $\pi^{Y}:Y\longrightarrow\Pi^{Y}$,
etc. defined verbatim for $Y$. Assume also that there exists a rational
symplectic map $\phi:Y\dashrightarrow X$. Then the constructions in 2.1 and
2.3 yield a map $\Phi:\Pi^{Y}\longrightarrow\Pi^{X}$, given by some PL
functions with $\mathbb{Z}$ \- coefficients,3)3)3) We use the notation
$\Pi^{X}=:\Pi$ and $\pi^{X}:=\pi$ in what follows. so that the following
diagram commutes:

(2.8)
$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{Y}}$$\scriptstyle{\phi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{X}}$$\textstyle{\Pi^{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\Pi^{X}.}$

Note that $\Phi$ _need not_ necessarily be a map of simplicial complexes.

The following lemma describes $\Phi$ as a map of _measure spaces_ :

######  Lemma 2.9.

There exists a positive number $\delta_{\phi}\in\mathbb{Z}$, depending only on
$\phi$, such that
$\Phi^{*}\pi^{X}_{*}\Omega^{n-1}=\delta_{\phi}\pi^{Y}_{*}\Omega^{n-1}$.4)4)4)
Here $\Phi^{*}$ is defined with respect to the (limiting) affine structure on
$\Pi$ induced from the complex one on $X$ (cf. 2.1).

###### Proof.

We have

$\pi^{Y}_{*}\phi^{*}\Omega^{n-1}=\delta_{\phi}\pi^{Y}_{*}\Omega^{n-1}$

for some real $\delta_{\phi}>0$ (cf. Definition 1.2). Now, since $\phi$ is the
restriction of a rational self - map of $\mathbb{P}^{n}$ (see 1.1), it follows
from 2.1 and (2.8) that $\pi^{Y}_{*}\Omega^{n-1}$ (resp.
$\Phi^{*}\pi^{X}_{*}\Omega^{n-1}$) coincides with the measure induced by the
standard one $dy_{1}\wedge\ldots\wedge dy_{n}$ on $\mathbb{R}^{n}$ (resp. by
$dl_{1}\wedge\ldots\wedge dl_{n}$ for some piecewise linear functions
$l_{i}=l_{i}(y)$ with $\mathbb{Z}$ \- coefficients). It remains to observe
that $dl_{1}\wedge\ldots\wedge dl_{n}=\delta_{\phi}dy_{1}\wedge\ldots\wedge
dy_{n}$ by construction. ∎

##  3\. Proof of Theorem 1.5

###  3.1. The entropy

Let $\Pi\subset\mathbb{R}^{n}$ be a simplicial complex with the standard Borel
measure $d\mu$. Fix some real number $M>0$ and consider various measures
$d\mu_{\ell}$ on $\Pi$, supported at the corner locus of PL functions $\ell$,
such that $\displaystyle\int_{\Pi}d\mu_{\ell}\leq M$. Let
$\mathcal{S}:=\mathcal{S}(\Pi,M)$ be the set of all such measures (aka
functions).

Further, given an integer $N>0$ the measure space $(\Pi,Nd\mu)=:\Pi_{N}$ may
be regarded as $\Pi\subset\mathbb{R}^{Nn}$, embedded diagonally, with the
measure being $d\mu^{N}:=\displaystyle\sum_{i=1}^{N}\pi_{i}^{*}d\mu$ for the
$i^{\text{th}}$ factor projections
$\pi_{i}:\mathbb{R}^{Nn}\longrightarrow\mathbb{R}^{n}$. The affine structure
on $\Pi_{N}$ is defined by the functions
$\displaystyle\sum_{i=1}^{N}\pi_{i}^{*}\ell_{i}$ for various PL $\ell_{i}$.
Note that

$\frac{1}{N}\int_{\Pi_{N}}\sum_{i=1}^{N}\pi_{i}^{*}(d\mu_{\ell_{i}})\leq M,$

i. e.
$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\pi_{i}^{*}(d\mu_{\ell_{i}})\in\mathcal{S}$,
provided $d\mu_{\ell_{i}}\in\mathcal{S}$ for all $1\leq i\leq n$.

Define the measures $d\mu^{N}_{\ell}$ on $\Pi_{N}\subset\mathbb{R}^{Nn}$ and
the set $\mathcal{S}(\Pi_{N},M)\ni\displaystyle\frac{1}{N}\,d\mu^{N}_{\ell}$
similarly as above. Let also

(3.2)
$C:=\sup_{N,\,\ell\in\mathcal{S}(\Pi_{N},M)}\displaystyle\frac{1}{N}\int_{\Pi_{N}}d\mu^{N}_{\ell}.$

######  Proposition 3.3.

There exists a number $\mathrm{ent}(d\mu,\mathcal{S})<\infty$, _depending only
on $d\mu$ and $\mathcal{S}$_, such that $C=\mathrm{ent}(d\mu,\mathcal{S})M$.

###### Proof.

After normalizing we may assume that $M=1$.Let us also assume for simplicity
that $\Pi$ is a simplex.

All measures $\displaystyle\frac{1}{N}\,d\mu^{N}_{\ell}$ can be identified
with points (mass centers) in the dual simplex $\Pi^{*}\subset\mathbb{R}^{n}$
(compare with the proof of [5, 4.4.A]). Let
$\mathcal{H}_{\mu,\mathcal{S}}\subseteq\Pi^{*}$ be the convex hull of this
set. Then

$\displaystyle\int_{\Pi}\bullet:\
\mathcal{H}_{\mu,\mathcal{S}}\longrightarrow\mathbb{R}_{\geq 0}$

is a _bounded_ ($\leq 1$) linear functional. By definition we obtain
$C=\displaystyle\max_{\mathcal{H}_{\mu,\mathcal{S}}}\displaystyle\int_{\Pi}\bullet=:\text{ent}(d\mu,\mathcal{S})$
and the result follows. ∎

######  Remark 3.4.

The constant $C=C^{X}$ resembles the value of _logarithmic rate decay
function_ at $d\mu$ (see e. g. [7, Lecture 4]). This suggests $C$ to be equal
the “ Boltzmann entropy ” and the estimate $C^{Y}\geq C^{X}$ in the setting of
2.7 (compare with [4, p. 7]). In fact, taking $d\mu_{\ell}:=d\mu_{o,s}$ for
various $s$ as in 2.3, we will apply this probabilistic reasoning to
(birational) geometry of $X$ (see below).

###  3.5. The estimate

Let $\Phi:\Pi^{Y}\longrightarrow\Pi^{X}$ be as in 2.7. Although $\Phi$ need
not preserve the simplicial structures, we still can find a pair of $k$ \-
simplices $\Pi_{0}^{X}\subseteq\Pi^{X}$ and $\Pi_{0}^{Y}\subseteq\Pi^{Y}$,
$1\leq k\leq n$, such that $\Phi(\Pi_{0}^{Y})=\Pi_{0}^{X}$.

Identify both $\Pi_{0}^{X}$ and $\Pi_{0}^{Y}$ with a simplex $\Pi$, carrying
two (Borel) measures $d\mu$ and $\delta_{\phi}\,d\mu$, induced by
$\pi^{Y}_{*}\Omega^{n-1}$ and $\Phi^{*}\pi^{X}_{*}\Omega^{n-1}$, respectively
(see Lemma 2.9).

Let us assume from now on that $\mathcal{S}:=\mathcal{S}_{X}$ consists of PL
functions $\ell$, obtained from various sections $s=\pi^{*}\ell\in\mathcal{M}$
and mobile linear systems $\mathcal{M}\subseteq|L^{\otimes N}|$, so that
$d\mu_{\ell}=\displaystyle\frac{1}{N}\,d\mu_{o,s}$ for some $o\in X$
satisfying $\pi(o)\in\Pi$ (see 3.1 and (2.5)). It follows from (2.5) that $M$
in 3.1 can be assumed to coincide with the mobility threshold
$M^{X}:=m_{X}(L;o)$ (cf. 1.1). Same considerations apply to $Y$, with
$\mathcal{S}_{Y}$, $M^{Y}:=m_{Y}(L;o)$, etc.

######  Lemma 3.6.

In the previous setting, we have $\mathrm{ent}(d\mu,\mathcal{S}_{X})=1$, and
similarly for $\mathcal{S}_{Y}$.

###### Proof.

This follows from (3.2) (cf. Proposition 3.3), definition of $m_{X}$ (cf.
(2.5)), and the fact that $\pi^{X}_{*}\Omega^{n-1}=d\mu$ on $\Pi=\Pi_{0}^{X}$
(see 2.1). ∎

######  Proposition 3.7.

For every $\ell\in\mathcal{S}(\Pi,M^{X})$, we have
$d\mu_{\Phi^{*}\ell}=d\mu_{\tilde{\ell}}$, where
$\tilde{\ell}\in\mathcal{S}(\Pi,M^{Y})$.

###### Proof.

It follows from (2.5) and Lemma 2.4 that

$\int_{\Pi}d\mu_{\ell}=\displaystyle\frac{1}{N}\,\int_{\Pi}d\mu_{o,s}=\displaystyle\frac{1}{N}\,\int_{U}dm_{o,s}=\displaystyle\frac{1}{N}\,\int_{U\setminus
Z}dm_{o,s}$

for any closed subset $Z\subsetneq U$. Recall that the _rational transform_
$\phi^{-1}_{*}s$ is naturally defined as a member of the mobile linear system
$\phi^{-1}_{*}\mathcal{M}$. In particular, if $\phi$ is a morphism over
$U\setminus Z$, then

$\int_{\phi^{-1}(U\setminus
Z)}dm_{o,\phi_{*}^{-1}s}=\text{mult}_{o}\left\\{\phi_{*}^{-1}s=0\right\\}.$

This $\phi_{*}^{-1}s$ defines a PL function $\tilde{\ell}$ as earlier and we
have

$\int_{\Pi}d\mu_{\Phi^{*}\ell}=\int_{\Pi}d\mu_{\tilde{\ell}}$

(cf. (2.8)). The identity $d\mu_{\Phi^{*}\ell}=d\mu_{\tilde{\ell}}$ follows
and $\tilde{\ell}\in\mathcal{S}(\Pi,M^{Y})$ by construction. ∎

Let $C:=M^{Y}$ be as in Proposition 3.3 ($\text{ent}(d\mu,\mathcal{S})=1$ by
Lemma 3.6) and $\delta_{\phi}$ as in Lemma 2.9. Then it follows from
Proposition 3.7 (cf. Remark 3.4) that

$C\delta_{\phi}=\sup_{N,\,\tilde{\ell}\in\mathcal{S}(\Pi_{N},M^{Y})}\displaystyle\frac{1}{N}\int_{\Pi_{N}}\delta_{\phi}d\mu^{N}_{\tilde{\ell}}\geq\sup_{N,\,\ell\in\Phi^{*}\mathcal{S}(\Pi_{N},M^{X})}\displaystyle\frac{1}{N}\int_{\Pi_{N}}\delta_{\phi}d\mu_{\Phi^{*}\ell}^{N}=$
$=\sup_{N,\,\ell\in\mathcal{S}(\Pi_{N},M^{X})}\displaystyle\frac{1}{N}\int_{\Pi_{N}}\Phi^{*}d\mu_{\ell}^{N}=\text{ent}(\delta_{\phi}d\mu,\mathcal{S})M^{X}$

(the last equality is due to the projection formula
$\Phi_{*}\Phi^{*}d\mu=\delta_{\phi}d\mu$ and the change of variables in
$\int$). Finally, since
$\text{ent}(\delta_{\phi}d\mu,\mathcal{S})=\delta_{\phi}\text{ent}(d\mu,\mathcal{S})$,
we conclude that $M^{Y}\geq M^{X}$.

##  4\. Some examples and applications

###  4.1. Soft

Setting $Y:=\mathbb{P}^{n-1}=(x_{n}=0)$ we arrive at the following immediate

######  Corollary 4.2.

Suppose $X$ in Theorem 1.5 is symplectically unirational. Then there exists a
point $o\in X$ such that $m_{X}(L;o)=1$.

###### Proof.

It suffices to prove that $m_{X}\geq 1$. This is done by considering the
projection $X\dashrightarrow\mathbb{P}^{n-1}$ from $o$ and observing that the
linear system $|\sigma^{*}L-a\Sigma|$ is mobile for some $a\geq 1$ (cf. 1.1).
∎

Suppose $X$ is a _quadric_. Although we do not know whether $X$ is
symplectically unirational (cf. Example 1.3), it is obviously rational, and
Corollary 4.2 confirms that $m_{X}=1$ in this case (the latter equality can
actually be proved directly by considering families of lines on $X$ as in
Example 1.4).

######  Remark 4.3.

It would be interesting two find out whether any _birationally isomorphic_
hypersurfaces $X$ and $Y$ as in Theorem 1.5 always have
$m_{X}(L;o)=m_{Y}(L;o^{\prime})$ for some points $o$ and $o^{\prime}$. It
should also be possible to generalize all our considerations to the case of
_any smooth_ $X$ and $Y$.

Let us proceed with non - trivial examples distinguishing ordinary
unirationality from the symplectic one.

###  4.4. Hard

Suppose $\deg f=3$ (i. e. $X$ is a _cubic_). It is a classical fact that $X$
is unirational (see e. g. [10, Chapter 3, Corollary 1.18]). Fix a point $o\in
X$. We may assume that $o=[1:0:\ldots:0]$, and hence $f=q_{1}+q_{2}+q_{3}$ in
the affine chart $(x_{0}\neq 0)$, where $q_{i}=q_{i}(x_{1},\ldots,x_{n})$ are
forms of degree $i$. Arguing as in [18, Section 1] we obtain that $q_{1}$ and
$q_{2}$ are _coprime_. Thus the linear system $\mathcal{M}\subset|2L|$ spanned
by $q_{1}^{2}$ and $q_{2}$ is mobile. We conclude that $m_{X}\geq 3/2$, since
$\text{mult}_{o}\,\mathcal{M}=3$ (cf. 1.1), and so $X$ is _not_ symplectically
unirational by Corollary 4.2.

Now assume only that $X$ is smooth (cf. Remark 4.3). Then it is possible to
find a (_Eckardt_) point $o\in X$ for which $m_{X}(L;o)=1$ (see [10, Chapter
5]). It would be interesting to study whether such cubics are symplectically
unirational.

Further, consider the case $\deg f=4=n$, assuming again that the quartic $X$
is just smooth. Note that it is still unknown whether any such $X$ is
unirational.6)6)6) Although a smooth quartic hypersurface $X$ _is_ unirational
when $n\gg 4$ (see [9, Corollary 3.8]. Here is a classical unirational example
after Segre (cf. [12, 9.2]):

$X=(x_{0}^{4}+x_{0}x_{4}^{3}+x_{1}^{4}-6x_{1}^{2}x_{2}^{2}+x_{2}^{4}+x_{3}^{4}+x_{3}^{3}x_{4}=0).$

We claim that $m_{X}(L;o)=1$ for some $o\in X$. Indeed, take the hyperplane
$\Pi:=(x_{1}-\alpha x_{2}=0)$, where $\alpha:=\sqrt{3+2\sqrt{2}}$. Then
$X\cap\Pi$ is a _cone_ in $\mathbb{P}^{3}$ given by the equation
$x_{0}^{4}+x_{0}x_{4}^{3}+x_{3}^{4}+x_{3}^{3}x_{4}=0$. We take $o$ to be the
vertex of this cone.

At the same time, if $X$ is generic, then one can show that $m_{X}\geq 3/2$ by
exactly the same argument as in the cubic case.7)7)7) It is proved in [13,
A.24] that in fact $m_{X}=3/2$. Thus again symplectic version of the
unirationality problem for $X$ is settled here.

Acknowledgments. I am grateful to John Christian Ottem for valuable comments.
The work was partially supported by the Russian Academic Excellence Project 5
- 100.

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